Diss. ETH No. 23476
A Non-Probabilistic Framework for
Scientific Theories
A dissertation submitted to
ETH ZURICH
for the degree of
Doctor of Sciences
presented by
Daniela Frauchiger
MSc. Physics, ETH Zürich
born 31st July 1985
citizen of Switzerland
to the consideration of
Renato Renner, examiner
Artur Ekert, co-examiner
Rüdiger Schack, co-examiner
2016
To my parents.
Acknowledgements
I thank my supervisor Renato Renner for his interest in my questions that led
to this research project. I also want to thank him for supporting me when
things turned out to be more challenging than we initially thought.
My special thanks go to Artur Ekert for his special support and interest in this
project, and for encouraging me to continue during difficult periods.
I want to thank Lı́dia del Rio who spent a lot of time reading the chapters and
giving me helpful feedback. I am very grateful for her thoughtful comments
and challenging questions. Thanks to her for caring in general.
I thank David Deutsch for the time he took for discussions during my visit in
Oxford. He has the great gift of not just trying to convince me of his opinion
but instead making me think about my questions from a di↵erent point of
view. I also thank Vlatko Vedral for hosting me in Oxford as well as for his
interest and helpful input.
I have also benefited a lot from the discussions with Rüdiger Schack at various
conferences. I always had the feeling that he truly listened to my questions
and I am very grateful for his thoughtful answers. Similarly I have profited a
lot from discussions with Roger Colbeck.
I thank Rotem Arnon-Friedman, Philipp Kammerlander, Anne Milek and Jan
Bertini for reviewing part of the thesis, which helped a lot improving this final
version. I thank David Sutter for being a perfect office mate and patiently
explaining to me all kinds of things. Finally, I would like to extend my thanks
to all members of the Quantum Information group at ETH Zurich for making
the time at the institute enjoyable. In particular, I want to thank Normand
Beaudry and Lı́dia del Rio for organising various social events.
I am very grateful for the support and understanding of my friends. Thanks
for sharing joyful moments and for listening in less joyful ones.
My special thanks go to my parents for always supporting me. They did not
only make it possible for me to go the way I chose but they also provided a safe
harbour where I could always return in difficult times. Thanks to my dad for
teaching me not to take myself too serious. Thanks to my mum for teaching
me that the purpose of life is not simplicity. This thesis is dedicated to them.
v
vi
Zusammenfassung
Ziel dieser Doktorarbeit ist die Entwicklung eines neuen Frameworks um wissenschaftliche Theorien zu analysieren. Die Struktur des Frameworks ist motiviert von der Idee, dass die Aufgabe einer Theorie nicht nur ist Voraussagen zu machen, sondern auch zu erklären. Wir diskutieren warum es deshalb
problematisch ist, wenn die grundliegenden Aussagen einer Theorie in Form
von Wahrscheinlichkeiten sind. Diese Beobachtung motiviert unseren Ansatz
das Framework nicht auf Wahrscheinlichkeiten zu basieren. Als Beispiel für
eine Anwendung zeigen wir, wie man innerhalb des Frameworks die Bornsche
Regel als nicht-probabilistisiche Aussage (BornObj) in der Form eines objektiven Naturgesetzes formulieren kann. Für Anwendungen, in denen es nützlich
ist, können Wahrscheinlichkeiten zusätzlich eingeführt werden, wenn man sie
subjektiv im Bayesschen Sinne interpretiert. Wir zeigen, wie man konkrete
Wahrscheinlichskeitsverteilungen aus objektiven Naturgesetzen ableiten kann,
indem man zusätzliche Annahmen darüber macht, was ein Agent für vernünftig
hält. Insbesondere zeigen wir, wie man aus dem Gesetz (BornObj) die probabilistische Bornsche Regel zurückgewinnen kann.
Eine weitere Eigenschaft des Frameworks ist, dass sich darin Annahmen über
die Eigenschaften einer Theorie natürlich formulieren lassen. Dies erlaubt es,
dass man überprüfen kann, ob eine Theorie bestimmte Annahmen gleichzeitig
erfüllen kann. Als Beispiel zeigen wir, wie man innerhalb des Frameworks
verschiedene Annahmen über die Konsistenz einer Theorie mathematisch formulieren kann und dass mindestens eine dieser Eigenschaften von sogenannten
“single-world” Interpretationen der Quantenmechanik nicht erfüllt sein kann.
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viii
Abstract
This thesis introduces a novel framework to analyse scientific theories. The
main motivation for the structure of the framework is the idea that the purpose of a theory is explanation rather than mere prediction. We discuss why it
is problematic to base theories on probabilities – an observation that motivated
the introduction of a non-probabilistic framework. We give an example of how
the framework allows to make non-trivial objective statements about the world
by deriving a non-probabilistic formulation of the Born rule which we term
(BornObj). Probabilities may be introduced in the framework by interpreting
them subjectively in a Bayesian sense. We discuss how actual probability assignments can be derived from (i) assumptions about an agent’s belief in the
correctness of certain laws of nature (which might be true of wrong), as well
as (ii) assumptions about what he considers to be rational (for which the question of correctness is meaningless). In particular, we show how the standard
probabilistic interpretation of the Born rule can be retrieved by assuming that
an agent believes in the correctness of the objective law (BornObj) and that he
additionally has certain beliefs about rationality.
Another feature of the framework is that it allows to make assumptions explicit
that are usually made implicitly in a natural way. This allows to actually test
whether a theory can satisfy a set of given assumptions. As an example we
show how di↵erent notions of consistency can be expressed on a formal level
within the framework and that they cannot be satisfied all together by certain
types of single-world interpretations of quantum theory.
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Contents
Page
Contents
1
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 The purpose of scientific theories: prediction vs. explanation
1.1.2 The scientific method vs. the inductive method . . . . . . . .
1.1.3 Falsification . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Theories and “reality” . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Implicit assumptions within the framework . . . . . . . . . .
1.2 The problem with probabilities . . . . . . . . . . . . . . . . . . . . .
1.2.1 The objective interpretation . . . . . . . . . . . . . . . . . . .
1.2.2 The subjective interpretation . . . . . . . . . . . . . . . . . .
1.2.3 Probabilities are subjective – consequences . . . . . . . . . .
1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Interpretations
2.1 Interpretations of classical mechanics
2.2 Interpretations of quantum theory .
2.2.1 The measurement problem .
2.2.2 Multiverse interpretations . .
2.2.3 Single-world interpretations .
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3 Stories about experiments – a non-probabilistic framework
3.1 The framework . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Additional examples . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Remarks about the interpretation of the trajectories . . . . . .
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CONTENTS
4 General relations between
4.1 Extensions . . . . . . . .
4.2 Equivalence . . . . . . .
4.3 Compatibility . . . . . .
theories
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5 Natural properties of theories
5.1 Compatibility with deterministic quantum theory
5.2 Robustness under small perturbations (Robust) .
5.3 Compliance with quantum theory (QT) . . . . .
5.4 Compatibility of di↵erent point of views (C) . . .
5.5 Single-world (SW) . . . . . . . . . . . . . . . . .
5.6 Self-consistency (SC) . . . . . . . . . . . . . . . .
5.7 Falsification of the assumptions . . . . . . . . . .
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6 Deriving the Born rule from non-probabilistic axioms
6.1 Motivation: Linking the mathematical formalism and observations . . . . .
6.2 The experiment Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Deriving the objective Born rule . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Stories that predict the frequency with an arbitrary precision are forbidden
6.5 A remark about possible falsification of the Born rule . . . . . . . . . . . .
6.6 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Retrieving the subjective Born rule
7.1 Operational definition of subjective probabilities
7.2 Rational beliefs about repeated experiments . . .
7.3 Example: Is it rational to go to the casino? . . .
7.4 Main statement . . . . . . . . . . . . . . . . . . .
8 Single-world theories are not self-consistent
8.1 Motivation . . . . . . . . . . . . . . . . . . . .
8.2 Extended Wigner’s friend experiment . . . . . .
8.2.1 A bird’s eye view on the experiment . .
8.2.2 Formal description of the experimenters’
8.3 Proof . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Applying the compatibility assumption
8.3.2 Analysis of individual views . . . . . . .
8.3.3 Combining the views . . . . . . . . . . .
8.4 Related work . . . . . . . . . . . . . . . . . . .
(BornDet)
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9 Conclusions
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9.1 Applications of the framework . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2 Outlook and open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A Bell’s Theorem
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2
CONTENTS
B Typicality Lemma
127
Bibliography
131
3
List of Publications
[1] Frauchiger, D. & Renner, R. Deriving the Born rule from non-probabilistic axioms.
arXiv preprint in preparation (2016).
[2] Frauchiger, D. & Renner, R. Truly random number generation: an example (2013).
URL http://dx.doi.org/10.1117/12.2032183.
[3] Frauchiger, D., Renner, R. & Troyer, M. True randomness from realistic quantum
devices. arXiv:1311.4547 (2013).
[4] Frauchiger, D. & Renner, R. Single-world interpretations of quantum theory cannot
be self-consistent. arXiv:1604.07422 (2016).
i
Chapter 1
Introduction
“Philosophy [i.e. physics] is written in this grand book – I mean the universe
– which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters
in which it is written. It is written in the language of mathematics, and its
characters are triangles, circles, and other geometrical figures, without which
it is humanly impossible to understand a single word of it; without these, one
is wandering around in a dark labyrinth.”
Galileo Galilei [1]
“It may seem odd that this suggestion – that we should try to form a rational
and coherent world-view on the basis of our best, most fundamental theories –
should be at all novel or controversial. Yet in practice it is.”
David Deutsch [2]
Looking back at the history of science we find that physics and philosophy used to be
much closer related than it is the case today. For Galileo mathematics was merely the
language to express philosophical and physical questions – in fact he did not even distinguish between the two fields. Today a gap has opened between them. Even though
Stephen Hawking stated in his book “The Grand Design” that “scientists have become
the bearers of the torch of discovery in our quest for knowledge” [3], hardly any physicist
would call himself a philosopher. The general attitude among scientists towards philosophical questions is rather agnostic. One of the main reasons for this is probably that in
contrast to mathematical problems, philosophical questions are generally formulated more
vaguely – often they are about things that are difficult to grasp such as “reality”, “the
mind” or “consciousness”. Therefore, they are less appealing for those that are attracted
by the exactness of mathematical formulas. However, a mathematical statement without
an interpretation remains empty. Of course the interpretation of the formal statements
must be done with care – Newton’s warning “Physics, beware of metaphysics!” is not
without reason [4]. One of the aims of this work is to introduce a framework for scientific
theories that allows us to bridge the gap between philosophy and science in a mathemati-
5
1. INTRODUCTION
cally precise way. The motivation for its structure is deeply inspired by the ideas of David
Deutsch [2] and Karl Popper [5, 6] which we will review in the first part of this chapter.
In particular, we will discuss the motivation for our decision not to base the framework
on probabilities.
The remaining part of the thesis is organised as follows. In Chapter 2 we review di↵erent interpretations of quantum theory that will play a central role later. The framework
itself is presented and illustrated with the help of examples in Chapter 3. In Chapter 4, we
discuss how some general relations between theories can be expressed naturally within the
framework. In Chapter 5 we formalise a set of natural assumptions about nature within
the framework that will be used in the following. A first application of the framework is
discussed in Chapter 6 where we show how a non-probabilistic formulation of the Born rule
can be obtained from two operational assumptions. In Chapter 7 we recover the standard
probabilistic formulation by the introduction of two further assumptions about an agent’s
beliefs. A second application of the framework is presented in Chapter 8 where we show
that “single-world” interpretations of quantum theory cannot satisfy a set of assumptions
that we would consider to be natural.
1.1
Motivation
1.1.1
The purpose of scientific theories: prediction vs. explanation
”To say that prediction is the purpose of a scientific theory is to confuse means
with ends. It is like saying that the purpose of a spaceship is to burn fuel.”
David Deutsch [2]
In his autobiography [7] Heisenberg recalls a conversation with Pauli about what “understanding” in physics means. When Pauli asked Heisenberg whether he had understood
Einstein’s relativity theory his reply was that he certainly did not have troubles with the
mathematical framework – but this did not mean that he actually understood it. For
example he said that he did not understand what it means that “the time for a moving
observer is di↵erent from the time for an observer at rest”. Pauli replied that grasping the
mathematical framework would certainly allow to predict the outcome of any measurement – and what more could one ask for? Thereupon Heisenberg answered: “You might
even say that I grasped the theory with my brain but not with my heart.”
Almost 100 years later the opinion that prediction is the sole purpose of theories is still
widespread. However, that understanding cannot be the same thing as prediction can be
seen with the help of simple examples. Let us consider for instance Ptolemy’s planetary
model. It describes the motions of the planets correctly but it is obviously a wrong description of reality. It is based on the false assumption that the Earth rests at the center
of the universe and therefore, it cannot help us to actually understand the motion of the
planets.
6
1.1 Motivation
Another example illustrating the di↵erence between prediction and explanation is mercury’s perihelion shift. The reason why we prefer Einstein’s theory to Newton’s is certainly
not that it can predict the trajectory by a tenth of a thousand degree more precisely.
Rather it is a more accurate description of reality and therefore, it allows us to get a
better understanding of the structure of space and time. In fact, prediction is often a
consequence of understanding but not the other way around.
The idea that understanding is the purpose of science and that theories are explanations
lies at the heart of the framework that we will introduce in this work. Informally speaking
the idea is that we may tell stories about experiments, explaining why certain observations
can or cannot be made. Crucially, these experiments are not limited to the ones that can
actually be carried out in practice. We may also tell stories about thought-experiments,
for example about what would happen when an observer would fall into a black hole.
Theories are then identified with rules that limit the set of allowed stories. In Chapter 3
we will formalise this idea.
1.1.2
The scientific method vs. the inductive method
“Assume that we have deliberately made it our task to live in this unknown
world of ours; to adjust ourselves to it as well as we can; to take advantage of
the opportunities we can find in it; and to explain it, if possible, and as far as
possible, with the help of laws and explanatory theories. If we have made this
our task, then there is no more rational procedure than the method of trial and
error – of conjecture and refutation: of boldly proposing theories; of trying our
best to show that these are erroneous; and of accepting them tentatively if our
critical e↵orts are unsuccessful.”
Karl Popper [5]
If the goal of science is to find explanatory theories this raises the question: is there
a scientific method to obtain them? The widespread opinion is that theories are derived
by inductive interference, i.e., by formulating universal laws based on the results of observations. However, it is well known that this method is difficult to justify. David Hume
is famous for introducing the problem of induction [8] in the 18th century, observing that
even though induction cannot be justified we nevertheless perform it. Max Born writes
[9] that while induction cannot be valid in everyday life, science has worked out a “code,
or rule of craft” for its validity – but he leaves open what this rule is, calling induction a
“metaphysical principle” or “something beyond physics”. The belief that science proceeds
from observation to theory is still so widely held, that Popper was suspected “of denying
what nobody in his senses can doubt” for questioning it [5].
Russell’s chicken [10] is an inductive proof demonstrating that induction cannot justify
any predictions: The chicken observes that it is fed every day. Inductively it concludes
that it will continue to be fed and every feeding day adds evidence to that theory. Then
one day the farmer kills the chicken, thus falsifying the chicken’s theory. Now the same
7
1. INTRODUCTION
experience (that their inductive method did not work) was made by many other chickens
– inductively it follows that inductive inference cannot justify predictions.
If theories cannot be derived from observations – how should we get them? According
to Popper’s and Deutsch’s reasoning, theories cannot be derived from anything but they
are created. In that sense a scientist is not an engineer taking an existing theory and
improving it, but rather an artist conjecturing theories, refuting them and replacing them
by better ones. Popper summarises the scientific method as follows.
“Without waiting, passively, for repetitions to impress or impose regularities
upon us we actively try to impose regularities upon the world. We try to discover similarities in it, and to interpret it in terms of laws invented by us. [...]
These may have to be discarded later, should observations show that they are
wrong.”
Karl Popper [5]
Within this approach scientific theories are distinguished from non-scientific ones by the
property that they are falsifiable. Let us therefore discuss how theories can be falsified.
1.1.3
Falsification
The obvious method to falsify a theory is by an actual experiment. However, David
Deutsch stressed that experimental tests are not the only method to falsify a theory [11]
– he explains that in fact most theories are refuted by argument without ever being experimentally tested. For example the theory that “drinking one litre of olive oil will make
you super fast” will probably never be tested – precisely because it does not contain any
explanation.
But what distinguishes a bad from a good explanation? In Deutsch’s words [2] “we
deem an explanation to be better if it leaves fewer loose ends and requires fewer and simpler postulates”. The corresponding principle is Occam’s razor [12], saying that “among
competing hypotheses, the one with the fewest assumptions should be selected”.
Computational complexity is the branch of mathematics formalising this idea. A way
to think about it is to imagine that the world is rendered by a virtual reality generator. If
whoever programmed the corresponding device was rational, he or she would choose the
simplest algorithm requiring the least amount of computational power. Because introducing extra non-necessary assumptions or parameters into a theory amounts to increasing its
complexity such a procedure destroys the scientific character of a theory [5]. In the next
chapter we will discuss how Bohmian mechanics, a single-world interpretation of quantum
theory, may be refuted by this argument.
8
1.1 Motivation
A good theory is also characterised by the property that it is compatible with explanations from other fields. For example another critique against Bohmian mechanics is that
it is incompatible with special relativity (see Chapter 2).
Properties of a theory that we would find desirable may be formally expressed within
the framework that we will introduce in this work. In Chapter 5, we discuss common
examples for such features such as compatibility with quantum theory or the idea that the
statements of a theory should be robust. By robustness we mean the property that the
statements of a theory remain valid under small perturbations of the formal description,
which is a necessary criterium to be applicable in practice. We also give formal definitions
of some operational assumptions, e.g., compatibility, the idea that if theories are applied
from di↵erent observers’ perspectives they should not obtain contradicting statements.
As an application of the idea that a good theory should be compatible with such natural
assumptions we show in Chapter 8 that single-world interpretations of quantum theory
cannot satisfy a set of such properties that we would consider to be desirable.
1.1.4
Theories and “reality”
If the ultimate goal of science is to understand the world, we may conclude that we should
seek for a true theory of nature or a true description of reality. However, because we can
never verify a theory, the question of a true theory becomes meaningless – all we can do is
to falsify theories. In Popper’s words we may impose laws upon nature but “nature very
often resists quite successfully, forcing us to discard our laws as refuted; but if we live we
may guess again” [5].
One might try to modify the aim to seeking theories that are true with high probability, where probability is understood to quantify our belief in the correctness in a Bayesian
sense [13]. However, the less explanatory power a theory has the higher is the probability
that it is true. Therefore, it cannot be the ultimate goal of science to find the theory that
is true with the highest probability. A good theory is restrictive – the more it forbids the
better it is.
Accepting that we can never know whether or not a theory is a true description of reality we may still understand our best theories as the best available description of reality.
Thus, the elements, or the parameters, of a theory should be viewed to be real in the sense
that they are more than a mere mathematical statement – otherwise the theory would not
contain any explanatory power. If the formalism of a theory already imposes constraints,
or implicit assumptions, on the structure of these parameters this may be problematic,
especially if we are not aware of it. For example, today’s fundamental theories are based
on the idea that we can express what will happen based on a given initial state of the
system – or, as in the case of quantum theory, that we can calculate the probability distribution of future events. Thus, the structure of these theories is based on the idea that
reality contains a notion of time which flows and that has certain properties associated
to the time-parameter within the formalism of the theory. The problem with this is that
9
1. INTRODUCTION
understanding time is actually one of the most puzzling topics in science. However, it
seems impossible that we can gain a better understanding of time itself from a theory
that is based on time-related notions. What makes the issue even worse is that in general
the definition of “time” is not even unique within a theory. For example within quantum
theory it is well known [14] that the global time parameter cannot be identified with an
observer’s local time. Moreover, we tend to overload the formal time parameters with
our everyday intuition. Finally, the idea that we can calculate what will happen for all
initial states is generally not even true on a formal level. For example Norton’s dome [15]
describes a setup within classical mechanics in which the initial state allows for di↵erent
solutions of the equations of motion.
Summarising, basing a theory on parameters that are loaded with our everyday intuition prevents us from gaining a better understanding of the structure of reality. This
observation motivated us to make as few assumptions as possible about the structure of
our framework. For example, it does not rely on a specific time structure. Rather, the idea
is that time can be introduced operationally by modelling it with the help of an actual
clock. In general our aim was to avoid as many implicit assumptions as possible, hoping
that by making them explicit it might be possible to avoid being misled by some prejudice
about the world. As Dirac wrote, history has shown that progress in science has usually
involved giving up such prejudices:
“When one looks back over the development of physics, one sees that it can
be pictured as a rather steady development with many small steps and superposed on that a number of big jumps. [...] These big jumps usually consist
in overcoming a prejudice. We have had a prejudice from time immemorial;
something which we have accepted without question, as it seems so obvious.
And then a physicist finds that he has to question it, he has to replace this
prejudice by something more precise, and leading to some entirely new conception of nature.”
Paul Dirac [16]
1.1.5
Implicit assumptions within the framework
One may wonder about the implicit assumptions introduced by the structure of the framework itself. Indeed we do assume that there is an external reality, in the sense that we
do exclude any kind of solipsism. There is no way to disprove that this external reality is
only taking part in my or your mind or that we are living in some kind of Matrix world.1
But even if it were we may discover the rules of that virtual world. Moreover, we would
probably not be satisfied by finding out its program but also seek for explanations about
1
The Matrix (1999) is a science-fiction movie in which the world is simulated by machines while in
reality humans serve the machines as energy source.
10
1.2 The problem with probabilities
its origin.
Besides the postulate that there is an external world we also assume that it is indeed
possible to describe it by the means of a theory. This raises the question whether the
entities in a theory are real or not. According to Deutsch deciding that they are not real is
equivalent to rejecting the corresponding explanation [2]. He gives the example of gravity:
at the time of Newton gravitation (the force pulling us down towards the center of the
earth) was considered as real – after all everyone could feel it. Since the introduction
of general relativity we know that there is no such force. What we feel is the curvature
of space-time that causes us to fall downwards. Thus, Deutsch concludes, reality is that
space-time is curved.
This idea is also related to the discussion from above where we argued that a good
theory should have a small computational complexity. “Real is what kicks back if it’s
kicked” Dr. Johnson said [2] while kicking a rock. Recalling our discussion from the previous section, we may assume that the rock is real, if this is the simplest explanation for
all the e↵ects associated to Dr. Johnson kicking it – including for example pain he feels. If
simulating the rock would require a high amount of computational power, we may refute
that explanation by Occam’s razor.
In the last part of this chapter we discuss why we choose the framework to be nonprobabilistic.
1.2
The problem with probabilities
“It is unanimously agreed that statistics depends somehow on probability. But,
as to what probability is and how it is connected with statistics, there has seldom been such complete disagreement and breakdown of communication since
the Tower of Babel. Doubtless, much of the disagreement is merely terminological and would disappear under sufficiently sharp analysis.”
Leonard J. Savage [17]
The idea that scientific theories are based on probabilities is so deeply rooted in science
that it seems absurd to even question it. In the previous section we argued that it is desirable to find theories with a high explanatory power about reality. If we want to explain
this reality with the help of probability based theories, we must first give these probabilities an operational interpretation. Without an interpretation probabilities are a purely
mathematical concept and thus, they cannot explain anything about nature. What makes
this issue difficult is that our understanding of probabilities is overloaded from everyday
experience. For instance, we may say that it is very probable that it rains tomorrow or
that the probability that the universe came out the way it did is 0.0000034. But what
11
1. INTRODUCTION
exactly do we mean by such statements on an operational level?
In the following, we will discuss that it seems impossible to give an objective operational definition of probabilities. We will look at the standard approaches to this problem
and find that they either su↵er from the problem of being unphysical or making circular
statements. In contrast to this a subjective interpretation is possible without running
into inconsistencies. However, it seems rather unsatisfactory to base a theory about an
objective reality on a subjective concept.
The following discussion about probabilities is supposed to illustrate this problem in
more detail and to motivate the introduction of a non-probabilistic framework. For a more
complete overview on probabilities see for example [18].
1.2.1
The objective interpretation
Attempts to define probabilities in an objective sense generally rely on statements about
the frequencies of outcomes. They can be divided into two categories – those definitions
that are based on finite and those that are based on infinite sequences. The earliest
approach of the first type goes back to Laplace.
“The theory of chance consists in reducing all the events of the same kind to
a certain number of cases equally possible, that is to say, to such as we may
be equally undecided about in regard to their existence, and in determining the
number of cases favorable to the event whose probability is sought. The ratio of
this number to that of all the cases possible is the measure of this probability,
which is thus simply a fraction whose numerator is the number of favorable
cases and whose denominator is the number of all the cases possible.”
Pierre Simon de Laplace [19]
This definition of probability, also known as classical probability, is based on the assumption of indi↵erence, meaning that it is assumed that “equally possible cases” have the
same probability. Because probabilities are defined in terms of probabilistic statements
this approach is often criticised to be circular.
One of the first approaches that try to capture the idea of objective probabilities in
terms of infinite sequences is found in the work of Venn.
“Every problem of Probability, as the subject is here understood, introduces the
conception of an ultimate limit, and therefore presupposes an indefinite possibility of repetition.”
John Venn [20]
12
1.2 The problem with probabilities
Because it is impossible to repeat any experiment an infinite number of times definitions
of this type are criticised to be unphysical. The critique on both approaches is nicely
summarised in a dialogue that can be found in a paper of Chris Fuchs [21].
1.2.2
The subjective interpretation
“But the value of probability is no factual datum, it has no objective meaning.
The sentence ‘The probability is equal to a given number p’ expresses no fact,
is neither true nor false, can be no hypothesis, cannot be called more or less
reliable.”
Bruno de Finetti [22]
Motivated by the problems one faces when trying to give probabilities an objective meaning, advocates of a subjective interpretation, such as de Finetti, are ready to give up the
idea that probabilistic theories can be used to describe an objective reality.
“So no science will permit us say: this fact will come about, it will be thus and
so because it follows from a certain law, and that law is an absolute truth. Still
less will it lead us to conclude skeptically: the absolute truth does not exist...”
Bruni de Finetti [22]
This attitude towards science is often criticised as some kind of solipsism. However, from
de Finetti’s words it is not clear whether he really denied the existence of an external
reality or simply that it can be described by the means of physical laws.
The subjective interpretation treats the probability of an event as a degree of belief or
knowledge about its outcome. The physicist is seen in the position of a gambler weighing
his chances. His probability assignments to the individual events are subjective in the
sense that nothing can prove him right or wrong. The only thing that can be proven by
the principles of probability theory is that there is no internal contradiction within his
assignments. For example, a gambler believing that the outcomes 4 and 10 are very likely
must conclude that also the event “4 or 10” is very likely. Otherwise, de Finetti says, “we
do not hesitate to call him crazy, and to say that anyone who assigns di↵erent values to
the probabilities lacks common sense” [22].
One possibility to operationally define subjective probabilities is within a betting scenario [23]. Consider for example an agent assigning the probabilities P (“heads”) and
P (“tails”) to a coin flip. If the agent is o↵ered a bet in which he obtains $1 in the event
of “heads” and nothing otherwise the operational interpretation of his probability assignment is that he is willing to pay $ P (“heads”) to enter that bet. Analogously P (“tails”)
is the amount of money the agent would pay for the bet in which he gets $1 in the event
of “tails” and nothing otherwise.
13
1. INTRODUCTION
Obviously this definition does not yet restrict actual probability assignments in any
way. In fact, an agent may even pay more than $1 to accept any of the bets described
above. However, in this case the agent would lose money with certainty and it may be
taken as an assumption that such a behaviour is irrational. It follows from the Dutch book
argument that if the prices assigned to bets do not result in losses with certainty then they
must satisfy the axioms of probability theory. In particular, it follows that they are between 0 and 1. In Chapter 7 we review the Dutch book argument and give a possible proof.
The question arises now whether one can find stronger constraints on the probability
assignments than that they should satisfy the axioms. Given that it is meaningless to ask
if a particular assignment is true or wrong, one may wonder if it is at least possible to
decide whether one particular assignment is better than another one? It turns out that one
can arrive at actual probability assignments by introducing additional assumptions about
the structure of repeated runs of the bet described above. In particular an agent may
believe for example that the outcome in a particular round is independent of how often
the experiment is carried out. The idea of “exchangeable events” and its implications on
probability assignments has been studied by de Finetti [24] and we will review his ideas in
Chapter 7 within the formalism of the framework. In particular, we will show how actual
probability assignments can be derived as a combination of laws about nature and certain
beliefs of an agent.
1.2.3
Probabilities are subjective – consequences
We argued that all the standard objective definitions of probability do not have an objective operational physical meaning and thus, that probabilities should be interpreted
subjectively. Let us discuss the consequences of this observation for existing theories.
Let us consider first a coin flip. When we say that the probability to obtain “heads”
in a coin flip is 1/2 it seems obvious that the probability must be subjective, because the
underlying theory (classical mechanics) is deterministic. In this case a precise description
of the initial conditions would allow to predict the outcome with certainty. However, there
are also cases for which the laws of classical mechanics are not deterministic [15]. In these
cases classical mechanics does not assign a probability distribution to the set of possible
events either and thus, also in this situation the probabilities are introduced by additional
assumptions expressing an agent’s knowledge about the system.
In the case of thermodynamics the issue is more subtle. We are used to think of phenomenological thermodynamics as an objective theory. However, the corresponding laws
can be derived from a statistical description, based on probabilities, by averaging over all
particles where it is assumed that all configurations are equally likely, i.e., by assigning a
uniform probability distribution to them. Thus, the laws of thermodynamics are subjective in the sense that they hold under the assumption that the described system is typical,
i.e., that it does not belong to the special class of systems for which the laws actually do
not hold. That such systems exist at least in theory has been demonstrated for instance
14
1.3 Summary
in [25]. Also Maxwell’s Demon [26] corresponds to a scenario in which the second law of
thermodynamics is violated.
What about quantum theory? We tend to think about probabilities within quantum
theory as more fundamental than within classical mechanics, in the sense that the outcomes
cannot be predicted any better than within the quantum formalism [27] – even if all the
information in the whole universe would be available. How should we deal with this
having found that probabilities cannot be defined objectively? Indeed, in order to be
consistent also in the case of quantum theory probabilities have to be defined subjectively.
We would like to stress that this does not necessarily imply that all the statements of
quantum theory are only subjective. One possibility to interpret quantum theory as an
objective description of reality while giving the probabilistic statements in it a subjective
interpretation is within a multiverse interpretation. Here the unitary quantum formalism
is taken to be a one-to-one description of the structure of the world – from which it
follows that there is some kind of universe splitting, giving many of us an uncomfortable
feeling because it is against our perception of one single reality. However, under the
assumption that this description is accurate, probabilities may be interpreted as a selflocating uncertainty [28] (see Chapter 2).
1.3
Summary
- Assuming that we are driven by the urge to understand nature, the purpose of a
scientific theory must be understood to be explanation as opposed to mere prediction.
This idea lies at the heart of the framework that we will introduce in the following.
- The scientific method according to Popper consists of replacing existing theories by
better theories. Theories are refuted either by direct falsification (i.e. observations
that contradict their predictions) or else by showing that they are “bad” theories
in the sense that they contradict some assumptions that we would consider to be
desirable. The structure of our framework allows to capture both ideas. In particular,
desired properties of theories can be made explicit naturally which makes it possible
to test whether a theory can satisfy all of them.
- The possibility to make assumptions explicit in the framework (for example assumptions about the time structure) is intended to avoid possible flaws resulting from
prejudices.
- Probabilities are mysterious: It is well-known that existing interpretations of objective probabilities either su↵er from circularity problems or else that they are
unphysical. Subjective probabilities on the other hand seem to be insufficient to express laws of nature. This motivates our approach to start with a non-probabilistic
framework. Probabilities may be introduced to serve as consistency requirements
expressing an agent’s beliefs.
15
1. INTRODUCTION
16
Chapter 2
Interpretations
“A physical theory remains an empty shell until we have found a reasonable
physical interpretation.”
Peter Bergmann [29]
In the previous chapter we motivated the approach to identify physical theories with
explanations about nature. We discussed that in order to achieve this purpose it is necessary to provide the mathematical formalism of a theory with an interpretation – because
otherwise the theory remains an empty statement. In the case of quantum theory it is not
obvious how the formalism should be interpreted and in fact all existing interpretations
seem to be unsatisfactory in some way. Because di↵erent interpretations correspond to
di↵erent explanations about nature we will identify them with di↵erent theories. It is
then an interesting question if one of the di↵erent interpretations of quantum theory is
preferable to the others because it is a better theory in the sense discussed in the previous
chapter. In particular, we will be interested in the question whether certain properties
that we would consider as natural can be satisfied within a given interpretation. In Chapter 3 we will introduce the framework and discuss how the di↵erent interpretations can
be expressed within it on a formal level and in Chapter 5 we will formalise the mentioned
properties a theory may have. This will allow us in Chapter 8 to test if an interpretation
can satisfy the properties. In this chapter we will give a short review on the di↵erent
interpretations of quantum theory. As a start we will briefly discuss that also in the case
of classical mechanics it is not as obvious as we tend to think how the formalism should
be interpreted.
17
2. INTERPRETATIONS
2.1
Interpretations of classical mechanics
“Classical mechanics is deceptively simple. It is surprisingly easy to get the
right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings, which depend on context, and often even change within a
given context.”
from “Structure and Interpretation of Classical Mechanics” [30]
In contrast to quantum theory there has hardly been any interest in the interpretation
of classical mechanics. The very idea of the formalism is that it is supposed to describe
systems exactly the way we expect them to behave. Therefore, the question about interpreting it seems to be superfluous. However, experience has taught us that this is not
always true. Considering for example the time parameter within Newtonian mechanics
we know that it cannot describe “real time”. With the introduction of special relativity it
has become clear that real time (whatever it is) certainly does not have the property that
it is absolute as it is the case in classical mechanics.
Another example illustrating that the interpretation of the classical equations of motion is not obvious was introduced by Norton [15]. While we expect from our intuition that
the classical equations of motion are deterministic it illustrates that this is not necessarily
the case. In Norton’s example a mass rests on top of a dome with shape (2/3)gr3/2 (see
Fig. 2.1). The equations of motion have an infinite number of solutions, i.e., the time as
well as the direction of the mass when it starts moving are not deterministic. A possible approach to operationally interpret the scenario would be to introduce a probability
distribution on the time as well as the direction. However, this distribution is not given
by the laws of classical mechanics. Rather, additional assumptions about our knowledge
would have to be made, for example, that the distribution is uniform for all directions.
?
Figure 2.1: Norton’s dome is an example illustrating that the classical equations of
motion may not be deterministic.
18
2.2 Interpretations of quantum theory
2.2
Interpretations of quantum theory
“Quantum theory is the most useful and powerful theory physicists have ever
devised. Yet today, nearly 90 years after its formulation, disagreement about
the meaning of the theory is stronger than ever. New interpretations appear
every day. None ever disappear.”
N. David Mermin [31]
In contrast to classical mechanics the interpretation of quantum theory is much more controversial. Even though it is generally accepted that the formalism needs an interpretation
the issue is met rather sceptically by many physicists. A common attitude is that this
is not the task of a serious scientist – an approach which was summarised by Mermin as
“Shut up and calculate”. In fact he attributed this attitude to advocates of the Copenhagen interpretation, which is taught in most standard text books about quantum theory.
Sometimes it is also argued that the quantum formalism provides the probability distributions for outcomes in measurements and this is all we can ask for. However, this line
of thinking reduces the power of quantum theory to mere predictions – and, as Bell said,
this would be really disappointing.
“However, the idea that quantum mechanics, our most fundamental physical
theory, is exclusively even about the results of experiments would remain disappointing. [...] The aim remains: to understand the world.
John Bell [32]
Moreover, we discussed in Chapter 1, it is not possible to define probabilities on an operational level such that they have an objective meaning – and as we argued before this
seems to be a necessary criterion for any theory that is supposed to tell us something
about nature.
Sometimes it is also argued that reasoning about interpretations is meaningless because by construction they cannot be tested experimentally. However, as we argued in
Chapter 1, most theories are refuted not by experiment but because they are bad explanations in the sense that they are in contradiction with certain properties of nature that
we consider to be “reasonable”.
Unfortunately, all existing interpretations of quantum theory contradict in some way
with what we think reality “ought to be”. Single-world interpretations assume that there
is a unique reality, but they have other problems that we will discuss in more detail in the
following. In short words these problems either concern some kind of inconsistency within
the formalism or else incompatibilities with well-accepted principles such as non-signalling.
Many-world interpretations postulate the existence of many realities in parallel – an idea
that certainly challenges our intuition. Their advantage compared to existing singleworld interpretations is that they are based on a well-defined and consistent mathematical
19
2. INTERPRETATIONS
formalism. One of the main contributions of this work is to provide an additional argument
to prefer many-world to single-world interpretations. This will be the topic of Chapter 8.
Here we will discuss the well-known problems related to the di↵erent interpretations.
2.2.1
The measurement problem
Interpretations of quantum theory are all indented so solve what is known as the measurement problem. We will briefly illustrate it here.
Gedankenexperiment I
Consider a simple quantum experiment consisting of a single photon source and a polarising
beam splitter, which directs the photon into two di↵erent branches, depending on whether
the polarisation is horizontal or vertical. We denote the corresponding states by |hi and
|vi and assume that the source emits diagonally polarised light in the state
1
|di = p (|hi + |vi) .
2
Let us now place a toy-detector consisting of a two-level atom into the upper arm (corresponding to the vertical polarisation) of the beam splitter. The ground state of the atom
is denoted by |gi and the excited state by |ei. The atom is excited from the ground to the
excited state if and only if the photon passes through the upper arm.
By linearity of quantum theory the atom will become entangled with the photon
✓
◆
1
1
p (|hi + |vi) ⌦ |gi
! p (|hi|gi + |vi|ei) .
2
2
Here |hi|gi is the joint state of the photon+atom where the photon is horizontally polarised
and the atom is in the ground state and |vi|ei is the joint state of the photon+atom where
the photon is vertically polarised and the atom is in the excited state. Thus, the state
of the atom after the photon has passed the beamsplitter is a superposition – it is in
the ground and in the excited state at the same time. In the case of the atom it is well
accepted that the unitary description from above is correct and that there is no collapse
happening. Let us now see what happens when we replace the atom by a bigger system –
or even a system with consciousness.
Gedankenexperiment II
Let us replace the atom by an observer with the ability to see single photons.1 For convenience with later purposes let us call him Wigner’s friend. It might be argued that
1
Even though this is unrealistic it simplifies the argument. For an observer with realistic eye-sight
nothing fundamentally would change in the argument as one could simply introduce a photon-multiplier
and view it as part of the observer. Note also that frogs can actually see single photons. [2]
20
2.2 Interpretations of quantum theory
Wigner’s friend cannot be described within quantum theory. However, as we will discuss
below, this assumption would require further justification explaining which kind of systems can be described within quantum theory and which not. Generally, the assumption
that quantum theory can be applied to all systems, independently of their size or whether
or not they are conscious, is referred to as universality. However, this formulation is a bit
unfortunate, because it seems to be rather the assumption of non-universality that would
require further justification and in particular a specification for which systems quantum
theory can be applied.
We denote the state of the friend “having seen the photon” by |yes!i and the state
“having not seen the photon” by |no!i. Additionally we introduce a state |?i describing
the state of the friend before the photon enters the beam splitter. If quantum theory can
be applied to the friend just in the same way as to the atom, Wigner’s friend will also
become entangled with the photon
1
p (|hi + |vi) ⌦ |?i
2
!
1
p (|hi|no!i + |vi|yes!i).
2
(2.1)
But what is the “real” state of Wigner’s friend? Is he also simultaneously in the state of
having seen and not having seen the photon? Or is he rather in a definite state corresponding to only one of the two possibilities? But then the joint state is either |hi|no!i or
|vi|yes!i and there is some non-unitary evolution di↵erent from the case of the atom. Or is
there possibly a hidden variable, that is not part of the quantum formalism, characterising
the real state of the observer?
The question “What is the ‘real’ state of the friend after the photon has passed
the beam splitter” is one way to state the measurement problem.1 This is also what
Schrödinger’s famous cat experiment [34] illustrates. Di↵erent approaches to solve the
problem have become known as di↵erent interpretations – here we will call them di↵erent
theories, because they di↵er in how they explain what is going on beyond the formalism.
Interpretations can be divided into two di↵erent groups: Those that assume that there
is always one observer who has seen one definite outcome (single-world interpretations)
and those that assume that there is some kind of splitting of the universe (multiverse
interpretations).
Before we discuss these two approaches in more detail let us make a remark about
universality. A common “solution” to the problem is to argue that quantum theory can
only be applied at a microscopic scale and therefore it is not possible to describe Wigner’s
1
The notion “measurement problem” actually involves several questions. According to Brukner [33]
there are at least two measurement problems. The first (the “small” problem) is that of explaining why a
certain outcome – as opposed to its alternatives – occurs in a particular run of an experiment. The second
(the “big” problem) is that of explaining the ways in which an experiment arrives at a particular outcome.
21
2. INTERPRETATIONS
friend within the formalism. However, as it was pointed out by Bell this approach introduces an ill-defined split between “systems” and “measurement-devices” which does
not solve anything, but rather raises even more questions. We will refer to his famous
quote, expressing his unease with the idea of measurements that cause a collapse of the
wavefunction, several times later.
“What exactly qualifies some physical systems to play the role of ‘measurer’ ?
Was the wavefunction of the world waiting to jump for thousands of millions
of years until a single-celled living creature appeared? Or did it have to wait a
little longer, for some better qualified system . . . with a PhD? ”
John Bell [32]
2.2.2
Multiverse interpretations
All the di↵erent variants of the multiverse interpretation introduced since the original work
by Everett [35] have one thing in common: the key idea is to take quantum theory literally
as a true description of nature. The wavefunction evolves unitarily and no collapse ever
takes place. Crucially, all the di↵erent components of the wavefunction, or the di↵erent
branches, are assumed to be equally real. Thus, when Wigner’s friend is in a superposition
of having seen the photon and not having seen the photon according to the formalism the
multiverse interpretation tells us that there are now two equally real versions of Wigner’s
friend – one for each possibility. In other words the measurement process has caused a
split of the friend into two copies. Variants of the multiverse interpretation di↵er in how
they interpret where this split takes place. Whereas many-worlds interpretations assume
that there is a truly physical split [36] the split takes place in the friend’s mind within
many-minds interpretations [37, 38]. Sometimes, the fact that it is not clear what the
reality the wavefunction describes is supposed to be, is used as an argument against the
multiverse interpretation. However, within single-world interpretations it is not obvious
what reality consists of either. The crucial di↵erence between the two interpretations is
simply that whereas within single-world interpretations only one of the branches of the
wavefunction is real, within multiverse interpretations the wavefunction as a whole describes reality. But both interpretations remain silent about what exactly this reality is.
Therefore, we will simply identify the multiverse interpretation with the point of view that
all the branches of the wavefunctions are equally real – whatever this reality may be.
A common argument against the multiverse interpretation is that it does not contain
probabilities. After the discussion about the problems associated with the interpretation
of probabilities in Chapter 1 this may be seen rather as a strength than a weakness. However, it is indeed possible to make use of probabilities within multiverse interpretations if
they are interpreted subjectively – for example as a measure for self-locating uncertainty
[39]. The idea of this self-locating uncertainty can be illustrated by the following thoughtexperiment. There are two rooms, one with the number 0 and the other with the number
1 on the outside of the door. Wigner puts his friend first asleep and then measures a
22
2.2 Interpretations of quantum theory
q
q
2
1
quantum state in a superposition, e.g.
|0i
+
3
3 |1i, w.r.t. the computational basis.
He then puts his friend in the room with the same number as the outcome he has seen,
wakes him up and asks him to guess in which room he is. According to the multiverse
interpretation there are now two friends – one in each room. However, it makes still sense
for each of the two friends to place a bet on where he is. As shown by Deutsch [40] it is
rational to guess for the friend that he is in the room with the higher absolute value of
the amplitude (the one with the 0 on the door in this case).
For an exhaustive review about the multiverse-interpretation see for instance the book
by David Wallace [41]. The reasons whz scientists tend to like and dislike it cannot be
expressed any better than in Barnum’s words.
“...there are reasons for disliking the MWI, for instance that it implies the
view that our minds occasionally split into many minds [...]. While I certainly
don’t consider it a priori impossible that science should lead us to this belief,
I do consider this belief sufficiently bizarre a priori that it is worth investigating alternatives. However, it may be more scientifically satisfying to adopt
the splitting-consciousness view along with a physical theory that is rigorous,
precise, and economical, than to adopt the ‘reduction of the statevector’ view
which ins uneconomical [...] and fuzzy [...].”
Howard Barnum [42]
2.2.3
Single-world interpretations
By a single-world interpretation we mean any theory in which measurements have only
one outcome and there is only one version of the observer after the measurement. We also
require that it is compatible with quantum theory in the sense that its statistical predictions
agree with the probabilistic interpretation of the Born rule. Those kinds of interpretations
can be classified into three groups:
- collapse theories
- hidden variable theories
- QBism
Collapse theories and hidden variable theories correspond to an objective description of
(a single) reality. In contrast to this QBism interprets quantum theory as a guideline for
an agent to make rational decisions. In the following we will discuss these approaches in
more detail.
Collapse theories
Collapse theories, or the Copenhagen interpretation, postulate a non-unitary evolution of
the wavefunction whenever a measurement occurs. In the example from above the evolution of Wigner’s friend and the photon would not be described by a unitary transformation
23
2. INTERPRETATIONS
as done in Eq. (2.1) but rather at the moment when Wigner’s friend sees the photon the
state collapses to
1
p (|hi + |vi) ⌦ |?i
! |vi|yes!i.
2
Analogously if he does not see the photon, when he realises that the photon has passed
through the lower branch, the state undergoes the transition
1
p (|hi + |vi) ⌦ |?i
2
!
|hi|no!i.
Thus, it is guaranteed that there is always only one version of Wigner’s friend. Within the
Copenhagen interpretation probabilities are assigned to the individual events by the Born
rule [43] which in this case would be equal to 1/2 for each possibility. After our discussion
about probabilities in Chapter 1 a first problem with this interpretation is obvious because it is not clear how those probabilities are supposed to be interpreted operationally.
In Chapter 6 we will derive an alternative interpretation of the Born rule which is not
based on probabilities – however, this interpretation also does not rely on the assumption
that there is only one friend but it is compatible with both single-world and multiverse
interpretations. But let us first discuss another problem concerning collapse theories related to the question: “When does the collapse happen?”
A common answer to that question is that the wavefunction collapses at the moment
when Wigner’s friend perceives the outcome. However, if the photon and Wigner’s friend
are spatially separated this approach is incompatible with special relativity. Even though
the two events – Wigner’s friend seeing the outcome and the wavefunction collapsing –
are simultaneous in the friend’s reference frame, this is not the case in any reference frame
that moves relative to it.
Moreover, even for Wigner’s friend himself the event “perceiving an outcome” may not
be well defined. For example in the second case, when the photon does not pass the upper
branch and thus the friend never sees it he will not realise at once that the photon has
actually passed the lower branch, but rather this is a continuous process.
Another possible answer is that the wavefunction collapses independently of Wigner’s
friend’s perception of the outcome. The problem with this idea is expressed by Bell’s quote
[32] asking whether it requires a PhD to have the power to collapse the wavefunction because it is generally agreed, and experimentally tested, that without a measurement the
wavefunction would remain in the superposition. Thus, if the atom cannot collapse the
wavefunction but Wigner’s friend can – what is the property that makes up this di↵erence
between them?
Thus, it seems to be impossible to make the answer to the question “When does the
collapse happen?” mathematically precise – at least with the current state of knowledge.
24
2.2 Interpretations of quantum theory
Hidden variable theories
Hidden variable theories are based on the idea that the formalism of quantum theory is
incomplete in the sense that there are additional parameters that are not part of it. Generally the motivation for their introduction is to avoid the problems of collapse theories by
constructing a deterministic formalism. In contrast to multiverse interpretations, which
are also deterministic, not all the branches are assumed to be equally real but one of the
branches, characterised by the hidden parameter, is supposed to be the real branch while
the other branches are empty.
Bell’s theorem is probably the best known work showing that there are limitations on
the structure of hidden variable theories. It is generally understood to show that “there is
no local realistic extension of quantum theory”. Because there tends to be some confusion
about what exactly is meant by this we review Bell’s theorem in Appendix A.
The best known example for a hidden variable theory is Bohmian mechanics [44]. Because we will often use it for illustration purposes we discuss it briefly here. Let us first
look at its connection to Bell’s theorem.
It follows from Bell’s theorem that as a special kind of a hidden variable theory
Bohmian mechanics is what is usually referred to as a non-local theory. However, this
is a bit misleading because sometimes also quantum theory is said to be non-local, but
in this case it is a fundamentally di↵erent kind of non-locality. As it is explained in
Appendix A non-locality can formally be divided into the two properties outcome- and
parameter-independence. Quantum theory does not satisfy outcome-independence whereas
Bohmian mechanics violates parameter-independence, i.e., it allows for faster than light
signalling.
Bell’s theorem was made stronger only recently [27] showing that any hidden variable
theory that gives more accurate predictions than quantum theory must be incompatible
with free choice. As we explain in Appendix A free choice does not refer to any metaphysical concept but is made formally precise. The definition captures our intuitive idea
that measurement settings can be chosen freely, i.e., such that they are only correlated to
events in their future w.r.t. a causal order. The introduction of a causal order ensures that
the definition is also meaningful for theories that are incompatible with special relativity1
as it is for example the case for Bohmian mechanics.
Within Bohmian mechanics the evolution of the wavefunction of the universe
is
given by the Schrödinger equation, just like in standard quantum theory. In addition
it postulates the existence of hidden parameters q k which are in general interpreted to
1
For theories that are compatible with special relativity a choice is said to be free if it is only correlated
to events that are in its future light cone.
25
2. INTERPRETATIONS
correspond to the positions of the particles. Their time evolution is given by the guiding
equation
dq k
mk
= ~rk ln Im[ ].
dt
If follows from the non-local character of Bohmian mechanics that the wavefunction
in general must describe the whole universe. That makes it rather impractical as a theory
for predictive purposes. However, by construction it gives the same predictions as standard quantum theory. This is guaranteed by the equilibrium hypotheses saying that the
distribution of the hidden parameters is given by the amplitude square of the wavefunction
| |2 . This assumption was critisied to violate the no-fine-tuning assumption, saying that
the observed statistical independence of variables should not be explained by fine-tuning
of the causal parameters [45].
What about the explanatory power of Bohmian mechanics? If it is supposed to be
a description of reality the hidden parameter must be real in some sense. The obvious
interpretation is to identify it with the actual position of the particles. However, there
are several examples in which the hidden variable does not behave as we would expect it
to if this would indeed be the case. For example in the fooled detector experiment [46] a
setup analogous to the one described above in the first gedankenexperiment is considered.
However, there are certain initial positions resulting in a configuration in which the hidden variable is in the lower branch but the atom (placed in the upper branch) gets excited.
Another problem is that it is well known that knowledge of the hidden parameter would
allow for faster than light signalling [47]. As a way out of the dilemma it is sometimes
assumed that even though the particle position exists, it cannot be known and therefore
it cannot be used for signalling in practice. Now there seem to be two options. Either
the particle position cannot be known for technical reasons, but there is no fundamental
law prohibiting that in principle it could be known and thus nature would allow signalling
in principle. Or else the particle position cannot be known fundamentally, i.e., not even
“God” can know it. But then it cannot have any meaning but to be a mathematical
parameter introduced into quantum theory and has no connection to reality. In this
case the only thing we have “gained” is that the quantum formalism has become more
complicated but not a better understanding of reality.
QBism
Quantum Bayesianism, or QBsim, has evolved from the work of Caves, Fuchs and Schack
[48]. It draws its motivation from Bayesian probability theory: given that it seems to
be impossible to objectively define probabilities (see Chapter 1) they are interpreted subjectively as a measure for an agent’s belief. This even includes cases in which an agent
is certain about an event, i.e., probability assignments 0 and 1 also express subjective
beliefs and not objective facts [49]. More precisely probabilities are defined by an agent’s
willingness to accept a bet he believes to be favourable to him on the basis of his probabilities. We will discuss the idea of defining probabilities via bets in more detail in Chapter 7.
26
2.2 Interpretations of quantum theory
The advocates of QBism conclude that because probabilities are subjective and quantum states are used to assign probabilities through the Born rule it follows that quantum
states are subjective, too. Fuchs stresses that quantum theory should thus be seen as
a tool describing an agent’s beliefs about the outcomes of measurements rather than his
knowledge about an external world, because knowledge suggests the existence of an agentindependent external reality.
Thus, within QBism quantum theory is viewed as a tool that an agent can use to
calculate his probabilistic expectations for future events based on his past experience.
More precisely Fuchs summarises the role of quantum theory from the QBist perspective
as follows.
“A QBist takes quantum mechanics to be a personal mode of thought – a very
powerful tool that any agent can use to organise her own experience. That
each of us can use such a tool to organise our own experience with spectacular
success is an extremely important objective fact about the world we live in. But
quantum mechanics itself does not deal directly with the objective world; it deals
with the experiences of that objective world that belong to whatever particular
agent is making use of the quantum theory.”
Chris Fuchs [49]
Chris Fuchs then writes that “This means that reality di↵ers from one agent to another”.
The idea that quantum theory cannot be used to describe an objective external reality
is often criticised by opponents of QBism. However, in contrast to other single-world
interpretations of quantum theory QBism has the great advantage that it is formally
precise and consistent as well as in agreement with special relativity.
27
2. INTERPRETATIONS
28
Chapter 3
Stories about experiments – a
non-probabilistic framework
In this chapter we introduce a framework that will allow us to talk about “physical laws”
in a mathematically precise manner. Unlike many standard approaches, the framework
does not rely on a notion of probabilities or any related concepts to quantify the accuracy of predictions. Rather, the basic idea is to consider “stories” that can be told about
gedankenexperiments including experiments that can actually be carried out in practice
but also purely hypothetical ones. Thus, it avoids the problems related to the interpretation of probabilities discussed in Chapter 1. However, in Chapter 7 we will show that
probabilities may be introduced into the framework as a measuere representing an agent’s
beliefs. This allows us to formulate objective laws of nature in a consistent manner (which
would not be possible if these laws were based on probabilities), while still making use of
subjective probability theory that has proven to be useful in the past.
Another advantage of the framework compared to standard formulations of theories
is that it does not assume that the theories have a given mathematical structure. The
two main results of this work assert the non-existence of a physical theory with certain
properties. Hence, the looser the definition we use for characterising a theory the stronger
is the claim. A priori we simply view a theory as a set of certain rules that forbid certain
stories. Crucially, these rules may forbid a story on the grounds of explanations that are
part of the story. For instance, Kepler’s theory about the solar system would forbid the
story that the sun rises tomorrow because the it moves in a circular orbit around the
earth. In contrast to this the story that the sun rises tomorrow because the earth moves
around the sun would not be forbidden. This idea takes into account that the purpose of
a theory is explanation rather than mere prediction as we discussed in Chapter 1.
In this chapter we will make these ideas formally precise. In the following chapters we
will give examples for applications of the framework. For instance, we will show how the
Born rule can be formulated as a non-probabilistic statement and how it can be derived
from two more natural assumptions about nature. As a second application we will prove in
29
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
Chapter 8 that single-world interpretations of quantum theory cannot be self-consistent.1
3.1
3.1.1
The framework
Stories
We start by clarifying the concept of a “story”, which lies at the heart of the framework.
A story s is a finite “account of connected events”,2 which may be real or hypothetical.
Mathematically this is expressed by the idea that the set of all stories, ⌃, is a countable
set. The countability of the set ⌃ ensures that the stories s 2 ⌃ have a finite description
– operationally this means that a person can tell the story in finite time.3 The form or
language of this description may however be arbitrary. For example one may think of the
set of stories as the set of (finite) texts (in a given language).
Note that the requirement that the set of stories is countable, and therefore stories
are finite, is motivated operationally – something that cannot be told in finite time is not
really a story. Interestingly, it turns out that the requirement is also relevant for technical
reasons. For instance it will ensure in Chapter 7 that we can introduce a (subjective)
probability distribution on events that are in one to one correspondence with stories. In
Chapter 6, where we derive the Born rule without any reference to probabilities, it will
turn out, that the requirement of finiteness ensures that stories can be logically extended,
without running into contradictions.4
The set ⌃ must not usually be characterised precisely. Rather, it is sufficient to specify
a countable superset. For example, we may say that ⌃ is the set of all “English texts”
(which is however hard to characterise). But this set is obviously contained in the set of
all finite sequences of letters, which is countable.
1
As we will make formally precise in Chapter 5 self-consistency expresses the idea that a theory should
allow that something happens at any time.
2
For example, presented in a sequence of written or spoken words or moving images.
3
This can be seen as follows. The elements of a countable set can be enumerated by definition.
Therefore, one can represent each element by its corresponding number, which is finite.
4
It will turn out that any story that predicts that frequencies can be measured with arbitrary precision
is forbidden. Not explicitly forbidden are stories that say for instance that the frequency will be equal to
0.500 with a precision of µ = 10 3 (it holds for all µ > 0). This story can be logically extended, for example
to the story that the frequency will be equal to 0.500111111.... Any story of this kind, corresponding to an
arbitrary precise extension with finite description is forbidden. The requirement that infinite extensions
that do not have a finite description do not correspond to stories and are thus not forbidden ensures that
it is still possible to logically extend stories.
30
3.1 The framework
3.1.2
Theories
The idea is that a theory T consists of laws which imply that certain stories are “forbidden”. Mathematically this is expressed as the requirement that a theory T specifies a
subset of ⌃, denoted by T, called the stories forbidden by T. Operationally s 2 T means
that the laws of T rule out the story s.
For example a story s that tells us that a particle moves at twice the speed of light
is forbidden by special relativity SR, i.e., s 2 SR, but not by classical mechanics CM, i.e.
s 62 CM. Note that the rules of a theory may not necessarily be experimentally testable.
For example Logics forbids stories that are inconsistent such as the story “Alice is prettier
than Bob. Bob is prettier than Eve. Eve is prettier than Alice.”
We stress that a story s that is not forbidden by a theory T should not be interpreted
as being “allowed” by T. It could be that s is not forbidden because it does not make sense
or because the theory is not applicable to it. For example the story about the particle
that moves faster than light is not forbidden within electrodynamics simply because there
is no notion of particles within electrodynamics.
3.1.3
Experiments
The remaining requirements are devoted to stories that we can tell about “experiments”.
Just as before these experiments are not limited to the ones that can actually be carried
out in practice. The only requirement is that they specify the values of the variables (such
as choices of parameters and outcomes of measurements) that we would typically want
to report when carrying out the experiment. In the following we will call these variables
physical quantities and the set of possible values the event space. The physical quantities
should include all parameters that are directly observable in an experiment (e.g., through
measurements), but could also include variables that are not directly visible. For example,
the polarisation state of a photon that we prepare may be regarded as a physical quantity.
The physical quantities defined by an experiment Exp correspond to a set ⇥Exp . Each
x̂ 2 ⇥Exp has a certain range, denoted by range(x̂), and can take values x 2 range(x̂).1
Any assignment of values to all physical quantities of ⇥Exp is called an event of Exp. Operationally ⇥Exp is the set of quantities that we want to talk about when telling a story
about the experiment Exp.
The specification of the event space [Exp] of an experiment takes into account that
there may be constraints on the joint specification ranges, for example because they are
1
The distinction between the physical quantity x̂ and an actual value x is analogous to the case of a
random variable X and an outcome X = x.
31
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
imposed by a protocol. Formally it is given by
[Exp] ✓ {range(x̂1 ) ⇥ . . . ⇥ range(x̂k ) : x̂i 2 ⇥Exp , k = |⇥Exp |}.
Note that sometimes it will be useful to consider classes of experiments. For instance
experiments with the physical quantities b and x̂ describing a measurement on a quantum
state with outcome x may describe a measurement on a photon as well as a measurement
on an electron or, at least in principle, even a human being. The assumptions entering the
argument in Chapter 8 will be explicitly formulated for such classes of experiments and a
possible objection would be that the relevant experiment is not in the class for which the
assumptions apply.
Stories about an experiment
The idea is that a story about an experiment describes a run of the experiment and should
therefore specify which events are observed. The experiment Exp assigns to certain stories s 2 ⌃ an interpretation in terms of a trajectory, sExp , defined as a subset of events
sExp ✓ [Exp]. The stories s 2 ⌃ for which sExp is defined are called stories about Exp.
Operationally the trajectory sExp consists of all events of Exp that actually happen during
the run of Exp according to the story s.
The approach ensures that there exists a function s 7! sExp that “interprets” stories.
The function equips any story s about Exp (which may just be seen as sequences of letters)
with an actual meaning (in terms of observable events in Exp).1 The explicit form of this
map corresponds to an interpretation of a story. While we would intuitively distinguish
between correct and false interpretations we do a priori not impose any constraints on
this function. In Chapter 5 we will formulate certain consistency requirements on them,
for instance, that if the story talks about di↵erent observers’ perspectives their respective
trajectories should be compatible.
These ideas are best illustrated with an example. We start with a simple experiment,
Cool, where a small bottle, initially at room temperature, is put into a huge pool filled
with ice-cold water (see Fig. 3.1). The experiment Cool has the temperatures T̂B and T̂P
of the bottle and the pool, respectively, as physical quantities.
We may tell the following stories about what could happen in a run of Cool.
sc =
1
(
“The temperature of the bottle, initially at 20 C
continuously decreases to 0 C. The temperature of the pool always stays at 0 C.”
Note that the function is not injective, i.e., two di↵erent stories may describe the same trajectory.
32
3.1 The framework
TB
TP
Figure 3.1: Illustration of experiment Cool. A bottle of beer is put into a huge pool. We
may tell stories about what could happen in this experiment. Physical theories put
constraints on the set of possible stories.
or
sh =
8“The temperature of the bottle, initially at
<
20 C, continuously increases to 100 C. The
: thermometer of the pool always stays at
0 C.”
or
sm =
8“The temperature of the bottle, initially at 20 C
>
>
>
continuously decreases. This decrease is com>
<
pensated by an equal increase of the temper-
ature of the pool, which is initially at 0 C.
>
>
>
>
: The process stops when the two temperatures
are equal.”
We would naturally interpret these stories by assigning to them the following trajectories.
sCool
= {(TB , TP ) 2 R2 : 0  TB  20; TP = 0}
c
sCool
= {(TB , TP ) 2 R2 : 20  TP  100; TP = 0}
h
sCool
= {(TB , TP ) 2 R2 : 0  TP  TB ; TB + TP = 20} .
m
Let us now look at how a theory T can be viewed as a rule that forbids the stories sh
and sm .
33
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
Theories applied to stories about experiments
An experiment may specify physical quantities that are part of the formalism of a theory
T, providing us with explanations for what is going on. These quantities may not be
directly observable. For example, according to thermodynamics, we could assign internal
energies UB and UP to the bootle and the pool, which are related to their temperatures
via
dUB = CB dTB
and
dUP = CP dTP ,
(3.1)
where CB are the respective heat capacities. The corresponding experiment Cool0 specifies
0
the physical quantities ⇥Cool = {T̂B , T̂P , ÛB , ÛP }. Because it includes ⇥Cool but is more
specific we may call Cool0 a refinement of Cool.
A story about Cool0 that gives an explanation in terms of these concepts may then
read as follows.
s c1 =
8
“As long as the bottle is warmer, its internal en>
>
>
>
< ergy flows into the pool, so that dUP = dUB .
But since CB ⌧ CP , for the pool is large,
>
Eq. (3.1) implies that the temperature of the
>
>
>
: pool stays at 0 C , whereas that of the bottle
decreases to this value.”
In contrast to sc , this story tells us not only that the pool temperature remains unchanged, but also why. There could be alternative explanations, too, as for instance the
following.
s c2 =
8“As long as the bottle is warmer, its internal en>
>
> ergy continuously disappears and, hence, ac>
<
cording to Eq. (3.1), its temperature decreases.
But since no energy is supplied to the pool,
>
>
>
>
: Eq. (3.1) implies that the pool temperature
stays at 0 C.”
The two stories sc1 and sc2 can be regarded as more specific versions of story sc . In
particular, the account of pairs (TB , TP ) that are observed according to them are equal to
the corresponding account sCool
of story sc . However, they provide di↵erent explanations
c
for these observations, which are not necessarily equally plausible.
Whether or not a story makes physical sense ultimately depends on the laws of physics
we invoke. Take, for instance, the standard laws of thermodynamics TD, and let TD be
the set of all stories about the experiments Cool and Cool0 that are incompatible with
34
3.2 Additional examples
them. The first law “forbids” all stories s that talk about internal energies UB and UP
whose sum is not constant, i.e.,
dUB + dUP 6= 0 (according to s)
=)
s 2 TD .
Similarly, the second law of thermodynamics may be written as
dUB
dUP
+
< 0 (according to s)
TB
TP
=)
s 2 TD .
These laws can be directly applied to stories about Cool0 which specify the values of UB
and UP by definition. For example, the first law rules out story sc2 .
In contrast to this stories about the less specific experiment Cool do not even mention
the quantities UB and UP , so that neither of the two laws above applies. But this does not
mean that both stories should be regarded as compatible with thermodynamics. Rather,
we would say that the stories are not specific enough to decide this question.
To extend our statements to such less specific stories, we demand that the set TD
satisfies a consistency property. Note that a story s may be obtained from another story
s0 by dropping some information. If this is the case we say that s0 is at least as specific as
s, and write s0 ! s. For example, we have sc1 ! sc and sc2 ! sc . Consistency now means
that if all stories s0 that are at least as specific as s0 , i.e., s0 ! s, are in the set TD then
also the story s is in TD. In other words, a story is only not forbidden, if it is possible to
add to the story s all necessary details without running into a contradiction with the laws
of thermodynamics.
If we assume that TD satisfies this consistency property then our example story sm is
excluded. Indeed, any story s0m that makes sm more precise by talking about the corresponding internal energies UB and UP must either violate the first law or the relation (3.1).
Similarly, but invoking this time the second law, one can argue that sh is in the set TD,
too. Conversely, sc1 is specific about the values of UB and UP but still in agreement with
the above laws of thermodynamics. The less specific story sc is therefore not excluded,
either. Hence, in summary, we have
sc , sc1 62 TD
and
sh , sm , sc2 2 TD .
Let us illustrate these ideas with two more examples.
3.2
Additional examples
Glide: an experiment described by classical mechanics CM and special
relativity SR
The example experiment, termed Glide, consists of a sledge that can glide on a horizontal
rail (see Fig. 3.2). Apart from gravity in the vertical direction, there shall be no other
35
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
influences, such as friction, acting on the sledge. The experiment is also equipped with
a device that measures and displays the position x of the sledge, as well as a clock that
displays time t , i.e., the set of physical quantities to describe this experiment within our
framework may be chosen as ⇥Glide = {x̂, t̂} with range(x̂) = range(t̂) = R.
X
T
Figure 3.2: Illustration of experiment Glide. A sledge glides without friction on a
horizontal plane. We may tell stories about what can be seen on the displays showing
the position of the sledge and the time.
Analogously to our first stories about Cool, stories about Glide can be rather minimalistic, such as
sr =
(
“The sledge remains at rest. Therefore, the
measured position x equals 0 independently of
the clock time t.”
This story provides us with an account of the displayed values x and t, but does not
contain any additional information. It could therefore equivalently be specified in terms
of the set of observed pairs (x, t),
sGlide
= {(x, t) 2 Z2 : x = 0} .
r
Similarly, we define two more such stories, sf (the story that the sledge moves really fast)
and sj (the story that the sledge jumps), by the sets
sGlide
= {(x, t) 2 Z2 : t = bx 10
f
9
c}
sGlide
= {(x, t) 2 Z2 : x = 0 if t < 0; x = 103 if t
j
0} .
In order to describe these stories within classical mechanics CM, we consider the more
specific experiment Glide0 with the additional physical quantity v = dx/dt referring to the
0
velocity of the sledge, i.e., ⇥Glide = {x̂, t̂, v̂}. The following two examples are manifestly
more specific versions of sr and sf , i.e., s0r ! sr and s0f ! sf .
s0r =
8
“The sledge was put at position x = 0 with ve>
<
locity v = 0. Since no force acts on it, the
>
: velocity remains constant and the sledge rests
at x = 0 for all times t. ”
36
3.2 Additional examples
s0f =
8
9
>
<“The sledge has velocity v = 10 , which remains
constant as no force acts on it. Since it passes
>
: position x = 0 at time t = 0,9 it must pass any
position x at time t = x/10 .”
Which of our stories about Glide are allowed by the laws of physics? Let us first have a
look at classical (non-relativistic) mechanics, denoting the corresponding set of forbidden
stories by CM. Recall that there were no horizontal forces acting on the slide, therefore
Newton’s first law asserts that the object moves with constant velocity, which may be
formulated as the condition
v 6= const (according to s)
=)
s 2 CM.
(3.2)
The two stories s0r and s0f explicitly assert that the velocity is constant and they are therefore certainly not in contradiction to this law. Furthermore, by virtue of our consistency
requirement, the same is true for the less specific stories sr and sf .
The situation is a bit di↵erent for story sj . Note that it does not talk about velocity and
is thus not directly excluded by the above condition, either. However, it seems impossible
to make this story more precise (by adding a claim about the velocity of the sledge)
without running into a contradiction with the classical laws of motion. The story is thus
ruled out by the consistency requirement. We thus conclude that
sr , s0r , sf , s0f 62 CM and
✔
sj 2 CM .
✘
Figure 3.3: Allowed and forbidden stories for Glide within classical mechanics CM. Any
physical law provides a constraint on the set of all conceivable stories that can be told
about an experiment. The figure illustrates allowed and forbidden stories for the
experiment Glide: Forbidden are all other stories in which the velocity of the sledge is
not constant. For example stories where the sledge suddenly jumps or where it is at
more than one position at the same time.
In a similar way, we may study the compatibility of our stories with other theories,
such as special relativity SR. The corresponding set of forbidden stories, SR, is subject to
additional constraints. In particular, it also contains sf , for this story talks about a sledge
that moves faster than light sf 2 SR. Conversely, there does not seem to be any story
about the experiment Glide that is compatible with special relativity theory but forbidden
37
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
by classical mechanics.
These facts could be summarised by the relation
Glide
CM
⇢ SR
Glide
for all stories about Glide,
where ⇢ means strict inclusion. If it would hold for all experiments we would call special
relativity an extension of classical mechanics (see Chapter 4). However, one can easily
think about stories that are forbidden by classical mechanics and allowed by special relativity. For instance, for experiments involving observers with clocks moving relative to
each other classical mechanics would forbid stories saying that their respective times are
di↵erent. Obviously, such stories would not be forbidden within special relativity given
that they are described correctly by the relevant laws.
So far we only considered experiments described within classical theories. In this case
the interpretation of the events in a story is a one-to-one correspondence – each event
is a real point in space-time. In the next section we have a look at a typical quantum
experiment and we illustrate how di↵erent interpretations of quantum theory correspond
to di↵erent rules to decide which stories are forbidden.
PBS: an experiment described by quantum theory QM
The experiment PBS (see Fig. 3.4) consists of a single-photon source emitting linearly polarised light. The light is transmitted to a polarising beam splitter, which directs it to two
di↵erent detectors, depending on whether the polarisation is horizontal or vertical. The
detectors shall be equipped with displays that show the respective number of incoming
photons ch and cv respectively.
Cv
1
Ch
0
Figure 3.4: Illustration of experiment PBS. Linearly polarised photons pass through a
beam splitter, directing them to two detectors. Stories may talk about which detector
clicks.
In addition to the number of photons, which are directly observable, the experiment
PBS also has the quantum state among its physical quantities such that ⇥PBS = {ĉv , ĉh , b}.
38
3.2 Additional examples
The specification range of the photon counters is given by the set of positive integers such
that range(ĉv ) = range(ĉh ) = N0 . According to quantum theory QM the quantum state is
represented by a normalised vector of a Hilbert space H with orthonormal basis states
|hi and |vi. Therefore, the event space of PBS is given by
[PBS] = (ch , cv , ) 2 N20 ⇥ H .
Furthermore, we can model the joint action of the beam splitter and the detectors as a
measurement with respect to projectors ⇡h = |hihh| and ⇡v = |vihv|, corresponding to
outcomes h and v, respectively. Stories about PBS could then read as follows.
shh =
(
“Because the source emits a photon in state |hi,
the measurement gives outcome h, increasing
ch by one.”
svh =
(
“Although the source emits a photon in state |vi,
the measurement gives outcome h, increasing
ch by one.”
The corresponding accounts of events are1
2
sPBS
hh = {(ch , cv , ) 2 N0 ⇥ H : ch  1, cv = 0,
= |hi}
2
sPBS
vh = {(ch , cv , ) 2 N0 ⇥ H : ch  1, cv = 0,
= |vi} .
The observations that are made according to these two stories are the same. However, the
explanations they provide, which refer to the polarisation state , di↵er. While the first
sounds plausible, the second is in contradiction to standard quantum theory, i.e., we have
shh 62 QM
and
svh 2 QM .
Things become a bit more interesting if we consider stories that talk about a photon that
p
has diagonal polarisation |di = 1/ 2(|hi + |vi), e.g.,
sdh =
(
“Although the source emits a photon in state
|di, the measurement only gives outcome h,
increasing ch by one.”
sdv =
(
“Although the source emits a photon in state
|di, the measurement only gives outcome v,
increasing cv by one.”
1
Note that because the experiment PBS does not have time among its physical quantities the photon
counters may also are both equal to zero.
39
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
or, getting more adventurous,
sdk =
8
“Because the source emits a photon in state |di,
>
>
>
< the ‘universe’ splits in two: in one the mea-
surement gives outcome h, increasing c by
h
>
>
>
: one, and in the other it gives outcome v, in-
creasing cv by one.”
The respective accounts of events defined by these stories are
2
sPBS
dh = {(ch , cv , ) 2 N0 ⇥ H : ch  1, cv = 0,
= |di}
2
sPBS
dv = {(ch , cv , ) 2 N0 ⇥ H : ch = 0, cv  1,
= |di}
2
sPBS
dk = {(ch , cv , ) 2 N0 ⇥ H : ch + cv  1,
= |di} .
Whether or not these stories can be considered compatible with quantum mechanics
is a bit more controversial. For example, according to a many-worlds interpretation, only
sdk would be allowed. Conversely, a protagonist of the Copenhagen interpretation may
argue that each of sdh and sdv is a possible course of events, but not sdk .
We may therefore view single-world SWI and many-world MWI interpretations as different theories such that
sdk 2 SWI
and
sdv , sdh 2 MWI.
A pragmatic approach is to interpret quantum theory QM to be compatible with both
interpretations, i.e., none of the stories from above is explicitly forbidden
sdk , sdv , sdh 62 QM.
As we can see it holds that
SWI, MWI ⇢ QM ,
i.e., the set of forbidden states is strictly larger within both interpretations. In that sense
single-world as well as multiverse interpretations are extensions of quantum theory.
We can also see that by construction there cannot be a story about PBS that is
allowed by both single-world and multiverse interpretations. Therefore, we say that they
are incompatible. We will discuss these ideas, as well as other relations between theories,
on a more formal level in the next chapter.
40
3.3 Remarks about the interpretation of the trajectories
Quantum stories about repeated experiments
A story about the experiment PBS may also involve more than one single photon, as for
instance
(
“Because the source continuously emits photons
in state |hi, the measurement always gives
outcome h, steadily increasing ch .”
shhn =
sdhn =
(
“Although the source continuously emits photons in state |di, the measurement always
gives outcome h, steadily increasing ch .”
Both stories talk about the same observations, but as in our previous examples, they
provide di↵erent explanations. Note that the accounts of events of shhn and sdhn are
similar to those of shh and sdh , respectively, except that the condition ch  1 is dropped.
But despite this similarity, the conclusions we draw about their validity within quantum
mechanics come out di↵erently. While the first is obviously allowed by quantum mechanics,
this does not seem to be the case for the second. If the Born rule tells us anything, then
certainly that we will at some point see a measurement outcome corresponding to ⇡v ,
provided we wait long enough. Indeed, as we shall see in Chapter 6, the objective Born
rule, (BornObj), forbids story sdhn . We thus have
shhn 2
/ QM
3.3
and
sdhn 2 QM .
Remarks about the interpretation of the trajectories
As explained above we require that stories s about an experiment Exp are mapped uniquely
to trajectories sExp , otherwise the framework cannot be applied (for a given story and a
given experiment). Thus, stories referring to di↵erent possibilities, such as
s = “The outcome will be either 0 or 1.”
are not meaningful within the framework. Rather, we consider such a story to correspond
to two di↵erent stories
s0 = “The outcome will be 0.”
and
s1 = “The outcome will be 1.”
We stress that this does not imply that the framework is limited to deterministic theories.
What is generally understood as a probabilistic theory would simply not forbid any of the
two stories, whereas a deterministic theory would forbid exactly one of them.
In that sense the events of a trajectories are considered to be real within that story. In
particular we have seen above that a multiverse interpretation would forbid both stories
s0 and s1 , whereas the story
s01 = “The outcome will be 0 and 1.”
41
3. STORIES ABOUT EXPERIMENTS – A NON-PROBABILISTIC
FRAMEWORK
is not excluded.
We remark that our interpretation of reality of physical quantities is di↵erent from
Einstein’s interpretation, stating that a physical quantity is real if its value can be predicted
with certainty [50]. Rather, we consider an event to be real if it is in principle observable.
However, for the derivations in the following chapters it will not be necessary to specify
what reality means.
42
Chapter 4
General relations between theories
In this chapter we will discuss certain notions that relate di↵erent theories. Examples for
such notions include the ideas of extensions and compatible or equivalent theories. While
such expressions are frequently used in scientific contexts it is generally difficult to express
these notions formally within existing formalisms. Our framework allows to formalise
these ideas naturally.
4.1
Extensions
“Every “good” scientific theory is a prohibition: it forbids certain things to
happen. The more a theory forbids, the better it is.”
Karl Poper [5]
In the previous section we have seen two examples of extensions of quantum theory –
single-world and many-world interpretations. In general we call T0 an extension of a
theory T if it is more restrictive.
Definition 1. A theory T0 is said to be an extension of a theory T if the set of
forbidden stories is strictly larger
T ⇢ T0 .
(4.1)
Example
In Chapter 2 we gave a short review on Bohmian mechanics as an example for a hidden
variable theory HV. Let us consider again the experiment PBS described in the previous
chapter. A photon in a quantum state was directed towards a polarising beam splitter
with two detectors at the end of each arm. The detector labelled ch counted the horizontally polarised photons and the detector labelled cv counted the vertically polarised
43
4. GENERAL RELATIONS BETWEEN THEORIES
⌃
T0
T
Figure 4.1: Illustration of an extension. A theory T0 is said to be an extension of a
theory T if the set of forbidden stories is strictly larger.
photons.
Let us now consider a more specific experiment PBS0 that additionally has a hidden
variable , with range( ˆ ) = {h, v}, among its physical quantities. We assume that the
hidden variable theory has the following rules
= v ^ ch = 1
(according to s)
=)
s 2 HV
= h ^ cv = 1
(according to s)
=)
s 2 HV .
and
(4.2)
(4.3)
Note that quantum theory may also describe the experiment PBS0 by simply not taking
into account the actual value of .
We may tell the following stories about PBS0 .
(
“Because the source emits a photon in state |di and the hidden
variable is equal to h, the measurement gives outcome h, increasing ch by one.”
sdhh =
sdhv =
(
“Because the source emits a photon in state |di and the hidden
variable is equal to v, the measurement gives outcome h, increasing ch by one.”
The corresponding accounts of events are
0
2
sPBS
dhh = {(ch , cv , , ) 2 N0 ⇥ H ⇥ {h, v} : ch  1, cv = 0,
0
2
sPBS
dhv = {(ch , cv , , ) 2 N0 ⇥ H ⇥ {h, v} : ch  1, cv = 0,
= |di,
= h}
= |di,
= v} .
From the rule Eq. (4.2) it follows that the second story is forbidden by the laws of the
hidden variable theory, i.e.,
sdhh 62 HV
and
44
sdhv 2 HV .
4.2 Equivalence
If we describe both stories within quantum theory QT we simply would not take into
account the value of and hence both stories are actually identical and obviously not
forbidden within quantum theory, i.e.,
sdhh , sdhv 62 QT.
More generally we see that because the law (4.2) is formulated in terms of a parameter
that is not part of the quantum formalism, the set of forbidden stories must be larger
within the hidden variable theory than within quantum theory
QT ⇢ HV.
This is exactly the requirement for an extension given in (4.1), i.e., the hidden variable
theory HV is an extension of quantum theory QT.
4.2
Equivalence
If two theories forbid the same stories about an experiment we say that they are equivalent
for that experiment.
Definition 2. A theory T0 is said to be equivalent to a theory T for an experiment
Exp if the set of forbidden stories is identical for all stories s 2 ⌃ about Exp, i.e., the
following requirement holds
s 2 T () s 2 T0
for all stories s 2 ⌃ about Exp.
(4.4)
⌃Exp ⇢ ⌃
T0 = T
Figure 4.2: Illustration of two equivalent theories. A theory T0 is said to be equivalent to
a theory T for an experiment Exp if the set of forbidden stories is identical for all stories
s 2 ⌃Exp about Exp. Here ⌃Exp denotes the set of stories about Exp.
If two theories are equivalent for all possible experiments we call them isomorphic.
For example Bohmian mechanics is equivalent to quantum theory for all experiments
that do not include the hidden variable, e.g., for the experiment PBS. However, the two
theories are not isomorphic, because there is an experiment, for instance the experiment
45
4. GENERAL RELATIONS BETWEEN THEORIES
PBS0 , for which the set of forbidden stories is di↵erent within the two theories.
Note that the di↵erent interpretations of quantum theory are not isomorphic according
to the definition from above. This can be seen from our discussion from the previous
chapter. Within multiverse interpretations stories are forbidden if they say that only one
of many possible outcomes happens whereas within single-world interpretations stories are
forbidden if they say that more than one of the possible outcomes happen. Thus, the set
of forbidden stories is di↵erent for the two kinds of interpretations.
4.3
Compatibility
Two theories T and T0 are called compatible for an experiment Exp if they do not make
contradicting statements, i.e., if there is at least one story about that experiment that is
not forbidden by both theories.
Definition 3. Two theories T and T0 are compatible for an experiment Exp if there
is a story s 2 ⌃ about Exp such that
s 62 T and s 62 T0 .
(4.5)
⌃Exp ⇢ ⌃
T0
T
Figure 4.3: Illustration of two incompatible theories. Two theories T and T0 are called
incompatible for an experiment Exp if there is no story s 2 ⌃Exp that is not forbidden by
both theories. Here ⌃Exp denotes the set of stories about Exp.
For example Bohmian mechanics is compatible with quantum theory for the experiment PBS but also for PBS0 because the story sdhh is not forbidden by both theories.
Note that if two theories are incompatible for an experiment then (at least) one of
them must be wrong. However, it might be impossible to experimentally test for which of
the two theories this is the case. In this case one of the theories has to be refuted by other
criteria than experimental tests, i.e., by arguing that one of the theories is better than
the other one. One possibility for such an argument is that one of the theories satisfies
certain other properties that we might find desirable. In the next chapter we will discuss
some examples for such properties.
46
Chapter 5
Natural properties of theories
Equipped with a framework that allows us to capture the content of physical laws, we
can now turn to a precise formulation of the properties we might want a good theory to
have. Here we will limit our discussion to the properties that will be used later to derive
our main results. However, these examples also illustrate how other assumptions may be
formulated in general within the framework.
The first assumption, termed (BornDet), corresponds to a non-probabilistic statement
within the quantum formalism. Informally it says that outcomes corresponding to projectors that are orthogonal to the measured state are forbidden. The second assumption,
(Robust), is a general property of any theory requiring that the statements of the theory
remain valid under small perturbations within the formal description. In Chapter 6 we
will derive an objective interpretation of the Born rule from (BornDet) and (Robust).
The remaining assumptions will be applied in Chapter 8 where we show that no singleworld interpretation of quantum theory can satisfy certain consistency properties. We will
formalise the assumption that measurements only have one outcome as a condition termed
(SW). Furthermore, we will introduce two properties that represent certain requirements
on the interpretation of stories. First, we discuss the compatibility assumption (C) which
demands that stories talking about di↵erent experiments sharing certain quantities are
non-contradicting. Finally, the self-consistency assumption (SC) says that there is at least
one story talking about an experiment that is allowed.
Let us now turn to the precise formalisation of these assumptions within our framework.
5.1
Compatibility with deterministic quantum theory (BornDet)
It is hardly ever doubted that quantum theory is our best theory of nature. In Chapter 1 we motivated the point of view that theories are identified with explanations and we
discussed why a purely mathematical formalism remains an empty statement without an
interpretation explaining in which sense the formalism represents reality. Without know-
47
5. NATURAL PROPERTIES OF THEORIES
ing what exactly reality is, it seems to make sense to assume that it is objective in the
sense that not every observer has his or her own reality. Therefore, if quantum theory is
supposed to describe, or better to explain, this reality, its formalism must have an objective interpretation.1
The requirement that quantum theory can be interpreted objectively challenges its
“standard” probabilistic interpretation. We discussed in Chapter 1 that it seems to be
impossible to objectively define probabilities operationally without running into inconsistencies. Therefore, if we want to view quantum theory as a “law of nature”, it seems to be
rather unsatisfactory to base its interpretation on probabilities. Unfortunately, this seems
to be exactly the case, because the only axiom of quantum theory that has an operational
interpretation is the probabilistic one.
These observations motivate our approach to give up the probabilistic axiom while
keeping the non-probabilistic postulates – after all they have been very successful so far.
More precisely, given a physical system, we assume that we can associate to it a Hilbert
space H so that any normalised vector
2 H corresponds
P to a state of the system.
Furthermore, any family {⇡x }x2X of projectors such that x2X ⇡x = idH represents a
measurement on the system, with possible outcomes x 2 X .
In the remaining part of this section we give this formalism a minimal operational
interpretation which is not based on probabilities. Informally we require that an outcome
is forbidden if the corresponding measurement is described by a projector and a state that
are orthogonal. We will call this assumption (BornDet).
The idea is that the requirement (BornDet) can be applied not only to quantum theory but in general to any theory that allows to describe systems with the Hilbert space
formalism. For example Bohmian mechanics (see Chapter 2) is based on the Hilbert space
formalism but also has additional parameters.
In order to formalise the assumption (BornDet) within our framework we introduce
a set of experiments, termed QH,{⇡x }x2X . Any experiment Meas 2 QH,{⇡x }x2X simply
consists of a measurement described within quantum theory by a state
and a set of
projectors {⇡x }x2X (see Fig. 5.1). Note that two di↵erent experiments in this set may
correspond to di↵erent systems that are measured for example a photon or an electron.2
For simplicity, we assume that H has a finite dimension d, that X = {0, . . . , d 1}, and
that ⇡x = |xihx|, where {|xi}x=0,...,d 1 is a fixed orthonormal basis of H. Thus, Meas may
be described by the physical quantities ⇥Meas = {x̂, b} with range(x̂) = {0, . . . , d 1, ?},
where ? denotes the event that no outcome was observed, and range( b) = H. The event
1
It might be given a subjective interpretation in addition.
The fact that both of these experiments can be described by the following mathematical formalism
may be viewed as an interpretation.
2
48
5.1 Compatibility with deterministic quantum theory (BornDet)
space of any experiment Meas 2 QH,{⇡x }x2X is thus simply given by the product space of
the individual specification ranges
[Meas] = (x, ) 2 (X [ ?) ⇥ H .
Note that we may also include the projectors {⇡x }x2X among the physical quantities
but not necessarily have to. The idea of the framework is that any choice of physical
quantities may be used to describe an experiment, i.e., a particular setup does not force
us to make a specific choice. In this case the measurement projectors are fixed and hence,
we chose to consider them as part of the description of the experiment instead of including
them among the physical quantities.
| i
{⇡x}
X
Figure 5.1: Illustration of a generic experiment Meas. This is a generic experiment,
needed for the description of our non-probabilistic assumption (BornDet): If a system in
a quantum state is measured w.r.t a set of measurement projectors {⇡x }x2X an
outcome x is forbidden if ⇡x = 0.
Stories about Meas may talk about the state
(
s00 =
, as for instance
“The system is initially in state |0i. The subsequent measurement therefore gives outcome
0.”
The corresponding trajectory of this story is
sMeas
= {(x, ) : x = ? or x = 0,
00
= |0i} .
Consider now a theory T that is applicable to all experiments Meas 2 QH,{⇡x }x2X , i.e.,
the physical quantities ⇥Meas are part of the formalism of T. The axiom (BornDet) can
now be phrased as a condition on theories of this type.
49
5. NATURAL PROPERTIES OF THEORIES
Definition 4. When we say that a physical theory T satisfies (BornDet) then this
means that T forbids all stories s according to which for some Meas 2 QH,{⇡x }x2X for
some x̄ 2 X the prepared state is orthogonal to ⇡x̄ , i.e., ⇡x̄ = 0, yet x̄ occurs, i.e.,
( , x̄) 2 sMeas .
In (other) words, the theory T must forbid all stories that explain an observation x in
terms of a state that has no overlap with the projector ⇡x .
5.2
Robustness under small perturbations (Robust)
In statistics the idea of robustness is a well established concept (see for instance [51]). For
models derived from statistical data it is important to understand if they remain valid,
or robust, under small perturbations of the assumptions and the data. This idea was
formalised by the notion of open sets [52]. In general a property is said to be robust, if
all subsets with that property are open. For example the set of real numbers with the
property “being a rational number” is not robust with respect to the topology induced by
the real numbers (the Euclidian distance).
Although the idea of robustness has rarely been discussed in physics it seems to be at
least as important in this case. In general we would require from a good theory that it
can be applied in practice. In order to be applicable in actual experiments it is necessary
that the statements of the theory remain valid under small perturbations of the formal
description, i.e., that the theory is robust. Here we will formalise this idea within the
framework.
As discussed in Chapter 3 the idea of the framework is to view a theory T as a set
of rules that forbid certain stories s. Thus, the theory can be identified with the set of
forbidden stories T. Applying the ideas from statistics we define a theory to be robust if
the following condition holds.
Definition 5. When we say that a physical theory T satisfies (Robust) then this means
that the set of forbidden stories T is closed.
This condition does not yet specify the relevant topology, but instead the idea is that it is
induced by the topologies of the physical quantities of an experiment. We will formalise
this idea below.
The requirement ensures that the set of stories that is not explicitly forbidden within
a theory is open. Therefore, it can equivalently be expressed as the requirement that for
50
5.2 Robustness under small perturbations (Robust)
Noise?
X
stories about an experiment
subset of forbidden stories
S
Figure 5.2: Robustness. In general a property of a set is understood to be robust if all
subsets with that property are open (with respect to the topology induced by the
property). Here we are interested in the robustness of theories applied to stories about
experiments. Intuitively this is a necessary criterion for any theory to be applicable in
practice. The requirement is that the subset of stories that are not forbidden is open.
any story s that is not forbidden, s 62 T, there exists a neighbourhood N (s) such that all
stories s0 2 N (s) are also not forbidden by T
s 62 T
=)
9 N (s) : s0 62 T
8 s0 2 N (s).
Topology on the set of stories
The notions of open sets and neighbourhoods refer to the topology on the set of stories
s. We define two stories s, s0 to be "-close for an experiment Exp if their trajectories are
"-close
"
"
s ⇡ s0
()
sExp ⇡ s0Exp .
The distance between trajectories is induced by the topology of the specification ranges
specified by the physical quantities x̂ 2 ⇥Exp . We use the Hausdor↵ distance to quantify
the closeness of trajectories, i.e.,
"
sExp ⇡ s0Exp
8 x 2 sExp
"
9 x0 2 s0Exp : x ⇡ x0
()
^
8 x0 2 s0Exp
"
9 x 2 sExp : x ⇡ x0 .
Note that the events x, x0 are tuples of the form x = (x1 , . . . , xk ) with k = |⇥Exp |. The
topology of two events x, x0 is induced by the topologies of the physical quantities
"
x ⇡ x0
()
"
xi ⇡ x0i
8 i = 1, . . . , k.
Let us now look at the topology on the set of stories about experiments of the type
Meas.
51
5. NATURAL PROPERTIES OF THEORIES
Topology on the set of stories about Meas 2 QH,{⇡x }x2X
We will apply the robustness property to the experiments Meas 2 QH,{⇡x }x2X with the
physical quantities including the state and the outcomes x 2 X . We would like to stress
that this is only one possible choice to describe this experiment within the framework. For
our purposes it is a convenient choice because we formulated (BornDet) such that it can
be applied to experiments with these physical quantities. However, one may also choose
di↵erent physical quantities. The general formulation of the robustness requirement allows
its application to any choice of the physical quantities.
Because we choose the measurement projectors to be part of the description of the
experiment rather than including them among the physical quantities, one may worry
that the robustness requirement should also be applied to them. However, formally we
may view a perturbation of a measurement projector as a perturbation of the state and
therefore, it suffices apply the criterion to either the state or the projectors.
Stories s about an experiment Meas are mapped to trajectories sMeas whose accounts
of events are tuples (x, ) 2 sMeas . Here x comes from the discrete set X [ {?} (with the
trivial topology) and from the Hilbert space H (whose norm defines a topology). Hence,
the closeness of trajectories is defined via the relation
"
sMeas ⇡ s0Meas
()
9 (x0 ,
8 (x, ) 2 sMeas
0)
^
8 (x0 ,
where
0)
2 s0Meas
"
(x, ) ⇡ (x0 ,
0
"
0)
"
0)
2 s0Meas : (x, ) ⇡ (x0 ,
9 (x, ) 2 sMeas : (x, ) ⇡ (x0 ,
()
)
"
x ⇡ x0
^
"
⇡
0
.
The trivial topology on X implies that the outcomes x have to be identical
"
x ⇡ x0
x = x0 .
()
The relevant topology on the set of quantum states is induced by the inner product of
the Hilbert space H
p
k k= h | i
and therefore
0
d( ,
Two states
,
0
) := k
p
= 2(1
0
k
<h | 0 i).
are "-close if the following condition holds
⇡"
0
()
52
d( ,
0
)  ".
5.2 Robustness under small perturbations (Robust)
Summarising, for an experiment of the type Meas two stories s, s0 are "-close if the following
relation is satisfied
"
sMeas ⇡ s0Meas
8 (x, ) 2 sMeas 9 (x,
8 (x,
0)
()
0)
2 s0Meas : d( ,
^
2 s0Meas 9 (x, ) 2 sMeas : d( ,
0)
0)
<"
<".
One may wonder whether our existing theories are robust or not. Let us already stress
at this point that for classical mechanics as well as for quantum theory this is not the
case: for both theories there are experiments such that robustness implies that all stories
about this experiment are forbidden. For classical mechanics it is quite straightforward
to construct such an experiment and we will give an example below. In the case of
quantum theory it is less obvious. In particular, for the experiment Meas the robustness
requirement is not a problem. However, as we will discuss in the next chapter, it will
become problematic when we turn to repeated measurements. But let us first verify that
the robustness requirement is satisfied for Meas.
Quantum theory is robust for the experiments Meas 2 QH,{⇡x }x2X
In order to show that quantum theory is robust for any Meas 2 QH,{⇡x }x2X we have to
make sure that the set of forbidden stories is closed. Equivalently we can show that its
complement is open which is expressed as the following condition.
Lemma 1. QT identified with (BornDet) is robust for all Meas 2 QH,{⇡x }x2X , i.e., for
all stories s about Meas the following condition holds
s 62 QT
=)
"
9 " > 0 : s0 62 QT, 8 s0 ⇡ s.
Proof. In order to see this take any story s about Meas that is not forbidden by (BornDet)
s 62 QT
=)
> 0 8 (x, ) 2 sMeas .
⇡x
Now we observe that we can write
k k2 = h | i
= h |⇡x | i + h |(id
h |⇡x | i
=)
d( ,
0 2
)
h |⇡x | i + h 0 |⇡x | 0 i
53
⇡x )| i
h |⇡x | 0 i
h 0 |⇡x | i.
(5.1)
5. NATURAL PROPERTIES OF THEORIES
Assume now that ⇡x
(5.1) that
0
= 0 holds for all " > 0 and for all
"2
h |⇡x | i
"
0 ⇡
. Then it follows from
8">0
holds. It is obvious that this relation cannot hold for all " > 0, for example by choosing
"2 = h |⇡x | i/2.
Classical mechanics is not robust for the experiment Glide0
Recall the experiment Glide0 introduced in Chapter 2 described by classical mechanics CM.
We identified CM with the rule1
v 6= const
according to s
s 2 CM .
=)
(5.2)
We will now conclude this section by showing that CM is not robust for the experiment
Glide0 .
Lemma 2. CM identified with the rule (5.2) is not robust for Glide0 , i.e., for all
stories s about Glide0 the following condition holds
s 62 CM
=)
"
8 " > 0 : 9 s0 2 CM, 8 s0 ⇡ s.
The lemma tells us that for all stories s that might be allowed there is an arbitrary close
story s0 which is forbidden. Because the set of forbidden stories is closed it follows that
the story s must be forbidden, too. Therefore, all stories about Glide0 are forbidden.
Intuitively this is quite obvious because for a story with constant velocity v which is not
forbidden by the law (5.2) it is straightforward to construct a story saying that the velocity
varies by an arbitrary amount " from v. Because the velocity within that new story is not
constant it follows that the story is forbidden and because " was arbitrary it follows from
the requirement (Robust) that the original story must be forbidden, too. A formal proof
is given below. Figure 5.3 illustrates the idea.
0
Proof. Recall that the experiment Glide0 is specified by the physical quantities ⇥Glide = {x̂, t̂, v̂},
where x and t are the position and the time and v is the velocity. Let now s be a story
that is not forbidden by CM, i.e.,
v = v0
0
8 (x, t, v) 2 sGlide .
1
Because we only needed this example to illustrate the framework we have not made it formally explicit.
In order to be precise one would have to specify a set of experiments of the type Glide0 for which the rule
holds.
54
5.2 Robustness under small perturbations (Robust)
⇠
-neighbourhood
forbidden trajectory
trajectory with constant
velocity
⌧
Figure 5.3: Classical mechanics cannot satisfy assumption (Robust) for Glide0 : For all
allowed trajectories (with constant velocity) we can find an arbitrary close trajectory
that is forbidden (with varying velocity).
It it easy to construct for all " > 0 a story s0 that is forbidden as follows. Pick any t0 such
0
that (x, t0 , v) 2 sGlide and define
s
0Glide
0
:=
(
0
{(x, t, v0 + ") : (x, t, v) 2 sGlide } for  t0
{(x, t, v0
0
") : (x, t, v) 2 sGlide } for t > t0 .
Because in this case the natural topology is simply the Euclidian distance the two stories
are indeed "-close. However, s0 2 CM because its velocity is not constant.
Let us briefly discuss a possible solution to that problem because in Chapter 6 when we
turn to quantum theory describing repeated runs of an experiment we will find that quantum theory is not robust either for that experiment. The idea is to consider experiments
with limited precision. This is also relevant from a practical point of view because in an
actual experiment we will never be able to measure the velocity arbitrarily precisely but
always only with a limited precision µ that is determined by the setup of the experiment.
µ
The idea is to consider the experiment Glideµ with the physical quantities ⇥Glide = {x̂, t̂, v̂ µ },
where v̂ µ is the velocity with precision µ. Now the idea of the proof above does not work
anymore for " < µ, because in this case the trajectories corresponding to the two stories
are identical.
In the remaining sections of this chapter we will consider assumptions that are about
experiments that are described by protocols. These include a list of instructions to be
carried out at certain times t – which may be simply viewed as the value read from a clock.
The assumptions represent properties about nature that are related to our intuition about
55
5. NATURAL PROPERTIES OF THEORIES
this time – for instance that there exists only one value of a measurement outcome at
each instance in time. Therefore, we will include this time parameter among the physical
quantities of these experiments.
5.3
Compliance with quantum theory (QT)
Property (QT) corresponds to the assumption that the laws of standard quantum theory
are valid. Like in the case of the assumption (BornDet), it suffices for our purposes to
restrict to some particular rules that are implied by standard quantum theory, i.e., we
do not need to provide a full characterisation of the theory. Assumption (QT) is slightly
stronger than (BornDet) which only asserted that outcomes corresponding to projectors
that are orthogonal to the state are forbidden. In addition assumption (QT) requires that
if the probability P (x) = |hx| i|2 of an outcome x is positive then the outcome will occur
at some point if the measurement is repeated until this is the case. In fact, (QT) is a
consequence from the objective Born rule (BornObj) which we will derive in Chapter 6
from the assumptions (BornDet) and (Robust). Therefore, we could also assume (BornDet)
and (Robust) instead of (QT). However, because the requirement is weaker we state it as
a separate assumption here.
To specify the rule (QT), we consider a set of experiments termed QH,{⇡xH }x2X which
consists of repeated measurements of a quantum system according to the following protocol.
Protocol of Repeated Quantum Measurement Experiments, QH,{⇡xH }x2X
For a fixed quantum system with Hilbert space H, a fixed unitary U on H, a fixed
family of measurement projectors {⇡x }x2X on H, and a fixed value x̄ 2 X , repeat the
following steps for increasing n 2 N at most until the halting criterion is satisfied.
@ n:00
n:00 – n:01
@ n:01
@ n:02
Prepare a quantum system in state .
Let the system evolve according to U .
Measure the system w.r.t. {⇡x }x2X and record the outcome x.
Halt if x = x̄.1
Note that here and in the following, times indicated in an experiment should be understood as placeholders that could be replaced by any other points in time with the same
order.
1
The halting criterion is not strictly needed. However, provided that ⇡x̄ 6= 0 and that (QT) is
satisfied, it guarantees that the experiment will end after finite time. This avoids the need for stories that
talk about an infinite sequence of measurements.
56
5.3 Compliance with quantum theory (QT)
To simplify the presentation, we introduce the measurement projectors {⇡xH }x2X in
the Heisenberg picture, which are defined as
⇡xH = U † ⇡x U .
(5.3)
Standard quantum theory then implies that the following two rules must hold for any fixed
x̄ 2 X
(a) If ⇡x̄H
= 0 then the outcome x = x̄ does not occur.
(b) If ⇡x̄H 6= 0 then the outcome x = x̄ occurs at some point, provided we keep repeating
the experiment with the same state.
The definition of (QT) below captures the idea that any theory T that complies with
quantum theory must satisfy these rules.
On the formal level, the set QH,{⇡xH } may be chosen arbitrarily, provided that the
following requirements are met.
• Any Exp 2 QH,{⇡xH }x2X has the physical quantities ⇥Exp = {t̂, b, x̂}.
• Any Exp 2 QH,{⇡xH }x2X has an event space of the form
[Exp] ✓ (t, , x) 2 R ⇥ H ⇥ (X [ ?) .
• The set QH,{⇡xH }x2X includes all relevant parts of the Extended Wigner’s Friend
Experiment defined in Chapter 8, as specified by (8.3), (8.6), (8.8), and (8.10).
The last requirement ensures that we may describe measurements on observers within
quantum theory.
Definition 6. When we say that a physical theory T satisfies (QT) then this means
that T forbids all stories s according to which in some experiment Exp 2 QH,{⇡xH }x2X
and for some x̄ 2 X one of the following happens.
( a) For some n 2 N the prepared state is orthogonal to ⇡x̄H , i.e., (n:00, , ⇤) 2
sExp =) ⇡x̄H = 0. Yet x̄ occurs, i.e., 9 t 2 [n:01, n:02) : (t, ⇤, x̄) 2 sExp .
( b) The system is always prepared in the same fixed state 0 non-orthogonal to ⇡x̄H ,
i.e., ⇡x̄H 0 6= 0 and (n:00, , ⇤) 2 sExp ()
= 0 holds for all n 2 N. Yet x̄
never occurs, i.e., 9 t 2 [n:01, n:02) : (t, ⇤, x̄) 2 sExp holds for no n 2 N.
57
5. NATURAL PROPERTIES OF THEORIES
5.4
Compatibility of di↵erent point of views (C)
Two experiments, Exp and Exp0 , may correspond two di↵erent views on the same experimental setup and therefore describe it in two di↵erent ways. Consequently, their
interpretation of a given story s in terms of its trajectory will in general also be di↵erent.
Compatibility (C) requires that the two trajectories are still compatible, in a sense that
we are going to describe now. Intuitively, the requirement demands that outcomes that
are shared by both points of view must be identical.
Let us first consider a simple situation where an experiment Exp is a part of a larger
experiment Exp0 . An example of such a larger experiment could be an extension of an
experiment Exp 2 QH,{⇡xH }x2X described above with a second measurement with outcome
x0 carried out at t = 0:02. Therefore, Exp corresponds to simply ignoring the second
measurement of Exp0 .
It would be natural to demand that any event (t, , x, x0 ) that occurs according to a
story s in Exp0 must correspond to an event (t, , x) that occurs in the interpretation of
Exp, and vice versa. This corresponds to the compatibility constraint
(t, , x) 2 sExp
()
0
9 x0 : (t, , x, x0 ) 2 sExp ,
0
which should hold for any story s for which sExp is defined. To simplify the notation, we
will in the following abbreviate terms of the form 9 y : (x, y) by (x, ⇤). The compatibility
constraint can then be rewritten as
(t, , x) 2 sExp
()
0
(t, , x, ⇤) 2 sExp .
(5.4)
More generally, two experiments, Exp1 and Exp2 , may just have an overlap, but not be
contained in each other. Suppose for example that both of them, apart from time t, include
the same quantity x, but in addition also separate quantities, x1 and x2 , respectively. The
events in their trajectories thus consists of triples (t, x, x1 ) and (t, x, x2 ), respectively. The
compatibility constraint that models that the time t as well as the quantity x should be
the same in both experiments could then be formulated as the condition that
(t, x, ⇤) 2 sExp1
()
(t, x, ⇤) 2 sExp2 ,
holds for any story s for which both sExp1 and sExp2 are defined.
Formally, the requirement (C) is expressed for any two experiments with a time t
among their physical quantitates as the condition that the physical quantities they share
must be compatible.
58
5.5 Single-world (SW)
Definition 7. When we say that a physical theory T satisfies (C) then this means that
for any two experiments Exp1 and Exp2 with a time t among their physical quantities
the following condition holds for all shared physical quantities x̂ 2 ⇥Exp1 \ ⇥Exp2
(t, ⇤, . . . , ⇤, x, ⇤, . . . , ⇤) 2 sExp1
(t, ⇤, . . . , ⇤, x, ⇤, . . . , ⇤) 2 sExp2
()
(5.5)
for all s about Exp1 and Exp2 that are not forbidden by T.
5.5
Single-world (SW)
Property (SW) captures the idea that in any experiment of the following type, the value
x admits only one single value. To specify the rule (SW), we consider a set of experiments
termed OX which consists of repeated measurements according to the following protocol.
Protocol of Basic Observation Experiments, OX
For a fixed set X , perform the following steps at the corresponding times t.
@ 0:00
@ 0:01
@ 0:02
Start
Record the outcome z of a measurement with range X .
Halt.
In other words, a theory T that fulfils (SW) must forbid any story about this experiment
according to which multiple outcomes x occur. We remark that this is fundamentally
di↵erent from the requirement that T be deterministic. For example, T may prescribe
that the outcome of a spin measurement is either x = 12 or x = + 12 (not both), but
still assert that this outcome is not correlated to anything that can be known before the
measurement is carried out.
On the formal level, the set OX may be chosen arbitrarily, provided that the following
requirements are met.
• Any Exp 2 OX has the physical quantities ⇥Exp = {t̂, x̂}.
• Any Exp 2 OX has an event space of the form
[Exp] = (t, x) 2 R ⇥ (X [ ?) : x 6= ? if t 2 [0:01, 0:02) .
(5.6)
• The set OX includes all Quantum Measurement Experiments. That is, for any
Exp 2 QH,{⇡x } there exists Exp0 2 OX such that
(t, ⇤, z) 2 sExp () (t, z) 2 sExp
holds.
59
0
(5.7)
5. NATURAL PROPERTIES OF THEORIES
The single-world assumption asserts that there is only one outcome x as well as that
this outcome does not change during the time when it is recorded.
Definition 8. When we say that a physical theory T satisfies (SW) then this means
that T forbids all stories s according to which in some experiment Exp 2 OX the set
{x : 9 t 2 [0:01, 0:02) s.t. (t, x) 2 sExp }
has more than one single element.
5.6
Self-consistency (SC)
Property (SC) is the requirement of self-consistency. Generally speaking, it demands that
the theory allows at least one story that talks about an experiment. To express this
formally, let E be the set of all experiments. For our purposes, it is sufficient if E includes
the experiments F1, F2, A, and W, as defined in Chapter 8.
Definition 9. When we say that a physical theory T satisfies (SC) then this means
that the condition
sExp is defined for all Exp 2 E
does not imply that s is forbidden by T.
To illustrate this, let Exp and Exp0 be two experiments from the set E. If a theory T
is self-consistent then this means that there is at least one non-forbidden story s which
0
talks about both sExp and sExp . In other words, there is a “consistent” story about the
two experiments.
5.7
Falsification of the assumptions
In Chapter 1 we discussed that the scientific method consists in falsifying theories and
that good theories are falsifiable. We also argued that falsification may be direct through
experimental tests but it may also be by argument – most theories are refuted without
ever being experimentally tested simply because they are unreasonable. Let us therefore
discuss in what sense the assumptions from this chapter can be falsified.
It is obvious that (BornDet) (and similarly the assumption (QT)) can never be experimentally tested, because we cannot prepare perfect states and measurements in practice.
Nevertheless its validity is generally accepted, because it seems to be the best explanation
60
5.7 Falsification of the assumptions
for all experimental observations made so far.
As for the assumptions (Robust), (C) and (SC) the question whether or not they can be
falsified seems to be meaningless because they represent properties that any theory may or
may not have. In order to actually falsify them we would have to show that the true theory
of nature does not have this property. However, because we can never know whether we
have found such a theory, we can never know whether it satisfies the assumptions either.
61
5. NATURAL PROPERTIES OF THEORIES
62
Chapter 6
Deriving the Born rule from
non-probabilistic axioms
6.1
Motivation: Linking the mathematical formalism and
observations
“For if we want to say anything about nature – and what else does science try
to do? – we must somehow pass from mathematical to everyday language.”
Niels Bohr [7]
After a long discussion about the motivation of the framework we finally turn to its
first application: the derivation of the objective Born rule, (BornObj), from the assumptions (BornDet) and (Robust). Because the reader is (hopefully) eager to finally see what
can be done with the framework, we will discuss related work on this topic at the end of
the chapter.
One may wonder why we are interested in a new interpretation of the Born rule – after
all it is “clear” what the statement is. Also it is a postulate of quantum theory, hence why
would we be interested in deriving it? We claim that it is exactly the fact that it seems to
be obvious what the Born rule means that renders its standard interpretation problematic,
because it does not formally express what it is supposed to mean on an operational level.
Let us explain what we mean by this in more detail.
The standard formulation of the Born rule asserts that if a system in state undergoes
a measurement described by a set of projectors {⇡x }x then an outcome x is observed with
probability
P (x) = k⇡x k2 .
(?)
The problem with this statement is that it is based on probabilities and, as we discussed
in Chapter 1, it seems to be impossible to objectively define probabilities without running
63
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
into inconsistencies. And even though it is possible to consistently define subjective probabilities, this approach seems to be rather unsatisfactory in this case. After all the Born
rule is the only law of quantum theory that links the formalism with actual observations
– or that passes from mathematical to everyday language in Bohr’s words. If the Born
rule has only a subjective meaning, in what sense is quantum theory then supposed to be
a law of nature?
(a)
Actual
Observations
Mathematical
Formalism
Born Rule
X
(b)
Actual
Observations
X
Actual
Observations
X
| i
Mathematical
Formalism
Probabilities
Born Rule
Interpretation
PX
Goal
objective Born Rule
| i
Mathematical
Formalism
| i
Figure 6.1: (a) The Born rule is meant to provide a link between the mathematical
formalism of quantum mechanics and physical observations. (b) The traditional
formulation (?) refers to probabilities. However, unless they are equipped with a physical
interpretation, probabilities must be regarded as a purely mathematical concept. The
Born rule then merely connects two mathematical expressions, but does not have any
physical content. The goal of this chapter is to derive a non-probabilistic interpretation
of the Born rule from the natural assumptions (BornDet) and (Robust).
In this chapter we will show that within the framework it is possible to formulate
the Born rule without any reference to probabilities as a law about nature, denoted by
(BornObj). Because it is expressed as a law about stories rather than sequences it avoids the
difficulties of the standard formulations such as references to unphysical infinite sequences.
64
6.1 Motivation: Linking the mathematical formalism and observations
Informally it can be expressed the following statement.
(BornObj): Consider an experiment in which a measurement on a system, described by
a set of projectors {|xi}x2X and a quantum state , is repeated an arbitrary number
of times n. Any story about this experiment according to which the frequencies fx of
the outcome x do not approach |hx| i|2 with arbitrary precision, even if n is arbitrarily
large, must be forbidden.
Remarks about infinities
One may worry because the statement involves the notion of “arbitrary many” repetitions
of the experiment. In particular, one may feel alarmed as it is exactly the notion of infinite
sequences that lets us dismiss the standard frequentist interpretation of probabilities as
unphysical. Let us therefore make some remarks to illustrate why the objective Born rule
(BornObj) is a meaningful, physical statement about nature in contrast to the standard
frequentist interpretation of probabilities. To start, let us recall what the latter says on
an informal level.
“Let X be a random variable. The probability PX (x) of an event x is defined to
be the frequency of outcome x in an infinitely long sequence of measurements
of X.”
• Let us first observe that we dismiss this “definition” of probabilities to be unphysical
because it cannot be viewed as a property of the random variable X, or the system
that we describe by X, because it is impossible to actually measure the system infinitely many times. The objective Born rule does not have this problem because it
does to “define” probabilities.
One may worry now that (BornObj) makes a statement about a physical property of
a measurement on a system based on arbitrary many repetitions and thus, that this
property is not physical. Let us discuss why we claim that this is not problematic.
• We will not postulate (BornObj) as a law about nature but we will derive it from the
assumptions (BornDet) and (Robust) introduced in Chapter 5. These two statements
do not involve any notion of possible infinities and thus, we may view these two
assumptions as laws of nature. Here we want to compare them to the standard
interpretation of the Born rule and therefore, we actually have to formalise the
statement in a way that allows such a comparison. However, as a consequence of our
result, it follows that it is sensible to identify the Born rule with the two physical
laws (BornDet) and (Robust).
65
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
• (BornObj) captures exactly what the standard frequentist interpretation is supposed
to mean in practice:
– Any story telling us that a particular finite sequence will occur is not forbidden.
Analogously within the standard interpretation any finite sequence in general
has a positive probability, even if its frequency distribution fx is not close to
|hx| i|2 .
– As we will show, it follows from the assumptions (BornDet) and (Robust), which
imply (BornObj), that any particular infinite sequence is forbidden. Analogously within the standard interpretation any particular infinite sequence is
forbidden (because it has zero probability).
– Any story telling us that the frequency distribution fx of the observed sequence
will be arbitrary close to |hx| i|2 , without explicitly giving the sequence, is not
forbidden. Analogously within the standard interpretation the probability that
the observed sequence is a “typical” one is positive.
• In contrast to a definition of a physical property, like the frequentist interpretation
of probabilities, it is not problematic that a physical law may refer to a scenario that
is physically impossible – in particular, if the law forbids that scenario. An example
that may illustrate this is the following. Let us assume that, based on the fact that
the earth is a sphere, we derive a law which asserts that if we keep walking in one
direction we will after some finite time return to the point where we started. The
law thus forbids infinitely long walks in one direction. This is a sensible law, despite
the fact that there are other reasons that would prevent us from walking infinitely
long in one direction.
Examples and outlook
Let us look at some simple examples for stories that will be forbidden by (BornObj).
s=
8
“A system described by the state p12 (|0i + |1i)
>
>
>
< is measured w.r.t to the computational basis
{|0i, |1i}. No matter how often one repeats
>
>
>
: this measurement the outcome will always be
x = 0.”
or
s=
8
“A system described by the state p12 (|0i + |1i)
>
>
>
< is measured w.r.t to the computational basis
{|0i, |1i}. No matter how often one repeats
>
>
>
: this measurement the frequency of outcome
x = 0 will always be f0 = 0.6.”
66
6.1 Motivation: Linking the mathematical formalism and observations
or
s=
8
“A system described by the state p12 (|0i + |1i)
>
>
>
>
>
< is measured w.r.t the the computational basis
{|0i, |1i}. The frequency of outcome x = 0
>
will get closer and closer to f0 = 0.5 before
>
>
10
>
>
: n = 10 . Then it will deviate from f0 = 0.5
and not get close anymore. ”
In contrast to this a story that is not forbidden by (BornObj) is the following.
8
“A system described by the state p12 (|0i + |1i)
>
>
>
< is measured w.r.t to the computational basis
s=
{|0i, |1i}. If one repeats that measurement the
>
>
>
: frequency of outcome x = 0 will get closer and
closer to f0 = 0.5. ”
These examples illustrate the fundamental di↵erence of (BornObj) compared to standard formulations that are based on sequences. Finally, let us remark that we will show
later that the following story is also forbidden.
8
“A system described by the state p12 (|0i + |1i)
>
>
>
>
>
< is measured w.r.t to the computational basis
{|0i, |1i}. If one repeats that measurement the
s=
>
frequency of outcome x = 0 will become equal
>
>
>
>
: to exactly f0 = 0.5 at some point and not
deviate anymore. ”
Operationally to this is not surprising, because it is impossible to exactly measure a frequency within any experiment. This will motivate the idea to consider experiments with
a finite precision. We will discuss this issue in more detail later.
In order to formally express the idea illustrated by the examples above we will introduce a specific experiment, termed MeasFreqµ , and formulate (BornObj) as a rule about
experiments of this type. We stress that the specific choice of physical quantities used to
describe MeasFreqµ is not fixed in the sense that one is free to add others.
In contrast to the standard Born rule we will not postulate (BornObj) as an axiom but
derive it from the assumptions (BornDet) and (Robust). These are defined and discussed
in Chapter 5. Let us recall what they express on an informal level.
• (BornDet): If the system’s state
has no overlap with a projector ⇡x then the
corresponding measurement outcome x will not be observed.
• (Robust): The statements of a theory are robust under small perturbations.
Note that the measure to quantify perturbations was discussed in Chapter 5. The idea
is that the physical quantities of an experiment induce a topology on the set of stories.
67
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
In particular, for experiments involving a quantum state , the robustness assumption
implies that the statements remain valid for small perturbations of the state. This seems
to be a necessary requirement for a theory to be applicable in practice, because we can
never perfectly prepare a state.
We will show that if the two axioms (BornDet) and (Robust) are applied to an experiment in which a given measurement is repeated under identical initial conditions then the
frequencies will necessarily obey the Born law, i.e.,
(BornDet) ^ (Robust) =) (BornObj) .
(6.1)
The proof of this statement is provided in Section 6.3.
Before turning to an introductory example to illustrate the idea of the proof, we remark that the standard subjective interpretation of the Born rule can be derived from
the fundamental law (BornObj) by introducing additional assumptions about an agent’s
beliefs. This will be the subject of Chapter 7 but let us briefly explain the idea.
The idea is to define the standard interpretation of the Born rule that is based on
subjective probabilities as follows.
(BornBelief): Corresponds to Eq. (?) with PX (x) interpreted as an agent’s belief about
the event that the measurement outcome is x. Specifically, PX (x) may be defined as
the maximum price the agent would be willing to pay for a bet with payo↵ $ 1 if the
measurement outcome is x and no payo↵ otherwise.
Considering a scenario with repeated bets labeled by a time t, we will make assumptions
about what kind of behaviour an agent would consider as rational. First, we observe that
it follows from the Dutch book argument (see Chapter 7) that the probability assignments
must satisfy the axioms of probability theory. In order to show that they are actually given
by Eq. (?), if an agent believes in (BornObj), we introduce two additional assumptions
about the agent’s beliefs about rationality which are informally expressed as follows.
• (Repetitions): The outcomes up to time t do not depend on how long the agent
intends to continue gambling.
• (Indi↵erence): The bet on being successful at time t does not depend on t.
Suppose now that an agent thinks that believing in these two assumptions is rational.
In Chapter 7 we will show that if he also believes that the two non-probabilistic laws
(BornDet) and (Robust) are correct, then his beliefs about measurement outcomes are
correctly characterised by the Born rule, i.e.,
(BornDet) ^ (Robust)
(Repetitions) ^ (Indi↵erence)
68
=) (BornBelief) .
(6.2)
6.2 The experiment Light
We conclude this section by remarking that there has been a lot of interesting work
regarding the Born rule previous to our approach. We will discuss some of these results
and their connection to this work at the end of the chapter.
6.2
The experiment Light
To illustrate our argument, we consider an experiment, termed Light, as depicted in
Fig. 6.2. Light beams of increasing intensity are shined on a polarising beam splitter
(PBS) and the relative intensity of the reflected beam, R, is measured. For simplicity, we
assume that each incident light beam consists of a fixed number of photons, n, so that R
corresponds to the fraction of photons that are reflected. We remark that it is not relevant
for our argument whether the n incident photons are indistinguishable or not (the light
beam could consist of n di↵erent modes).
measurement
R
0.6
Rn
source
n
PBS
⌦n
Figure 6.2: Illustration of the experiment Light. A source emits light beams consisting of
n photons in a joint polarisation state . They are directed to a polarising beam splitter
(PBS) and the relative intensity R of the reflected beam is measured.
We are now going to describe this experiment within the usual formalism of quantum
mechanics. The e↵ect of the PBS can be characterised by two orthonormal vectors |0i and
|1i of a Hilbert space H = C2 , which indicate the polarisation directions of beams that
are fully transmitted or reflected, respectively. Hence, according to the standard Born
rule, a single photon with polarisation
2 H is reflected by the PBS with probability
|h1| i|2 . More generally, if we shine in an n-photon light beam whose photons all have
polarisation
then the probability that the relative intensity R of the reflected beam
equals r 2 [0, 1] is
PR (r) = h |⌦n M n,r | i⌦n ,
69
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
where1
M
n,r
=
X
n
O
x2{0,1}n i=1
|x|/n=r
|xi ihxi | .
We may now tell stories about the experiment Light. Let
that admits a finite description.2
(
(6.3)
2 H be a normalised vector
“A light beam consisting of n = 1 photon with
polarisation shines onto the PBS and is reflected completely.”
s1, =
or
s1,
⌦
=
8
“Light beams consisting of an increasing num>
<
=
8
>
<“Light beams consisting of an increasing number
or
s1,
2
⌦
>
:
ber n = {1, 2, . . .} of photons with polarisation
shine onto the PBS. They are all reflected
completely.”
n = {1, 2, . . .} of photons with polarisation
>
: shine onto the PBS. Each time exactly bn/2c
of the photons are reflected.”
or, for any
2 (0, 12 ),
s1±
2
,
⌦
=
8
“Light beams consisting of an increasing number
>
<
n = {1, 2, . . .} of photons with polarisation
>
: shine onto the PBS. Each1 time 1M photons are
reflected, where M/n 2 [ 2
, 2 + ].”
To analyse these stories within our framework, we need to specify what physical quantities
we are interested in. Of course one is free to make any choice of physical quantities to
describe the experiment. In the following we will make a minimal choice of the quantities
that are necessary for our purposes, but one is free to add additional quantities. In particular, one may also add the set of measurement projectors describing the e↵ect of the beam
splitter. Because it is fixed we regard it to be part of the description of the experiment
and do not include it among the physical quantities.
We choose ˆ to describe the joint polarisation state
in the incident
L1of all⌦nphotons
3
light beam. It has as its range the Hilbert space H = n=0 H . The second physical
1
In the case where the n photons in the light beam are indistinguishable, one may also consider the
projection of the operator M n,r onto the symmetric subspace of H⌦n .
2
This requirement is necessary because the description of will be part of the story.
3
If the light beams consist of indistinguishable photons, one may replace H by the Fock space of H.
70
6.2 The experiment Light
quantity we consider is the measured relative intensity of the reflected beam. We call it R̂
and define its range as [0, 1]. Finally we also include the number of incident photons n̂
among the physical quantities. Setting ⇥Light = { ˆ , n̂, R̂}, the event space will be of the
form
[Light] = ( , n, r) 2 H ⇥ N ⇥ [0, 1] .
The trajectories of the first three stories above, they are given by
sLight
= {( , 1, 1)}
1,
sLight
= {(
1, ⌦
⌦n
sLight
= {(
1
, ⌦
⌦n
2
, n, 1) : n 2 N}
, n, bn/2c
n ) : n 2 N} .
The trajectory of the story s 1 ± , ⌦ in the experiment Light is not well defined, because
2
the outcome of the intensity measurement is only specified up to finite precision . To take
this into account, we may consider a slightly modified (and more realistic) experiment,
Lightµ , where the intensity of the reflected beam is measured with finite precision only.
More precisely, the physical quantity R̂ is replaced by a quantity R̂µ which is rounded up
to precision µ for µ 2 (0, 1).1 Story s 1 ± , ⌦ then has a well defined trajectory, provided
that
is sufficiently small,2 namely
µ
sLight
1
± ,
2
⌦
2
= {(
⌦n
, n, 12 ) : n 2 N} .
For a quantum-mechanical analysis, two laws are relevant, (BornDet) and (Robust).
Note that Light is not of the form of an experiment Meas 2 QH,{⇡x }x2X for which (BornDet)
applies. However, we may assume that Light corresponds such an experiment
9 Meas 2 QH,{M n,rµ }n,rµ :
where
=
L1
n=0
⌦n
µ
and M n,r =
( , n, rµ ) 2 sLight
P
µ
() ( , (n, r)) 2 sMeas ,
M n,r̄ with M n,r defined in (6.3).
r̄2[r µ,r+µ]
Starting with (BornDet), it is straightforward to verify that the following relations hold
sLight
violates (BornDet) () h |1i = 0
1,
sLight
violates (BornDet) () h |1i = 0
1, ⌦
sLight
violates (BornDet) () h |0i = 0 or h |1i = 0
1
, ⌦
2
µ
sLight
1
⌦
±
,
2
1
2
violates (BornDet) () h |0i = 0 or h |1i = 0 .
The physical quantity R̂µ thus has range {r 2 [0, 1] : r/µ 2 N}.
The trajectory is uniquely defined if µ
.
71
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
if
These are still rather weak statements. The story s1,
is orthogonal to |1i, i.e.,
h |1i = 0
=)
s1,
⌦
⌦
, for instance, is only forbidden
2T.
However, as we shall argue now, taking into account the additional law (Robust) leads
to significantly strengthened statements, such as
|h1| i| < 1
=)
s1,
⌦
2T.
(6.4)
To explain this, we first introduce some notation. For any n0 2 N, let
✓nM
◆ ✓M
◆
1
0 1
⌦n
⌦n
n0
⌦n
⇧
=
idH
idH
|1ih1|
n=n0
n=0
be the projector onto the subspace of H that excludes all vectors |1i⌦n for n n0 . It is
easy to verify that, for n0 large, applying the projector ⇧ n0 leaves product states of the
form ⌦n almost unchanged, unless is parallel to |1i. More precisely, for any 2 H
with |h1| i| < 1 and for any " > 0 there exists n0 such that
n0
⇧
⌦n "
⌦n
⇡
,
where the approximation is with respect to the metric of the Hilbert space.1
We now use the projectors ⇧ n0 to modify stories. Given a story s about Lightµ ,
let s|n0 be the story obtained by applying ⇧ n0 to the physical quantity ˆ , followed by
renormalisation. This may be expressed formally in terms of the trajectories, which for
the modified story is given by
⇢⇣
⌘
µ
⇧ n0
Lightµ
s|n0 =
,
n,
r
: ( , n, r) 2 sLight
.
n
0
k⇧
k
In the case of story s1, ⌦ , and assuming |h1| i| < 1, it follows from the above that for any
" > 0 there exists n0 such that
µ
"
µ
sLight
⇡ sLight
.
1, ⌦ |n0
1, ⌦
1
Let  := |h1| i|2 < 1.
h⇧
=)
d(⇧
n0
n0
⌦n
|
⌦n
i = h | in
=1
h1| in h |1in
|h1| i|n
= 1 n
p
⌦n
, ⌦n ) = 2(1 <h⇧
p
= 2n
n!1
! 0.
72
n0
⌦n |
⌦n i)
6.2 The experiment Light
Note that as discussed in Chapter 5 we use the Hausdor↵ norm to quantify the closeness
of trajectories.
By the definition of the projector ⇧ n0 , the vectors
occurring in the trajectory
are orthogonal to |1i⌦n for any n n0 . The trajectory therefore violates (BornDet),
i.e., the story s1, ⌦ |n0 is forbidden by any theory T that satisfies (BornDet). Hence,
provided that |h1| i| < 1, we have established that for any " > 0 there exists a story
s1, ⌦ |n0 2 T whose trajectory is "-close to that of s1, ⌦ . This is equivalent to say that the
trajectory of s1, ⌦ is contained in the closure of the set T. But because (Robust) demands
that T is closed, we conclude that the trajectory of s1, ⌦ is also an element of this set,
which proves (6.4).
sLight
1, ⌦ |n0
In the next section we generalise this statement and prove the objective law (BornObj),
which informally asserts that a story about a measurement described by a quantum state
w.r.t. the set of projectors {|xi}x2X is forbidden if it says that the frequencies of the
outcome x will never get close to |hx| i|2 .
Applied to the story s 1 ±
2
,
⌦
, (BornObj) will tell us that the following rule holds
|h1| i|2 2
/ [ 12
, 12 + ]
=)
s1±
2
,
⌦
2T.
We remark that the positive parameter > 0 in the story is crucial. We will show in
Section 6.4 that for = 0 actually all stories are forbidden.
This can be illustrated by the example of story s 1 , ⌦ , which asserts that exactly bn/2c
2
of the n photons are reflected, corresponding to setting = 0. As this story predicts
an exact frequency it is always forbidden, even if the polarisation direction
satisfies
1
2
|h1| i| = 2 . To see this one may proceed along the argument for deriving (6.4), replacing ⇧ n0 by a projector that excludes all vectors of the form |0i⌦n bn/2c ⌦ |1i⌦bn/2c and
permutations thereof, for n
n0 . Intuitively, this makes sense because the number of
reflected photons always fluctuates, i.e., it is not equal q
to bn/2c for all n. Thus, while
⇣
⌘n
⌦n
1
for n increasing the frequency distribution of the state
|0i + |1i
becomes ar2
bitrarily peaked around
1
2
there will always be contributions from other frequencies as well.
Note that in the case of story s1, ⌦ it is not necessary to require > 0, because for
| i = |1i all photons are reflected and there are no fluctuations. In the technical argument
above, the projector ⇧ n0 would map all states in s1, ⌦ with a photon number n
n0
to 0, i.e., the trajectory of the modified story would no longer be close to that of s1, ⌦ .
73
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
6.3
Deriving the objective Born rule
After this illustrating example we are now ready to precisely state and prove our main
statement, the objective Born rule (BornObj).
Whereas in the example Light we fixed the measurement projectors to be |1i and
|0i we consider now a more general set of experiments termed Qn,H,{⇡x }x2X . In any
MeasFreq 2 Qn,H,{⇡x }x2X an increasing number of quantum states is measured with respect
to a set of projectors {|xi}x2X for some alphabet X . More precisely, the frequency distribution of the di↵erent outcomes is measured. Note that di↵erent experiments in this set correspond to di↵erent physical setups. For instance, the measured system may correspond to a
photon, an electron or, in principle, even a human being. For any MeasFreq 2 Qn,H,{⇡x }x2X
we denote by MeasFreqµ the experiment in which the frequency distribution is measured
with precision µ.
We chose to describe the experiments using 2 + |X | di↵erent physical quantities. Note
that one may also chose di↵erent quantities, but our choice will be convenient to formulate
our main claim. This is not a limitation of the statement, because a story is forbidden
if there is any experiment, corresponding to a specific choice of physical quantities, such
that the story is forbidden for that experiment.
Like in the previous example we choose the first of the physical quantities
to be ˆ ,
L1
⌦n .
the joint state of all systems with range equal to the Hilbert space H =
n=0 H
µ
Additionally we denote by n̂ the number of systems and by F̂ the
P frequency distribution
|X
|
of the outcomes with precision µ with range [0, 1] such that
fx = 1 and fx /µ 2 N
x2X
µ
for all x 2 X . Therefore we have ⇥MeasFreq = {H, n̂, F̂ µ } with an event space given by
[MeasFreqµ ] = ( , n, f ) 2 H ⇥ N ⇥ [0, 1]|X | .
Note that because the measurement projectors are assumed to be fixed, we may include
them in the description of the experiment instead of adding them to the physical quantities.
We are now ready to formally express our main statement.
74
6.3 Deriving the objective Born rule
Definition 10. When we say that a physical theory T satisfies (BornObj) then this
means that T forbids all stories s according to which for some MeasFreq 2 Qn,H,{⇡x }x2X
the frequency of an outcome x does not approach |hx| i|2 , even if the number of
repetitions n is arbitrarily large, must be forbidden. More precisely, it means that the
following condition holds
lim sup
n!1
n
(s) > µ
=)
s2T
for all µ > 0, where
n
(s) = sup{|fx
x
with x 2 X ,
|hx| i|2 | : (
⌦n
µ
, n, f ) 2 sMeasFreq }
2 H and lim sup xn := inf sup xk .
n!1
n2N k n
Note that lim sup guarantees that the limit is also defined for stories that do not talk
about arbitrary many repetitions. In this case a story may always be valid according to
the definition and in fact this is how we apply quantum theory in practice: If our resources
are limited to 1000 copies of a quantum state all frequencies may be observed. In other
words, (BornObj) tells us that all stories according to which the frequencies fx deviate by
more than the precision µ from |hx| i|2 , even if n is arbitrarily large, must be forbidden.
With all these definitions, implication (6.1), which we described informally in the introduction, is now finally turned into a well-defined theorem.
Theorem 1. Any theory T that satisfies (BornDet) as well as (Robust) also satisfies
(BornObj).
Proof. Let us first observe that the experiments MeasFreqµ 2 Qn,H,{⇡x }x2X are a priori
not of the form such that we can apply the assumption (BornDet). Therefore, we have
to assume that any such experiment corresponds to an experiment Meas 2 QH,{⇡x }x2X by
dropping the quantity n. To make the statement more precise we first specify the relevant
set of measurement projectors.
For a given sequence s 2 X n we denote the induced frequency distribution by
Psn (x) :=
|k : sk = x|
.
n
Furthermore, we define the projector onto all sequences of length n with frequency f to
75
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
be
⇡f,n =
s:
X O
Psn =f
⇡s ,
(6.5)
s
where ⇡s is defined as
⇡s := |s1 , . . . , sn ihs1 , . . . , sn |.
For an experiment with precision µ the set of measurement projectors is then given by
µ
{⇡f,n
}f,n ,
where
µ
⇡f,n
:=
f 0 2[f
X
⇡f 0 ,n .
µ
,f + µ
]
2
2
The assumption that we can apply (BornDet) to an experiment MeasFreqµ can then
be expressed as the condition
9 Meas 2 QH,{⇡µ
f,n }
( , n, f ) 2 sMeasFreq
:
µ
() ( , (n, f )) 2 sMeas .
(6.6)
µ
Let now s be a story about an experiment MeasFreqµ with trajectory sMeasFreq that
is not explicitly forbidden by T, i.e., s 62 T holds.
We will show that
n
lim sup
n!1
µ
(sMeasFreq )  µ
must hold.
We first note that assumption (Robust) implies that there is a " > 0 such that all
"-close stories s̄ are also not forbidden, i.e.,
(Robust) =)
9 " > 0 : s.t. s̄ 62 T
"
8 s̄ ⇡ s.
Next we use assumption (BornDet) via relation (6.6) applied to the experiment MeasFreqµ
h
i
µ
µ
µ
0
(BornDet) =)
s̄MeasFreq 62 T =) ⇡f,n
8 ( n0 , n, f ) 2 s̄MeasFreq ,
n > 0,
µ
where the set of projectors {⇡f,n
} is defined in Eq. (6.5).
Combining the two statements we obtain
(Robust) + (BornDet) =)
h
µ
0
9 " > 0 : ⇡f,n
8(
n > 0,
0
n , n, f )
2 s̄MeasFreq
76
µ
i
µ
"
µ
8 s̄MeasFreq ⇡ sMeasFreq .
(6.7)
6.3 Deriving the objective Born rule
Using the Hausdor↵ norm (see Chapter 5) we can rewrite this as the condition
(Robust) + (BornDet) =)
h
µ
0
9 " > 0 : s.t. ⇡f,n
n > 0,
8
0
n
⌦n
⇡"
i
8(
⌦n
µ
, n, f ) 2 sMeasFreq .
(6.8)
This can be seen from the definition of the Hausdor↵ norm
"
µ
s̄MeasFreq ⇡ sMeasFreq
8 (n̄, f¯,
0
n)
and
8(
⌦n
µ
2 s̄MeasFreq
()
µ
⌦n
9(
, n, f ) 2 sMeasFreq
µ
9(
0
¯
n , n̄, f )
"
µ
⌦n
µ
0
¯ "
n , n̄, f ) ⇡
, n, f ) 2 sMeasFreq : (
2 s̄MeasFreq : (
0
¯
n , n̄, f )
, n, f ) ⇡ (
(
⌦n
, n, f ),
where
(
0
¯ "
n , n̄, f ) ⇡
()
"
n̄ ⇡ n
⌦n
(
, n, f )
"
f¯ ⇡ f
^
0 "
n ⇡
^
⌦n
.
µ
0
MeasFreq
Assume now 9 n0 ⇡" ⌦n such that ⇡f,n
n = 0. Then we can construct a story s̄
µ
from sMeasFreq by simply replacing ⌦n by n0
µ
µ
⌦n
s̄MeasFreq := (sMeasFreq \ {(
, n, f )}) [ {(
µ
0
n , n, f )}.
µ
This story is "-close to sMeasFreq by construction, but violates Eq. (6.7). Thus Eq. (6.8)
follows.
Next we show the following relation
µ
⇡f,n
0
n
> 0,
8
0
n
⌦n
⇡"
=)
h
⌦n
µ
|⇡f,n
|
This can be seen as follows. Assume by contradiction that h
0
n
µ
⇡f,n
)
:= (id
77
⌦n
.
⌦n
i > "0 :=
"2
.
2
⌦n |⇡ µ | ⌦n i
f,n
(6.9)
 "0 and let
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
⌦n
This state is "-close to
d(
0
n,
because
⌦n
⌦n
) = k n0
k
p
= 2(1 <h n0 | ⌦n i)
q
µ
= 2(1 <h ⌦n |(id ⇡f,n
)|
q
µ
= 2<h ⌦n |⇡f,n
| ⌦n i
q
µ
= 2h ⌦n |⇡f,n
| ⌦n i
p
< 2"0
⌦n i)
= ".
But we also have
µ
⇡f,n
0
n
=0
µ
obtaining a contradiction to the assumption ⇡f,n
0
n
0
n
> 0 for all
⇡"
⌦n .
Combining (6.8) and (6.9) yields
(Robust) + (BornDet) =)
9 "0 > 0 : h
⌦n
µ
|⇡f,n
|
⌦n
i > "0
⌦n
8(
µ
, n, f ) 2 sMeasFreq .
Let now Q be the probability distribution defined by
Q(x) = |hx| i|2 .
In Appendix B we show the following typicality Lemma
kf
Qk
"+µ
=)
h
⌦n
µ
|⇡f,n
|
⌦n
i < nd
1
·2
n 2
"
2
8n
(6.10)
where k · k corresponds to the trace distance and d = |X | is the dimension of the Hilbert
space H.
Hence we obtain
(Robust) + (BornDet) =)
0
9 " > 0 : s.t. kf
Qk <
r
2
((d
n
1) log n
78
log "0 ) + µ
8(
⌦n
µ
, n, f ) 2 sMeasFreq .
6.4 Stories that predict the frequency with an arbitrary precision are
forbidden
Now we have
lim sup
n!1
n
n!1
x
 lim sup
X
= lim sup
kf
n!1
x
n!1
⌦n
Qk : (n, f,
) 2 sMeasFreq
) 2 sMeasFreq
µ
µ
2
((d 1) log n log "0 ) + µ
n
r
2
((d 1) log n log "0 )
n
n!1
= inf sup
n 0 m n
r
n 0
⌦n
|hx| i|2 | : (n, f,
|fx
µ
) 2 sMeasFreq }
r
 lim sup
= inf
⌦n
|hx| i|2 | : (n, f,
sup{|fx
(s) = lim sup
2
((d
n
log "0 ) + µ
1) log n
=µ
Which finally implies
(Robust) and (BornDet) =)
lim sup
n
n!1
(s)  µ.
Because we assumed that s 62 T this can be rewritten as
s 62 T
=)
lim sup
n
lim sup
n!1
n
(s)  µ
which is equivalent to
n!1
(s) > µ
=)
s 2 T.
This concludes the proof.
6.4
Stories that predict the frequency with an arbitrary precision are forbidden
Let us have another look at the story s 1 ± , ⌦ from Section 6.2.
2
8
“Light beams consisting of an increasing number
>
<
n = {1, 2, . . .} of photons with polarisation
s1± , ⌦ =
2
>
: shine onto the PBS. Each1 time 1M photons are
reflected, where M/n 2 [ 2
, 2 + ].”
We saw that the trajectory is uniquely defined for all experiments MeasFreqµ with µ
by
µ
sLight
1
± ,
2
⌦
= {(
⌦n
, n, 12 ) : n 2 N} .
79
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
We already mentioned in Section 6.2 that it is crucial that
story s 1 , ⌦ would be forbidden for all states – even for
> 0 because for
= 0 the
2
1
= p (|0i + |1i)
2
which we would expect to be allowed because for that state the overlap with |1i, modelling
the polarisation direction of reflected beams, is exactly one half |h1| i|2 = 12 . We will now
show that this is true in general: any story that predicts a frequency with arbitrary
precision is forbidden.
Lemma 3. Let T be a theory with the rule (BornObj). Within such a theory any story
s that predicts the frequency with arbitrary precision is forbidden
9(
⌦n
, n, f ) 2 sMeasFreq
µ=0
and x 2 X : |hx| i|2 < 1
s 2 T.
=)
Note that for stories that do not predict the frequency with arbitrary precision the traµ=0
jectory sMeasFreq
is not defined.
Proof. Analogously to the illustration in the experiment Light we can construct for all " > 0
"
a story s̄ ⇡ s that is forbidden by (BornDet). From the assumption (Robust), requiring
that the set of forbidden stories must be closed, it follows that s must be forbidden as well.
Fix now one particular x and let |hx| i|2 := . We define for n0 2 N and fx : (
the projector
⇧
n0
=
✓nM
0 1
n=0
For all
id⌦n
H
◆
there is an n0 such that ⇧
h⇧
n0
✓M
1
n=n0
n0
d(⇧
n0
X
s2X n
|{i: si =x}|=fx
|sihs|
"
⇡
◆
.
for all " > 0 because
X
⌦n ⌦n
|
i = h | in
|hs| i⌦n |2
1
=)
id⌦n
H
⌦n
,
⌦n
n
q
2(1
p
 2n
)=
n!1
! 0.
80
s2X n
|{i: si =x}|=fx
<h⇧
n0
⌦n | ⌦n i)
⌦n , n, f )
2 sMeasFreq
µ=0
6.5 A remark about possible falsification of the Born rule
Therefore, the story s̄ that is mapped to the trajectory
s̄
MeasFreqµ=0
=
⇢⇣
⇧
k⇧
n0
⌦n
n0
⌦n k
⌘
, n, f : (
⌦n
, n, f ) 2 sMeasFreq
µ=0
is "-close to s. However, s̄ is forbidden
P by (BornDet) because it does not have any overlap
with the frequency projector
|sihs| by construction. Hence, because of the
s2X n
|{i: si =x}|=fx
requirement (Robust), s must be forbidden as well.
Let us make a final remark about logical extensions of stories. By a logical extension
we mean a more detailed version of a story. It seems desirable that any story that is
not forbidden s 1 ± , ⌦ 62 T can be made more precise by telling the frequency a bit more
2
accurate. For example for = 0.01 we can simply extend s 1 ± , ⌦ by a story s 1 ± 0 , ⌦ with
2
2
0 = 0.001 or 0 = 0.0001 and so on. However, we have just seen that it is impossible to tell
a story about the true frequency, i.e., we cannot extend s 1 ± , ⌦ by setting 0 = 0. Luckily
2
the requirement that stories are finite ensures that stories still can be logically extended
– as long as the extension cannot be told in finite time. Because such an extension does
not correspond to a story within the framework, it is not forbidden.
6.5
A remark about possible falsification of the Born rule
Let us make a remark about the possibility to falsify the law (BornObj). Informally, the
it says, that there is a number n0 such that after n0 trials the frequency of the outcome
x stays arbitrarily close to |hx| i|2 . However, the law does not tell us what this number
is. Therefore, it cannot be falsified in an actual experiment and one may argue that this
renders the law unphysical. Let us explain why it is not a problem that (BornObj) is not
experimentally testable.
Let us first remark that obviously any law that is directly falsifiable would be preferable.
However, we do not have such a law. In particular, none of the standard interpretations
of the Born rule are falsifiable either, because they are probabilistic statements.
We derived (BornObj) from the assumptions (BornDet) and (Robust) and we discussed
in Chapter 5 why these two laws are not falsifiable and why this is generally not considered
to be problematic. Regarding the assumption (BornDet) it is well accepted that it cannot
be falsified because we cannot prepare perfect quantum states. For the robustness requirement (Robust) the question about falsification is not meaningful because it is a property
of a theory and not a law of nature. As it is impossible to derive something falsifiable
from non-falsifiable statements it is not surprising that (BornObj) is not falsifiable by an
experimental test either.
81
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
6.6
Related work
Various approaches to derive the Born rule from more fundamental principles have been
proposed in the literature. One of them is to use principles that refer to probabilities
as a purely mathematical concept. This means that probabilities are simply regarded
as values P (x) that can be assigned to the possible outcomes x of a measurement. A
prominent example for such an approach is Gleason’s theorem [53]. It asserts that Eq. (?)
follows essentially from the assumption that each value P (x) is uniquely determined by
the projector ⇡x corresponding to x, that for any measurement defined by a partition of
the identity into projectors these values sum up to 1, and that H is at least 3-dimensional.
More recently, Saunders [54] showed that Eq. (?) can also be obtained from more operationally motivated assumptions on the values P (x), dropping in particular the assumption
on the dimension of H. Furthermore, Zurek [55] demonstrated that Eq. (?) is implied by
a symmetry principle called “environment-assisted invariance”. The common feature of
these arguments as well as Gleason’s theorem is that they connect quantum states to a
mathematical notion of probabilities. In that respect they are somewhat orthogonal to
our objective, which is to establish a connection to physical observations rather than to
probabilities.
One of the best known approaches to obtain the Born rule without resorting to probabilistic axioms is the decision-theoretic argument proposed by Deutsch [40] and later
refined by Wallace [56]. They showed that under certain assumptions about rationality,
if a system in state is measured, a rational agent will bet on the outcome x with the
maximal value P (x), as given by (?). The statement they derive is thus similar in spirit
to the Bayesian interpretation of the Born rule (BornBelief) that we will discuss in more
detail in Chapter 7.
A rather di↵erent line of reasoning, based on typicality arguments, was proposed by
Everett [57], DeWitt [58], Graham [59], and Hartle [60], and later strengthened by Farhi,
Goldstone and Gutmann [61]. The idea is to start from an axiom similar to (BornDet)
and apply it to an experiment that is repeated infinitely often. It is then shown that
(BornFreq) holds for “typical” sequences of measurement outcomes, i.e., for a set of sequences of weight 1 according to some probability measure on the set of infinite sequences.
However, as pointed out by Caves and Schack [62], additional assumptions are needed in
order to define this probability measure, and the present arguments are therefore incomplete.
A possible additional assumption that could be used to complete the typicality arguments was proposed by Buniy, Hsu and Zee [63]. Their idea is to postulate that the quantum state space is fundamentally discrete. This postulate is related to our axiom (Robust),
in the sense that a theory with discrete state space is by definition robust against small
perturbations of the states. Also de Raedt, Katsnelson and Michielsen [64] discuss the
82
6.6 Related work
idea to require that quantum theory is robust.
Finally, there are several derivations of quantum theory from natural axioms that are
based on a probabilistic framework such as the work by Hardy, Müller, Masanes and
d’Ariano [65, 66, 67]. Their approach is somewhat orthogonal to our work here. In
particular we assume that the formalism of quantum theory is true in first place (such
as that systems are described by hermitian operators defined on Hilbert spaces1 ) but
we do not assume the probabilistic axiom (?). In contrast to this approaches by Hardy,
Müller, Masanes and d’Ariano are intended to derive this formalism but do not question
its relation to probabilities.
1
Of course we agree that these assumptions are quite unnatural and that the problem of replacing
them by more operational axioms is an interesting question for research. However, we stress that this is
not the goal of this work.
83
6. DERIVING THE BORN RULE FROM NON-PROBABILISTIC
AXIOMS
84
Chapter 7
Retrieving the subjective Born
rule
“Part of our knowledge we obtain direct; and part by argument. The Theory
of Probability is concerned with that part which we obtain by argument, and it
treats of the di↵erent degrees in which the results so obtained are conclusive or
inconclusive.”
John Maynard Keyens [68]
“That point of view is only one of the possible points of view but I would not be
completely honest if I did not add that it is the only one that is not in conflict
with the logical demands in my mind.”
Bruno de Finetti [24]
Let us summarise what we have discussed so far. In Chapter 1 we motivated the point of
view to identify theories with explanations and we concluded that it is necessary to provide
the mathematical formalism of a theory with an explanatory interpretation. In order to
have explanatory power it seems to be a minimal requirement that the interpretation of a
theory has an objective meaning. This observation makes it problematic to base a theory
on probabilities because it seems to be impossible to objectively define probabilities as
pointed out for example by de Finetti [24] (see also Chapter 1).
“There are two procedures that have been thought to provide an objective meaning of probability: the scheme of equally probably cases, and the consideration
of frequencies. Indeed it is on these two procedures that the evaluation of probabilities generally rests and the cases where normally the opinions of most
individuals coincide. However, these same procedures do not oblige us at all to
admit the existence of an objective probability; on the contrary, if one wants
to stretch their significance to arrive at such a conclusion, one arrives at wellknown difficulties, which disappear when one becomes a bit less demanding,
85
7. RETRIEVING THE SUBJECTIVE BORN RULE
that is to say when one seeks not to eliminate but to make more precise the
subjective element in all this.”
Bruno de Finetti [24]
The observation that theories should not be based on probabilities motivated the introduction of our non-probabilistic framework which is based on the idea that theories
provide rules to decide if stories about experiments are forbidden. As a first application
we derived in Chapter 6 an objective interpretation of the Born rule as a non-probabilistic
statement about nature which we termed (BornObj).
When we say that it is problematic to base theories on probabilistic statements we
do not mean that probabilities should not be used at all – after all they have proven to
be useful in many applications. In this chapter we discuss the ideas of de Finetti [24] in
the context of our framework: we will reinvestigate the problem of the definition and the
operational meaning of probability, as well as the question of how one arrives at actual
probability assignments. In particular, we are interested in the question of how the ideas
of subjective probabilities can be introduced in the framework.
Because probabilities are considered to be subjective, it is not meaningful to ask
whether a specific probability assignment of an agent is true or false. However, this does
not necessarily mean that the assignments are arbitrary, because the question whether
they are rational may indeed be meaningful. What an agent considers to be rational is
based on his beliefs about what he finds rational. In particular, a di↵erent agent may
have di↵erent beliefs and thus may arrive at di↵erent probability assignments. In practice
however, many agents will share certain assumptions about rationality, for example, that
it is irrational to accept bets that result in losses with certainty.
We will use the idea of bets to operationally define the probability P (x) of an event
x as the maximal amount of money an agent would be willing to pay to enter the bet in
which he gains $ 1 if x occurs and nothing otherwise. As pointed out by Ramsey [69] this
idea “will not seem unreasonable when it is seen that all our lives we are in a sense betting.
Whenever we go to the station we are betting that a train will really run...”. Obviously
this definition does not yet restrict probability assignments in any way – in fact the agent
may even pay more than $ 1 to accept the bet. To arrive at stronger statements we have
to make assumptions about what an agent finds rational. We will first discuss the Dutch
book argument that was inaugurated by Ramsey [69] showing that if the agent believes
that it is irrational to enter bets that result in losses with certainty then his probability
assignments must satisfy the axioms of probability theory. In particular, it follows that
they are between 0 and 1.
The requirement that in order to be coherent an agent’s probability assignments should
be between 0 and 1 is still very weak. De Finetti expressed it as follows: “...each of these
86
7.1 Operational definition of subjective probabilities
evaluations corresponds to a coherent opinion, to an opinion legitimate in itself, and every individual is free to adopt that one of these opinions which he prefers, or, to put
it more plainly, that which he feels.” He then raises the question of whether it makes
sense to ask: “Is there among the infinity of evaluations that are perfectly admissible in
themselves, one particular evaluation which we can qualify, in a sense as yet unknown, as
objectively correct? Or, at least, can we ask if a given evaluation is better than an other
one?” In [24] de Finetti demonstrates how the concept of “exchangeability” within trials
of the same phenomenon can be used to arrive at actual probability assignments for the
individual events. Here we will resume his ideas and formalise them within our framework.
In order to illustrate the subjective character of the beliefs that lead to a particular
assignment, we will introduce a gambler, Rachel, who is o↵ered the bet described above
repeatedly and we will introduce assumptions about her beliefs about the structure of
these bets. More precisely, we will enumerate the rounds Rachel plays by a label t and
assume that Rachel believes (i) that the individual outcomes do not depend on how long
she intends to continue playing and (ii) that her bet on being successful in round t does not
depend on t. Furthermore, we assume that Rachel believes in the correctness of certain
laws of nature.
We will first illustrate this idea by the help of an example before turning to the main
statement of this chapter: the claim that a rational agent would assign the probability
P (x) = |hx| i|2 to the outcome x described by a measurement on a quantum state
w.r.t. a set of projectors {|xi}x2X .
7.1
Operational definition of subjective probabilities
Most of the ideas of this chapter are independent of the framework and discuss generally
how probabilities can be defined operationally. The crucial idea is that probability assignments result from an agent’s beliefs about rationality for which it is meaningless to ask if
they are true or wrong. Additionally, an agent may also believe in the correctness of certain laws of nature for which the possibility of falsification may indeed make sense. When
we say that an agent believes in the correctness of such a law, we actually mean that he
believes that it is his best available description of nature. In general he will be convinced
that there is a better description – either (i) because he lacks certain information or (ii)
because he believes that the theory will be replaced by a better one. An example illustrating the first situation is thermodynamics. It is an emergent theory in the sense that its
rules result from averaging over the unknown particle distributions. Regarding the second
point we discussed in Chapter 1 that it is actually the very idea of the scientific method
that theories are falsified and replaced with better theories. Thus, any agent believing in
the scientific method of falsification is sure that any theory is wrong. We also stress at
this point that the idea is not that the agent assigns probabilities quantifying his belief
that a theory is correct. In fact, it was pointed out by Popper that seeking theories that
are highly probable is a mistake.
87
7. RETRIEVING THE SUBJECTIVE BORN RULE
“... I explained why it is a mistake to conclude from this that we are interested
in highly probable theories. I pointed out that the probability of a statement
(or set of statements) is always the greater the less the statement says: it is
inverse to the content or the deductive power of the statement, and thus to its
explanatory power.”
Karl Popper [6]
Within the framework theories are about such laws of nature. In particular, as discussed in the previous chapters, they forbid certain stories about experiments. In order to
introduce probabilities in the framework the idea is to assume that an agent who believes
in the correctness of a theory, in the sense explained above, will assign probability zero
to stories that are forbidden by that theory. His probability assignments to events then
result from the combination in his beliefs about rationality and the correctness of certain
laws of nature (see Figure 7.1).
Let us now operationally define probabilities within the framework. The idea is to
assign probabilities to the outcomes x 2 range(x̂) of a physical quantity x̂ as follows.
Definition 11. The probability PX (x) of an outcome x 2 range(x̂) is defined as the
maximal amount of money a rational agent is willing to pay for the following game: x̂
is measured once – if outcome x occurs the agent obtains $ 1 and otherwise nothing.
Note that the idea is that at the moment when the agent places the bet he does not know
whether he will eventually change sides with the bookkeeper. This will ensure for example
that his probability assignments must be positive.
As discussed above this definition of probabilities is not yet very meaningful, because
a priori the numbers PX (x) may be anything. In order to arrive at further restrictions we
will introduce a gambler, Rachel, and see how certain rational beliefs of hers influence her
probabilities. So far, Rachel does not have any beliefs and thus, nothing even prevents
her from paying more that $ 1 to accept the bet. However, in this case Rachel would lose
money with certainty and this seems irrational to her. From the Dutch book argument
it follows that the distribution PX must satisfy the three axioms of probability theory, if
Rachel wants to avoid bets that result in losses with certainty.
The Dutch book argument
A Dutch book is a list of bets that guarantee a nett loss for the agent. An agent is called
rational if the amount of money he is willing to bet does not allow for such a Dutch book.
88
7.1 Operational definition of subjective probabilities
Laws about nature
Rational beliefs
For example:
For example:
- Energy conservation
- Bets that result in certain losses
- Non-signalling principle
are irrational (no Dutch book)
- General relativity
- Choice of sample space
- Ptolemey’s planetary model
- Beliefs about properties
- Determinism
of repeated trials
- Objective Born rule (BornObj)
- Beliefs about correctness of laws
- ...
- ...
Probability assignments
- P (“heads”) =
- P( ) =
1
2
1
6
- PX (x) = |hx| i|2
- ...
Figure 7.1: Probability assignments resulting from beliefs in laws of nature and
rationality. Whereas laws about nature can be tested (experimentally or by argument) it
is not meaningful to question whether beliefs about rationality are correct.
89
7. RETRIEVING THE SUBJECTIVE BORN RULE
The Dutch book argument states that if the prices an agent is willing to pay for a list of
bets violate the axioms of probability theory then there exists a Dutch book against him.
The axioms of probability theory are taken to be the laws introduced by Kolmogorov [70].
He begins with a collection of elementary events ⌦, an algebra F of subsets on ⌦ (with
elements called random events) and a real valued function P : F ! R on F. The laws of
probability theory are then the following three axioms about P .
The axioms of probability theory
(I) Non-Negativity: P (x)
0,
8x2F
(II) Finite Additivity: P (x1 [ x2 ) = P (x1 ) + P (x2 ),
8 x 1 , x2 2 F : x 1 \ x 2 = ;
(III) Normalisation: P (⌦) = 1
The Dutch book argument
Let P (x) be a probability assignment to an event x in a sample space ⌦ with the
interpretation that an agent is willing to pay $ P (x) for the bet in which he obtains
$ 1 in the case that x occurs and nothing otherwise. If the agent is rational, i.e., if
the assignments do not allow for a Dutch book against him, they satisfy the axioms
of probability theory (I)-(III).
Note that alternative proofs can be found in the literature (see for instance [71]).
Proof.
(I) Non-Negativity:
Let x̄ be the event “not x”. We first observe that P (x̄)  1 holds, because otherwise
the corresponding bet would result in loss with certainty. Assume now that the
agent is o↵ered two bets: One where he receives $ 1 if x occurs and one where he
receives $ 1 if x̄ occurs. Let P (x) and P (x̄) be the amount of money he is willing to
pay for each of the bets. Additionally, assume that after he placed his bets, he may
change sides with the bookkeeper.
90
7.1 Operational definition of subjective probabilities
event
paid
received in
case of success
x
$ P (x)
$1
x̄
$ P (x̄)
$1
total amount paid
$ P (x) + P (x̄)
total amount received
$1
If the agent is rational the total amount of money paid from the agent should be
smaller than the amount he wins
P (x) + P (x̄)  1.
Because he does not know whether he changes sides with the bookkeeper it actually
follows that
P (x) + P (x̄) = 1
and therefore,
)
P (x) = 1
P (x̄)
0
holds, where we used that P (x̄)  1.
(II) Finite Additivity:
Let us first observe that it follows from the proof in (I) that
P (x1 [ x2 ) + P ((x1 [ x2 )) = 1
(7.1)
holds. Consider now the following list of bets.
event
paid
received in
case of success
x1
$ P (x1 )
$1
x2
$ P (x2 )
$1
(x1 [ x2 )
$ P ((x1 [ x2 ))
$1
total amount paid
$ P (x1 ) + P (x2 ) + P ((x1 [ x2 ))
total amount received
$1
Like before a rational agent would avoid bets that result in certain losses, i.e.,
P (x1 ) + P (x2 ) + P ((x1 [ x2 )) = 1.
91
7. RETRIEVING THE SUBJECTIVE BORN RULE
Using Equation (7.1) it follows that
P (x1 [ x2 ) = P (x1 ) + P (x2 ).
(III) Normalisation:
For any event x we have
⌦ = x [ x̄.
Using Axioms (I) and (II) we find
1 = P (x) + P (x̄)
= P (x [ x̄)
= P (⌦).
7.2
Rational beliefs about repeated experiments
So far we only discussed “single shot bets” and concluded from the Dutch book argument that Rachel would assign probabilities satisfying the axioms of probability theory.
In order to derive more interesting constraints on her probability assignments we will now
consider the case where Rachel is o↵ered the bet repeatedly. Note that a priori we do
not impose any constraints on Rachel’s beliefs about the structure of these bets – she
may assign di↵erent probabilities in each run and in particular, she may also make her
probability assignments dependent on the outcomes of previous runs. In the following we
will consider scenarios in which it seems reasonable for Rachel to believe that the outcome
of an individual run does not depend on previous outcomes. We will discuss how the
resulting properties of the distribution can be used in combination with certain laws of
nature to derive actual probability assignments, which in turn can serve as a guide for
rational behaviour.
First, we will illustrate the idea by the help of an example, in which Rachel goes to
the casino to play roulette. The casino has di↵erent odds every day and Rachel believes
that it is a law of nature that there is a critical value of these odds, for which she would
be deemed to run out of money with certainty, if she would continue playing until she has
no money left. We will show that it follows that it is unreasonable for Rachel to go to the
casino at all on days for which the odds are worse than this critical bound – even if she
would intend to return home before she runs out of money.
After this illustrating example, we will turn to the main result of this chapter, which is
to show that a rational agent would assign probabilities P (x) = |hx| i|2 to the outcomes
of a measurement on a quantum state w.r.t. a set of projectors {|xi}x2X . Because the
assumptions about the structure of the probability distribution are identical to the ones
from the casino example, we only have to consider a di↵erent law of nature which will be
the objective Born rule (BornObj) in this case.
92
7.3 Example: Is it rational to go to the casino?
7.3
Example: Is it rational to go to the casino?
Rachel goes to the casino to play roulette. At the beginning she has $ m0 to start and in
each round she bets on either “red” or “black”. The casino has fixed the odds such that a
player has to pay $ p in each round where p 2 [0, 1], with the value of p varying each day.
Considering only the axioms of probability theory Rachel cannot draw any conclusions
about on which days it is reasonable to go to the casino and when it is better to stay at
home. Intuitively it is clear to her however, that as the value of p increases it becomes
less rational to play.
In order to find out whether it is a good day to play roulette, she describes her visit
to the casino within our framework as an experiment, termed ExpGamble, with a physical
quantity t̂ with range(t̂) = N+ , indicating the time Rachel has already spent gambling
(more precisely, we take t to be the number of rounds she has played) and a physical
quantity m̂ with range(m̂) = R+ corresponding to Rachel’s amount of money (we assume
that Rachel cannot make debts). An additional physical quantity p̂ 2 [0, 1] describes the
odds fixed by the casino on this day.
We may tell various possible stories about Rachel’s success in ExpGamble.
sloser =
8
>
<“Rachel starts at time 0 with $ 100. On this
day the casino has set the odds to p = 0.5.
>
: Rachel plays until he has no money left, but
never wins.”
or
swinner01 =
8
“Rachel starts at time 0 with $ 100. On this
>
>
>
< day the casino has set the odds to p = 0.1.
Rachel keeps playing. Sometimes she wins and
>
>
>
: sometimes she loses. But she never runs out
of money.”
or
swinner09 =
8
“Rachel starts at time 0 with $ 100. On this
>
>
>
< day the casino has set the odds to p = 0.9.
Rachel keeps playing. Sometimes she wins and
>
>
>
: sometimes she loses. But she never runs out
of money.”
or
shome =
8
“Rachel starts at time 0 with $ 100. On this day
>
<
the casino has set the odds to p = 0.9. Rachel
>
: plays 1000 times and always wins. She then
stops playing and returns home.”
93
7. RETRIEVING THE SUBJECTIVE BORN RULE
The corresponding trajectories are
sExpGamble
= {(p, t, m) 2 [0, 1] ⇥ N+ ⇥ R+ : p = 0.5, m = $ 100
loser
0.5 · t, t  200}
+
+
sExpGamble
winner04 = {(p, t, m) 2 [0, 1] ⇥ N ⇥ R : p = 0.1, m > 0}
+
+
sExpGamble
winner06 = {(p, t, m) 2 [0, 1] ⇥ N ⇥ R : p = 0.9, m > 0}
sExpGamble
= {(p, t, m) 2 [0, 1] ⇥ N+ ⇥ R+ : p = 0.9, m = $ 100 + 0.1 · t, t  1000}.
home
Rachel believes that whether or not these stories describe a possible reality depends
on the value of p. Intuitively she believes for instance that while it is possible to win on
the long run for p = 0.1 she will be deemed to go home at some point for p = 0.9. Thus,
for her the story swinner01 describes a possible reality while the story swinner09 does not.
More precisely, Rachel believes that there is a critical value p0 such that if she would
decide to continue playing until she has no money left, she would run out of money at
some point for all days with p p0 . Within the framework this law can be expressed as
follows.
Definition 12. When we say that a physical theory T satisfies the (CasinoRule) then
this means that T asserts the existence of a value p0 2 (0, 1] such if the odds are fixed
to p p0 and if an agent intends to play until he has no money left, then he is deemed
to lose, i.e., the condition
9 (p
p0 , t, m) 2 sExpGamble 8 t 2 N+ : m > 0
=)
s2T
holds.
For instance Rachel may believe that the (CasinoRule) is valid for p0 = 0.5 because this
is the setting in “usual” casinos and she is convinced that real casinos will always win on
the long run. In this case the story swinner09 would be forbidden. However, it does not
directly follow that it is unreasonable to play roulette even on those days when the odds
are fixed to p > p0 , because Rachel is free to go home while she has still money left. In
particular, the story shome is not forbidden by (CasinoRule) for any p and thus, it may
describe a possible scenario.
In the following we will show that if Rachel believes that the (CasinoRule) is valid for
some p0 , then it is irrational to play on any day with p > p0 . More precisely, we will show
that it follows from the (CasinoRule) that we can introduce a probability distribution P
on the events “success” and “loss” with the property P (“success”)  p0 .
94
7.3 Example: Is it rational to go to the casino?
In order to be able to introduce a probability distribution, we first have to define the
sample space and a map between stories and events of this sample space. Assumption
(SampleSpace) basically states that Rachel believes that in each round she will either win
or lose. Thus, she excludes certain events by assumption such as for example, that the
roulette board is destroyed or that the sun blows up. She also believes that in principle
she could continue playing forever, or in other words, that there is no law of nature setting
an upper bound on the number of rounds she can play.
Rational belief 1. (SampleSpace). An agent believing in the assumption (SampleSpace)
will chose the sample space to be
⌃ := {(x, m0 , p) 2 {1, 0}⇤ [ {1, 0}1 ⇥ R+ ⇥ [0, 1]},
where xt indicates success (xt = 1) or loss (xt = 0) in the t-th round and m0 is the
starting money according to a story s with trajectory
sExpGamble (x, m0 , p) := {(p, t, m) : m = m0 +
t
X
(xi
p)}.
(7.2)
i=1
Hence, we may treat x, p and the initial amount of money m0 as random variables
X = X1 . . . Xn , P and M0 and assign probabilities to them.1
More precisely, Rachel assigns probabilities PX,M0 ,P (x, m0 , p) to sequences x 2 {0, 1}⇤ [ {0, 1}1 ,
where |X| is the time when she goes home, i.e., the number of rounds she played. Note
that |X| = 1 describes a fictive story in which Rachel never goes home – even though this
obviously refers to a scenario which is physically impossible, this story may be told (in
finite time). The operational interpretation of PX|M0 =m0 ,P =p (x) according to Definition 11
is that it corresponds to the amount of money Rachel is willing to pay for the bet in which
she obtains $ 1 in the case that her visit to the casino is described by the sequence x and
nothing otherwise (given that Rachel startet with $ m0 and the casino has fixed the odds
to be $ p).
As a property of the mathematical definition of probabilities we can take the partial
trace for any x such that |x| < 1
PX1 (1) = trX2 ...Xn PX1 X2 ...X|X| ||X|=n (1).
The operational interpretation of this probability distribution is that it corresponds to the
amount of money Rachel is willing to pay for the bet in which she obtains $ 1 in the case
1
This is possible because even though the underlying set has infinitely many elements only countable
many of them correspond to stories, because, as we explained in Chapter 3, the requirement that stories
are finite implies that there are only countable many stories.
95
7. RETRIEVING THE SUBJECTIVE BORN RULE
that she wins in the first round (knowing that there are n
goes home) and nothing otherwise.
1 rounds to follow before she
Next, we need a rule to express a law of nature, such as our (CasinoRule), as a property
of the probability distribution. For a general rule (Rule) within a theory T an agent
believing the correctness of this rule will assign probabilities as follows.
Rational belief 2. (Correctness). Let (Rule) be a law of a theory T defining a set of
forbidden trajectories FExp for an experiment Exp, i.e.
(Rule) : sExp 2 FExp
=)
s 2 T.
A rational agent believing in the (Correctness) of (Rule) assigns probabilities such that
PX (x) = 0
8 x : sExp (x) 2 FExp .
As a result of the (CasinoRule) and the (Correctness) assumption we can restrict ourselves for p p0 to finite sequences x 2 {1, 0}⇤ in the following, i.e.,1 for all x 2 {1, 0}1
PX|P
p0 (x)
=0
(7.3)
holds. In order to state Rachel’s remaining beliefs we introduce an additional event,
denoted by ⌦, indicating that Rachel decided to continue playing until she has used up
all her money. Under the condition ⌦ and p p0 it then holds that
|X|
X
⇥
P M0 +
Xt
t=1
Note that P [|X| < 1 P
law (CasinoRule) is correct.
p · |X| =
6 0 ⌦, P
⇤
p0 = 0.
(7.4)
p0 ] = 1 holds according to the assumption that the casino
Rachel has two more beliefs about gambling. First she believes that the outcomes up
to time t do not depend on how long she intends to continue playing afterwards nor on
the odds fixed by the casino. Mathematically this can be expressed as follows.
Rational belief 3. (Repetitions). For all t, x 2 {1, 0}⇤ , m0 , p 2 [0, 1] such that
|x| t and PX t 1 ,M0 ,P,⌦ (x, m0 , p) > 0 it holds that
1
PXt |X t
1
1
1
=xt1
1
= PXt |X t
1
1
=xt1
1
,M0 =m0 ,P =p,⌦
.
Note that in the proof this will be used whenever we restrict sums to be finite.
96
7.3 Example: Is it rational to go to the casino?
Here we used the notation X1t
1
:= X1 . . . Xt
1.
Finally, Rachel also believes that her bet on being successful at time t does not depend
on t. Formally this is expressed as the following condition.
Rational belief 4. (Indi↵erence). For all x 2 {1, 0}⇤ , m0 , p and t, t0 2 {1, . . . , p1 · m0 }
it holds that
PXt |M0 =m0 ,P =p,⌦ = PXt0 |M0 =m0 ,P =p,⌦ .
Note that the requirement t, t0 2 {1, . . . , p1 · m0 } ensures that the distributions are
well-defined, because the condition ⌦ (7.4) makes sure that
m0 +
|x|
X
p · |x| > 0
xt
t=1
=) |x|
1
· m0
p
holds.
We are now ready to formally state our claim that it is irrational to go to the casino on
the days with odds p > p0 even if Rachel intends to go home before she runs out of money.
According to the operational definition of probabilities this is expressed as the fact that
PX1 (1)  p0 because it means that it is irrational to pay more than $ p0 for the bet in
which Rachel obtains $ 1 in the case of success and nothing otherwise, which corresponds
to the rules fixed by the casino.
Proposition 1. The amount of money Rachel is ready to pay for a bet with payout
$ 1 if she wins in the first round is bounded by
PX1 (1)  p0
provided that Rachel has the four rational beliefs (SampleSpace), (Correctness) of
(CasinoRule), (Repetitions) and (Indi↵erence).
⇤ over infinitely long sequences X̄ = (X̄ )
Proof. We define a distribution PX̄
t t2N via
8 t, x 2 {1, 0}⇤ s.t. , |x|
PX̄⇤
t 1
=xt1 1
t |X̄1
:= PXt |X t
1
t and PX t 1 (xt1 1 ) > 0 :
1
1
=xt1 1
.
Using the (Repetitions) assumption, this may be expressed as:
8 t, m0 , x 2 {1, 0}⇤ s.t. |x|
PX̄⇤
t 1
=xt1 1
t |X̄1
= PXt |X t
1
1
=xt1
t and PXt1
1
1 ,M0 ,P,⌦
,M0 =m0 ,P =p0 ,⌦
97
.
(xt1 1 , m0 , p0 ) > 0 :
(7.5)
7. RETRIEVING THE SUBJECTIVE BORN RULE
Let now m̂0 be such that PM0 ,⌦ (m̂0 , p0 ) > 0. Let furthermore ⇥n , for any n 2 N, be the
set of sequences x = (x1 , . . . , xn ) of length n defined by
x 2 ⇥n
()
8t 2 {0, . . . , n} : m̂0 +
t
X
xr
r=1
p0 · t > 0 .
Under the condition ⌦
M0 +
|X|
X
p0 · |X| = 0
Xt
t=1
holds with certainty and therefore, we also have for any n 2 N
|X|
X1n
()
n
1
2 ⇥n
(7.6)
1
under the condition ⌦ ^ (M0 = m̂0 ) ^ (P = p0 ). Using Eq. (7.6) it follows that Eq. (7.5)
can be expressed as (replacing t by n)
8 n, x 2 {1, 0}⇤ s.t. xn1
PX̄⇤
n 1
1
=xn
n |X̄1
1
1
= PXn |X n
2 ⇥n
1
1
=xn
1
1
1
and PX n
1
1
n 1
, m̂0 , p0 )
,M0 ,P,⌦ (x1
,M0 =m̂0 ,P =p0 ,⌦
>0:
.
(7.7)
Next we show that the following relation holds
(7.7)
=)
8 t, x 2 {1, 0}⇤ s.t. xt1
1
2 ⇥t
1
:
PX̄⇤ t = PX1t |M0 =m̂0 ,P =p0 ,⌦ .
(7.8)
1
This can be seen as follows. We will apply Eq. (7.7) for n 2 {1, . . . , t} using
xt1 1 2 ⇥t 1 =) xn1 1 2 ⇥n 1 for any n  t. It is the requirement PX n 1 ,M0 ,P,⌦ (xn1
1
we have to be careful about.
1
, m̂0 , p0 ) > 0
We first note that it follows from (7.7) and PM0 ,P,⌦ (m̂0 , p0 ) > 0 that for all x1
PX̄⇤ 1 = PX1 |M0 =m̂0 ,P =p0 ,⌦
(7.9)
holds.
Now we observe that if the marginal is equal to zero the joint distribution has to vanish,
too
PX1 |M0 =m̂0 ,P =p0 ,⌦ (x1 ) = 0
=)
PX1t M0 =m̂0 ,P =p0 ,⌦ = 0
and
PX̄⇤ 1 (x1 ) = 0
=)
PX̄⇤ t = 0 .
1
Therefore, the equality (7.8) is satisfied in this case
PX̄⇤ t = PX1t ,M0 =m0 ,P =p0 ,⌦ = 0 8 x1 : PX1t |M0 =m̂0 ,P =p0 ,⌦ (x1 ) = 0.
1
98
7.3 Example: Is it rational to go to the casino?
If x1 is such that PX1t |M0 =m̂0 ,P =p0 ,⌦ (x1 ) > 0 it follows that PX1 ,M0 ,P,⌦ (x1 , m̂0 , p0 ) > 0,
hence in this case we can write
PX̄⇤ t = PX1t |M0 =m̂0 ,P =p0 ,⌦
1
()
PX̄⇤ t |X̄1 =x1 · PX̄⇤ 1 (x1 ) = PX2t |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ · PX1 |M0 =m̂0 ,P =p0 ,⌦ (x1 )
2
8 x1 : PX1 ,M0 ,P,⌦ (x1 , m̂0 , p0 ) > 0
and because of Eq. (7.9) we are left to show that
PX̄⇤ t |X̄1 =x1 = PX2t |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦
2
8 x1 : PX1 M0 =m̂0 ,P =p0 ,⌦ (x1 , m̂0 , p0 ) > 0
holds.
Because we assume that PX1 ,M0 ,P,⌦ (x1 , m̂0 , p0 ) > 0 holds, we can apply (7.7) again, yielding that for all x2
PX̄⇤ 2 |X̄1 =x1 (x2 ) = PX2 |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ (x2 )
(7.10)
holds.
Just like before we first consider the case that x2 is such that PX2 |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ (x2 ) = 0
(and therefore also PX̄⇤ |X̄ =x (x2 ) = 0) and observe that if the marginal vanishes, then so
2
1
1
does the joint distribution
PX̄⇤ t |X̄1 =x1 (x2 ) = PX2t |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ (x2 ) = 0
2
8 x2 : PX2 |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ (x2 ) = 0.
If x2 is such that PX2 |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ (x2 ) > 0 it follows that PX2 ,X1 ,M0 ,P,⌦ (x1 , x2 , m̂0 , p) > 0,
hence in this case we can write
PX̄⇤ t |X̄1 =x1 = PX2t |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦
2
()
PX̄⇤ t |X̄2 =x2 ,X̄1 =x1 · PX̄⇤ 2 |X̄1 =x1 (x2 ) = PX3t |X1 =x1 ,X2 =x2 ,M0 =m̂0 ,P =p0 ,⌦ · PX2 |X1 =x1 ,M0 =m̂0 ,P =p0 ,⌦ (x2 )
3
8 x2 : PX1 ,X2 ,M0 ,P,⌦ (x1 , x2 , m̂0 , p0 ) > 0 .
Using Eq. (7.10) we are left to show the following relation
PX̄⇤ t |X̄1 =x1 ,X̄1 =x1 = PX3t |X1 =x1 ,X2 =x2 ,M0 =m̂0 ,P =p0 ,⌦
3
Going on like this we find the desired result (7.8).
99
8 x1 , x2 : PX1 ,X2 ,M0 ,P,⌦ (x1 , x2 , m̂0 , p0 ) > 0 .
7. RETRIEVING THE SUBJECTIVE BORN RULE
We have that t  m̂0 /p0 =) xt1
1
2 ⇥t
8 t s.t. t  m̂0 /p0 :
and therefore, we can write (7.8) as
1
PX̄⇤ t = PX1t |M0 =m̂0 ,P =p0 ,⌦ .
1
The only requirement was PM0 ,P,⌦ (m̂0 , p0 ) > 0 hence we can rewrite this as the condition
8 m0 , t s.t. t  m0 /p0 and PM0 ,P,⌦ (m0 , p0 ) > 0 :
PX̄⇤ t = PX1t |M0 =m0 ,P =p0 ,⌦
1
which can be rewritten as
8t:
PX̄⇤ t = PX1t |M0
1
p0 ·t,P =p0 ,⌦
,
(7.11)
where we assumed that
P [M0
p0 · t] > 0
8t
holds, i.e., that Rachel does not have an upper bound on the amount of money she will
bring to the casino.
Next we show that this implies
8t:
PX̄⇤ t = PX1 |M0
p·t,P =p0 ,⌦ .
This can be seen as follows.
Taking the marginal we find from (7.11)
8t:
PX̄⇤ t = PXt |M0
p0 ·t,P =p0 ,⌦
.
Because 1, t  M0 /p0 holds we can apply assumption (Indi↵erence)
PXt |M0
p0 ·t,P =p0 ,⌦
= PX1 |M0
1
p0 ·t,P =p0 ,⌦
yielding
8t:
PX̄⇤ t = PX1 |M0
p0 ·t,P =p0 ,⌦ .
If we assume that Rachel does not know an upper bound on the amount of money she
brings to the casino, i.e.,
⇥
⇤
P M0 p0 · t ⌦ = 1 8 t,
1
PXt |M0
t/p0 ,P =p0 ,⌦
=
X
PXt |M0 =m0 ,P =p0 ,⌦ =
p0 ·t:
PM0 ,P,⌦ (m0 ,p0 )>0
m0
X
PX1 |M0 =m0 ,P =p0 ,⌦ = PX1 |M0
p0 ·t:
PM0 ,P,⌦ (m0 ,p0 )>0
m0
100
p0 ·t,P =p0 ,⌦ .
7.3 Example: Is it rational to go to the casino?
we obtain from the assumption (Repetitions)
X
PX1 |M0 p0 ·t,P =p0 ,⌦ =
PX1 |M0 =m0 ,P =p0 ,⌦
m0 p0 ·t:
PM0 ,P,⌦ (m0 ,p0 )>0
X
=
m0 p0 ·t:
PM0 ,P,⌦ (m0 ,p0 )>0
⇥
PX 1 = PX 1 · P M 0
⇤
p0 · t ⌦ = PX1
yielding
PX̄⇤ t = PX1 .
8t:
(7.12)
Assume now that PX1 (1) > p0 holds. Then a symmetrisation argument implies
8 m0 9
⇥
P ⇤ 8n 2 N : m0
>0:
n · p0 +
n
X
X̄r > 0
r=1
⇤
.
In order to show Eq. (7.13) we define the random variable Fn via
Fn := {i  n : X̄i = 1}.
This allows us to write
P
⇤
⇥
p0 · n +
n
X
r=1
X
⇤
⇥
⇤
X̄r > 0 = P ⇤ Fn > p0 · n =
PFn (f ) .
f >p0 ·n
Now we define the symmetric probability distribution for all fixed n via
PX̃1 ...X̃n :=
1 X ⇤
P
n! ⇡ X̄⇡(1) ...X̄⇡(n)
for which
PX̃1 = PX⇤ 1
holds.
Therefore, we can write for any n
PX⇤ 1 (1) = PX̃1 (1)
=
n
X
f =0
PF̃n (f )PX̃1 |F̃n =f (1),
where
F̃n := {i  n : X̃i = 1}.
It holds that
PF̃n = PFn
101
(7.13)
7. RETRIEVING THE SUBJECTIVE BORN RULE
because
PF̃n (f ) =
X
PX̃1 ...X̃n (x)
x: |{i: xi =1}|=f
=
=
=
=
=
1 X
n! ⇡
X
PX̄⇤
x: |{i: xi =1}|=f
1 X
n! ⇡
X
PX̄⇤ 1 ...X̄n (x⇡
1 X
n! ⇡
1 X
n! ⇡
⇡(1) ...X̄⇡(n)
(x)
1 (1)
. . . x⇡
1 (n)
)
x: |{i: xi =1}|=f
X
PX̄⇤ 1 ...X̄n (y)
y: |{i: y⇡(i) =1}|=f
X
PX̄⇤ 1 ...X̄n (y)
y: |{i: yi =1}|=f
X
PX̄⇤ 1 ...X̄n (y) .
y: |{i: yi =1}|=f
Therefore, we have
PX⇤ 1 (1)
=
n
X
f =0
PF̄n (f )PX̃1 |F̄n =f (1) .
(7.14)
Now we observe that
PX̃1 |Fn =f (1) =
f
n
holds. This can be seen as follows.
PX̃1 |Fn =f (1) =
X
x̄: x̄1 =1
|{in: x̄i =1}|=f
PX̃1 ...X̃n |Fn =f (x̄) .
From the definition it follows that PX̃1 ...X̃n (x̄) is uniformly distributed on all sequences
with the same frequency. Thus, for any x̄ such that |{i  n : x̄i = 1}| = f
✓ ◆
n
PX̃1 ...X̃n |Fn =f (x̄) =
f
holds. There are
n 1
f 1
1
such sequences with fixed x1 = 1. Therefore, we find
PX̃1 |Fn =f (1) =
✓
n
f
102
1
1
◆✓ ◆
n
f
1
=
f
.
n
7.3 Example: Is it rational to go to the casino?
Going back to Eq. (7.14) we find
PX⇤ 1 (1) =
=
n
X
f =0
n
X
PFn (f )PX̃1 |Fn =f (1)
PFn (f )
f =0
X
f
n
X
f
f
+
PFn (f )
n
n
f p0 ·n
f >p0 ·n
X
X
<
PFn (f ) · p0 +
PFn (f ) · 1
=
(?)
PFn (f )
f p0 ·n
< p0 + P ⇤
n
⇥X
r=1
f >p0 ·n
⇤
X̄r > 0 .
Therefore, we obtain assuming PX1 (1) > p0 and using PX⇤ 1 (1) = PX1 (1) as well as the fact
that the result is valid for all n
⇥
P ⇤ 8n 2 N : m0
p0 · n +
n
X
r=1
⇤
X̄r > 0 > 0.
The left hand side is independent of n and therefore, there exists a
n) with
n
X
⇥
⇤
P ⇤ 8n 2 N : m0 p0 · n +
X̄r > 0
.
> 0 (independent of
r=1
This shows Eq. (7.13).
Let us now return to Eq. (7.8). For a fixed m̂0 such that PM0 ,P,⌦ (m̂0 , p0 ) > 0 we found
that
8 t, x 2 {1, 0}⇤ s.t. xt1
1
2 ⇥t
1
:
PX̄⇤ t = PX1t |M0 =m̂0 ,P =p0 ,⌦
1
(7.15)
holds. From this we conclude that
8 t, x 2 {1, 0}⇤ s.t. xt1 1 2 ⇥t 1 :
⇥
⇤
⇥
P ⇤ X̄1t 1 2 ⇥t 1 = P X1t 1 2 ⇥t
1 | ⌦, M0
⇤
⇥
= m̂0 , P = p0 = P |X|
t | ⌦, M0 = m̂0 , P = p0
(7.16)
holds, where we have used (7.6).
Note also that the left hand side is lower bounded by
⇥
P ⇤ X1t
1
2 ⇥t
1
⇤
⇥
P ⇤ 8 n 2 N : m̂0
103
p0 · n +
n
X
r=1
⇤
X̄r > 0 .
⇤
7. RETRIEVING THE SUBJECTIVE BORN RULE
However, as we argued before in Eq. (7.13), if we assume by contradiction PX1 (1) > p0
then this value is strictly larger than zero, i.e.,
⇥
⇤
P |X| t | ⌦, M0 = m̂0 , P = p0
(7.17)
holds for arbitrarily large t. But because under the condition P = p0 it follows from
(Correctness) that |X| is always finite, i.e., for all m0 we have
X ⇥
⇤
1=
P |X| = t|M = m̂0 , P = p0 , ⌦
t2N
⇥
⇤
= lim P |X| = t|M = m̂0 , P = p0 , ⌦
t!1
⇥
⇤
= lim 1 P |X| = t|M = m̂0 , P = p0 , ⌦
t!1
⇥
⇤
= 1 P 8 t : |X| =
6 t|M = m̂0 , P = p0 , ⌦
=)
⇥
⇤
P 8t : |X| > t M = m̂0 , P = p0 , ⌦ = 0 .
Hence, we obtain a contradiction to Eq. (7.17). This concludes the proof.
7.4
Main statement
In Chapter 6 we derived the objective Born rule (BornObj) as a law of nature. Informally,
it says that stories about a quantum measurement that is repeated an arbitrary number of
times are forbidden, if they say the frequencies do not converge to |hx| i|2 . Let us recall
the formal statement.
Definition. When we say that a physical theory T satisfies (BornObj) then this means
that T forbids all stories s according to which for some MeasFreq 2 Qn,H,{⇡x }x2X the
frequency of an outcome x does not approach |hx| i|2 , even if the number of repetitions
n is arbitrarily large, must be forbidden. More precisely, it means that the following
condition holds
lim sup
n!1
n
(s) > µ
=)
s2T
for all µ > 0, where
n
(s) = sup{|fx
x
with x 2 X ,
|hx| i|2 | : (
⌦n
2 H and lim sup xn := inf sup xk .
n!1
n2N k n
104
µ
, n, f ) 2 sMeasFreq }
7.4 Main statement
Here we will show that an agent believing in the correctness of this rule will assign
probabilities to the measurement outcomes that are given by |hx| i|2 .
Theorem 2. Any probability assignment P to x that is compatible with the rational
beliefs (SampleSpace), (Correctness) of (BornObj), (Repetitions) and (Indi↵erence) must
be of the form PX (x) = |hx| i|2 .
Proof. We consider an experiment analogous to ExpGamble in which Rachel goes to a
“quantum-casino” in which a system described by a quantum state is measured w.r.t. a
set of projectors {|xi}x2X . Let us fix one outcome x and assume that in each round Rachel
has to pay $ p and that she will win $ 1 if the outcome is x an nothing otherwise. Like
in the previous example the odds p 2 [0, 1] fixed by the casino vary each day. Rachel now
reasons on which days it is rational to go to the casino and when she better stays at home.
In contrast to the previous example she does not believe anymore in the law (CasinoRule)
but instead in (BornObj). Informally, this law states that if she would continue playing the
frequencies of outcome x would converge to |hx| i|2 . Unfortunately this observation alone
does not yet help Rachel to make a decision. In the following we will discuss how Rachel
comes to the conclusion that it is irrational to go to the casino on days with p > |hx| i|2 .
As a first step of the proof we will show that it follows from (BornObj) that if Rachel
would decide to stay until she runs out of money, she would be deemed to go home on
those days when the odds are fixed to p |hx| i|2 . This means that we have identified
(BornObj) with (CasinoRule) for p0 = |hx| i|2 and can proceed like in the proof from the
previous example. However, it also means that we have only established P (x)  |hx| i|2
and not yet equality as desired.
In order to show that actually P (x) = |hx| i|2 holds we have to consider another scenario in which Rachel does not win if the outcome is x but instead she wins if the outcome
is not x. Proceeding as in the case in which she wins if the outcome is x it then follows
that P (x) |hx| i|2 must hold and thus we find the desired equality. Let us start with
the case in which Rachel wins if the outcome is x.
Because the experiment has the same physical quantities as in the previous example
we will also call it ExpGamble. Recall that the physical quantities include the number of
rounds t̂ with range(t̂) = N+ , Rachel’s amount of money m̂ with range(m̂) = R+ , as well
as p̂ with range(p̂) = [0, 1] corresponding to the odds fixed by the casino.
Like before Rachel expresses her beliefs about the sample space, e.g., that in each round
she will either lose or win and that there is no law of nature imposing an upper bound on
the number of rounds she can play, by the assumption (SampleSpace). Rachel thus assigns
105
7. RETRIEVING THE SUBJECTIVE BORN RULE
to every trajectory sExpGamble a unique sequence x 2 {1, 0}⇤ [ {1, 0}1 of numbers which
indicate whether Rachel wins (xt = 1) or loses (xt = 0) in the tth round. This allows
Rachel to assign probabilities PX,M0 ,P (x, m0 , p) to the sequences x describing her success
in the casino.
The problem is now that (BornObj) applies to stories about the experiment MeasFreq
and not to ExpGamble. Recall that in MeasFreq an increasing number of copies of the
quantum state was measured w.r.t. the set of projectors {|xi}
and that the physical
Lx2X
1
⌦n , the number of
quantities included the quantum state
2 H, with H =
H
n=1
|X
|
copies n 2 N as well as the frequency distribution f 2 [0, 1] . In order to be able to
apply (BornObj) to the experiment ExpGamble (with events corresponding to triples of the
form (t, m, p)) we have to find a function that maps this experiment to an experiment
MeasFreq 2 Qn,H,{⇡µ } (with events corresponding to triples of the form ( , n, f )). The
f,n
state is fixed and the number of repetitions n simply corresponds to the time t. In order
to determine the frequency of the outcome x, we observe that for each such event Rachel
would win $ p whereas she loses $ 1 otherwise. Hence, after t rounds Rachel has $ m with
m = m0 + fx t t. Therefore, the frequency of outcome x is given by f = 1t (m m0 + p)
and the assumption that we can apply (BornObj) to the experiment ExpGamble can be
expressed as the condition
9 MeasFreq 2 Qn,H,{⇡f,n } :
( , t, f =
1
(m
t
m0 + p)) 2 sMeasFreq () (t, m, p) 2 sExpGamble ,
where the projectors ⇡f,n are given in Eq.(6.5).
The (Correctness) assumption can then be mathematically formulated as the condition
t
X
⇥
1
P lim sup
(m
t
t!1
i=1
with
⇤
m0 + p)) > p0 = 0 8 µ > 0
(7.18)
p0 := |hx| i|2 + µ.
Recalling that we defined in Eq. (7.2) the following map between the random variable X,
characterising Rachel’s success in the casino, and stories about ExpGamble
s
ExpGamble
(x, m0 , p) := {(p, t, m) : m = m0 +
t
X
(xi
p)}
i=1
we may rewrite the (Correctness) assumption as follows
t
X
⇥
⇤
Xi
P lim sup
> p0 = 0 8 µ > 0
t
t!1
i=1
with
p0 := |hx| i|2 + µ.
106
(7.19)
7.4 Main statement
Let now as before ⌦ denote the event that Rachel indents to stay at the casino until
she runs out of money. We now show that it follows from (BornObj) that Rachel will be
deemed to lose under the condition ⌦ on those days when p
p0 . This can be seen as
follows.
The probability that Rachel will never run out of money is mathematically given by
(using p p0 )
⇥
P 8 t 2 N : M0
t
X
p·t+
i=1
t
⇤
⇥
⇤
M0 X X i
Xi > 0 = P 8 t 2 N :
+
>p
t
t
i=1
k
⇥
M0 X X i
 P inf sup
+
> p]
t2N k t k
k
i=1
t
⇥
⇤
M0 X X i
= P lim sup
+
>p
t
t
t!1
i=1
⇥
= P lim sup
⇥
t!1
 P lim sup
t!1
t
X
Xi
⇤
>p
t
X
Xi
> p0
i=1
i=1
t
t
⇤
= 0.
Hence, we obtain under the condition p > p0 the analogue of Eq. (7.4)
⇥
P M0 +
|X|
X
t=1
Xt
p · |X| =
6 0 ⌦, P
⇤
p0 = 0.
The rest of the argument is analogous to the previous section, yielding
PX1 (1)  p0 .
(7.20)
Recall now that x1 = 1 denoted the event “success”, i.e., that the outcome of the measurement on was x. Denoting by X the random variable corresponding to the outcome
of the quantum measurement we can therefore write
PX1 (1) = PX (x)
and it follows from (7.20) and the definition of p0 that
PX (x)  |hx| i|2 + µ
holds.
107
(7.21)
7. RETRIEVING THE SUBJECTIVE BORN RULE
Now we run the argument again, but this time Rachel bets on the event “not x”. In
this case Xi = 0 (“loss”) denotes the event that the outcome was x, i.e.,
PX1 (0) = PX (x)
and Xi = 1 (“success”) denotes the event that the outcome was not x.
Let now
p00 := |hx| i|2
µ
and note that in this case (BornObj) results in the mathematical requirement
t
X
⇥
Xi
P lim sup
> (1
t
t!1
i=1
⇤
p00 ) = 0 .
By the same argument as before it follows that the probability that Rachel will never run
out of money on those days with p 1 p00 vanishes
⇥
P M0 +
|X|
X
Xt
t=1
p · |X| =
6 0 ⌦, P
1
⇤
p00 = 0.
Therefore, we also obtain a contradiction to the assumption PX1 (1) > 1
finding
PX1 (1)  1 p00 =) PX1 (0) p00 .
Inserting PX1 (0) = PX (x) this implies
PX (x)
|hx| i|2
µ.
Summarising the two relations (7.22) and (7.21) we obtain
|hx| i|2
µ  PX (x)  |hx| i|2 + µ
and because this holds for all µ > 0 we find the desired result
PX (x) = |hx| i|2 .
108
p00 in this case,
(7.22)
Chapter 8
Single-world theories are not
self-consistent
8.1
Motivation
“I’m sure that quantum theory will be proved false one day, because it seems
inconceivable that we’ve stumbled across the final theory of physics. But I
would bet my bottom dollar that the new theory will either retain the parallel
universe feature of quantum physics, or it will contain something even more
weird.”
David Deutsch [72]
In Chapter 2 we discussed the two main interpretations of quantum theory: single-world
and multiverse interpretations. While many physicists share suspicious feelings towards
the multiverse interpretation it is also often argued that it seems to be preferable to have a
rigorous formalism at the price of giving up the prejudice of a single reality. For instance,
the quote by Howard Barnum in Chapter 2 expresses the point of view that while the
idea of a splitting universe is sufficiently bizarre that it is worth to investigate other alternatives, it seems to be preferable to have a well-defined theory than to adopt the rather
fuzzy formalism of collapse theories.
Here we will make this point even stronger by showing that if one wants to hold on to
the idea of a single reality one has to give up at least one other assumption about reality
that seems to be natural. More specifically, we prove the following theorem.
Theorem 3. No physical theory T can satisfy (QT), (C), (SW) and (SC).
The assumptions (QT), (C), (SW) and (SC) were introduced in Chapter 5. Let us briefly
recall what they say on an informal level.
109
8. SINGLE-WORLD THEORIES ARE NOT SELF-CONSISTENT
(QT) Compliance with quantum theory: T forbids all measurement outcomes that are
forbidden by standard quantum theory (and this condition holds even if the
measured system is large enough to contain an experimenter).
(C) Comaptibility: T’s statements about di↵erent experiments must be compatible.
(SW) Single-world: T rules out the occurrence of more than one single outcome if a
measurement is performed once.
(SC) Self-consistency: T allows at least one story about each experiment.
Standard quantum theory, to which (QT) refers, should be understood as the plain
theory that does not impose any constraints on the complexity of objects it is applied to.
In particular, no element of the theory precludes measurements on a macroscopic system
that contains, for instance, a physicist who herself carries out measurements on a subsystem. This occurs, for instance, in the well-known Wigner’s Friend gedankenexperiment
[73], where one experimenter measures another experimenter, and from which our argument draws inspiration. Property (QT) therefore corresponds to the requirement that the
theory T, even when applied to such situations, does not contradict the laws of quantum
theory.
It is certainly legitimate to question whether a theory that accurately describes nature
must respect this requirement – we are still lacking experimental evidence for the validity
of quantum theory on macroscopic scales. Conversely, self-consistency and compatibility,
as defined by (SC) and (C), are arguably unavoidable requirements to any reasonable
physical theory. The result proved in this work thus leaves two options.
• Option 1: Quantum theory does not accurately describe the behaviour of macroscopic systems. (It could be, for instance, that the behaviour of such systems is
governed by a, yet undiscovered, physical mechanism that leads to an “objective
collapse” of the wavefunction.) In this case, it may be possible to retain a singleworld view.
• Option 2: Quantum theory is valid on macroscopic scales. In this case, the singleworld view must be rejected and replaced, for instance, by a many-worlds view.
8.2
Extended Wigner’s friend experiment
Our argument is based on an extension of the Wigner’s Friend gedankenexperiment [73].
We first describe the experiment informally from an overall perspective. Then, using the
story-based framework introduced in Chapter 3, we provide a formal characterisation. For
this, we consider the di↵erent views that experimenters can have on the experiment and
specify how they are related to each other.
110
8.2 Extended Wigner’s friend experiment
8.2.1
A bird’s eye view on the experiment
In contrast to the original Wigner’s Friend experiment, the extended version features two
friends, F1 and F2, who shall be located in separate labs. We assume that they can be
treated as isolated quantum systems. This means that the time evolution of their state is
described by a unitary.1 We also assume that F1 is fully informed about the state of F2
(including her entire lab) at the time when the experiment starts, and that this state is
therefore pure. In addition, we consider two experimenters, Wigner W and his assistant A,
who are not only informed about the friends’ initial states, but can also carry out arbitrary
quantum measurements on them.2
The friends and the experimenters carry out the following protocol.
Protocol of the Extended Wigner’s Friend Experiment WF
Repeat the following steps for increasing n 2 N until the halting criterion in the last
step is satisfied.
@ n:00
@ n:10
@ n:20
@ n:30
@ n:40
@ n:50
F1 invokes a quantum random number generator and memorises the
output r 2 {head, tail}.
Depending on whether r = head or r = tail, F1 sets the spin of an
electron E to state |#iE or |!iE , respectively, and hands it over to F2.
F2 measures the spin of E with respect to the basis {|#iE , |"iE } and
memorises the outcome z 2 { 12 , + 12 }.
A measures F1 with respect to a basis {|okiF1 , |failiF1 }, records the
outcome x 2 {ok, fail}, and informs W about it.
W measures F2 with respect to a basis {|okiF2 , |failiF2 } and records the
outcome w 2 {ok, fail}.
Halt if w = ok and x = ok.
The following definitions of the states and measurements correspond to a specific choice
of the states introduced by Hardy [74]. The quantum random number generator used in
the first step shall be designed such that the probabilities of the outcomes r = head and
r = tail are 1/3 and 2/3, respectively. For concreteness, we may think of a mechanism that
prepares a qubit C, the “quantum coin”, in state
p
p
0
1/3|headi +
2/3|taili
(8.1)
C =
C
C
1
Basically the same assumption has already been made by Schrödinger when he famously proposed to
consider the wavefunction of a cat sitting in an isolated box [34].
2
The reader shall not be worried about the technological feasibility of this experiment. It serves its
purpose well as long as none of its steps is forbidden by a basic law of physics.
111
8. SINGLE-WORLD THEORIES ARE NOT SELF-CONSISTENT
and measures it with respect to the basis {|headiC , |tailiC }. Furthermore, for the spin state
of the electron E prepared in the second step, which lives in the space spanned by the two
orthonormal vectors |#iE and |"iE , we use the convention
p
p
|!iE = 1/2|#iE + 1/2|"iE .
The basis {|okiF1 , |failiF1 } with respect to which the measurement by A on F1 is carried
out shall be defined as
p
p
1/2|taili
|okiF1 = 1/2|headiF1
F1
p
p
|failiF1 = 1/2|headiF1 + 1/2|tailiF1 ,
where |headiF1 and |tailiF1 are the states of F1 in the case where r = head and r = tail,
respectively. Similarly, the basis {|okiF2 , |failiF2 } of the measurement carried out by W on
F2 shall be defined as
p
p
1/2|+ 1 i
|okiF2 = 1/2| 12 iF2
2 F2
p
p
1
|failiF2 = 1/2| 2 iF2 + 1/2|+ 12 iF2 ,
where | 12 iF2 and |+ 12 iF2 are the states of F2 in the case where z =
respectively.
1
2
and z = + 12 ,
Following the spirit of the framework, we will consider stories that one can tell about
this experiment. An example could be a story that starts as follows.
8
“At time t = 0:00 the output of F1’s random number generator
>
>
>
is r = head. She therefore prepares the electron E in state |#iE .
>
>
>
< When F2 measures E at t = 0:20, she gets outcome z = 21 .
s=
The subsequent measurements carried out by A and W at times
>
>
t = 0:30 and t = 0:40 result in outcomes x = fail and w = fail,
>
>
>
>
: respectively. They therefore continue with a second round. At
time t = 1:00 ...”
Note that while we may view WF as one big experiment defined by the physical quantities corresponding to the di↵erent measurement outcomes it is not possible to analyse
this experiment within quantum theory, because there is no joint description of all the
involved systems. For example, it is impossible to model within quantum theory the outcome z of the measurement performed on the electron E by F2 together with the outcome
w which is the outcome of the measurement performed on F2 himself. In the following we
will therefore consider di↵erent sub-experiments that may be analysed within quantum
theory.
8.2.2
Formal description of the experimenters’ views
For the formal description of the Extended Wigner’s Friend Experiment, we provide separate definitions for the four “sub-experiments” F1, F2, A, and W, as viewed by F1, F2,
112
8.2 Extended Wigner’s friend experiment
A, and W, respectively. Note that we use the same label, e.g. F1, both for the experimenter and for the experiment as viewed by this experimenter. While there is in principle
a lot of freedom for how to choose these definitions, the guiding principle here is that
we only specify the elements of the experiment which will be relevant to the argument of
Section 8.3 below.
Definition of Experiment F1
We define the experiment F1 to have the physical quantities ⇥F1 = {t̂, r̂, bE , ŵ} with event
space
[F1] = (t, r,
E , w)
2 R ⇥ {head, tail, ?} ⇥ HE ⇥ {ok, fail, ?} :
if t = n:10 then r = head ,
E = |#i
and r = tail ,
E = |!i
.
(8.2)
This definition takes into account that F1 prepares the states of the electron depending
on the value of r. The trajectory sF1 of the story s described above, for instance, would
then include the elements
(t, head, 0, ?)
(t, head, |#i, ?)
(t, head, 0, ?)
(t, ?, 0, ?)
(t, ?, 0, fail)
where we set
measured.
E
:
:
:
:
:
t 2 [0:00, 0:10)
t 2 [0:10, 0:20)
t 2 [0:20, 0:30)
t 2 [0:30, 0:40)
t 2 [0:40, 0:50) ,
= 0 for the state of E before it has been prepared and after it has been
In order to apply assumption (QT) we have to specify the isometry according to which
the system E evolves until the second measurement that is included in F1 takes place at
n:40.
Let therefore U = U n:20 (for any n 2 N) be the isometry from E to F2 such that
U |#iE = |
1
2 iF2
and
U |"iE = |+ 12 iF2
H = U † |wihw|U for w 2 {ok, fail}.
and let ⇡w
Note that assumption (QT) is expressed as a rule about experiments Exp 2 QHE ,{⇡wH }
and F1 is a priori not in this set, because its event space is di↵erent. However, we may
require that the experiment F1 corresponds to a Repeated Quantum Experiment of the
form QHE ,{⇡wH } by simply dropping the quantity r
9 Exp 2 QHE ,{⇡wH } :
(t, ⇤,
E , w)
2 sF1 () (t0 ,
E , w)
2 sExp ,
(8.3)
with a relation t $ t0 such that n:10 $ n:00, n:40 $ n:01, and n:50 $ n:02. In addition,
we demand that r is an observation, in the sense that
9 Exp 2 O{head,tail} :
(t, r, ⇤, ⇤) 2 sF1 () (t0 , r) 2 sExp ,
113
(8.4)
8. SINGLE-WORLD THEORIES ARE NOT SELF-CONSISTENT
where t 7! t0 is such that n:10 7! 0:00 and n:40 7! 0:01. This will allow us to apply the
assumption (SW) to the experiment F1.
Definition of Experiment F2
Similarly to the above, we define the experiment F2 to have the physical quantities
⇥F2 = {t̂, r̂, bE , ẑ} with event space
[F2] = (t, r,
E , z)
1
1
2 , + 2 , ?}
2 R ⇥ {head, tail, ?} ⇥ HE ⇥ {
if t = n:10 then r = head ,
E = |#i
and r = tail ,
:
E = |!i
.
(8.5)
To specify how z arises as the outcome of a measurement, we define ⇡zH = |zihz| for
z 2 { 21 , + 12 } and in order to be able to apply assumption (QT) we require that
9 Exp 2 QHE ,{⇡zH } :
(t, ⇤,
2 sF2 () (t0 ,
E , z)
E , z)
2 sExp ,
(8.6)
with a relation t $ t0 such that n:10 $ n:00, n:20 $ n:01 and n:40 $ n:02.
Definition of Experiment A
For the views of A and W, we include the initial state, C of the quantum coin C in our
definitions. More precisely, we define the experiment A to have the physical quantities
⇥A = {t̂, bC , ẑ, x̂} with event space
[A] = (t,
C , z, x)
2 R ⇥ HC ⇥ {
if t = n:00 then
C
0
C
=
1
1
2 , + 2 , ?}
,
⇥ {ok, fail, ?} :
(8.7)
where C0 is the state defined by (8.1). To specify the measurement of z and x, we let
V = V n:00 n:10 be an isometry from C to F1 ⌦ E such that
V |headiC = |headiF1 ⌦ |#iE
and
V |tailiC = |tailiF1 ⌦ |!iE
H = V † |xihx| ⌦ |zihz|V .
and define the projectors ⇡z,x
Like in the case of F1 and F2 the experiment A is a priori not in the set QHC ,{⇡z,x
H }
– instead of one measurement outcome it involves two outcomes described by a joint
quantum state. In order to be able to apply assumption (QT) we require that
9 Exp 2 QHC ,{⇡H
(z,x)
}
:
(t,
C , z, x)
2 sA () (t0 ,
C , (z, x))
2 sExp ,
(8.8)
with a relation t $ t0 such that n:00 $ n:00, n:20 $ n:01, n:30 $ n:01, and n:50 $ n:02.1
1
In order to take into account that the measurement of z and x should be at the same time according
to the protocol of the Basic Observation Experiment O{ 1 ,+ 1 }⇥{ok,fail} one may consider a coarse grained
2
2
clock. More precisely, the situation is mapped to an experiment with a less accurate clock such that the
two di↵erent times n:20 and n:30 cannot be distinguished but are both mapped to n:01.
114
8.3 Proof
Definition of Experiment W
The experiment as viewed by W shall have the set of physical quantities ⇥W = {t̂, bC , x̂, ŵ}
with event space
[W] = (t,
C , x, w)
2 R ⇥ HC ⇥ {ok, fail, ?} ⇥ {ok, fail, ?} :
if t = n:00 then
C
=
0
C
.
(8.9)
To specify the measurement of x and w, we define the projectors
H
⇡y,w
= V † |xihx| ⌦ U † |wihw|U V ,
where U and V are the isometries specified above. Finally, in order to apply assumption
(QT) we then require that
9 Exp 2 QHC ,{⇡H
(x,w)
}
:
(t,
C , x, w)
2 sW () (t0 ,
C , (x, w))
2 sExp ,
(8.10)
with a relation t $ t0 such that n:00 $ n:00, n:30 $ n:01, n:40 $ n:01, and n:50 $ n:02.
The condition that the experiment is repeated until the halting criterion is satisfied
(which can be tested by W) may be written as the requirement that the implication
n = 1 or ((n
1):50, ⇤, ⇤, ⇤) 2 sW and ((n
=)
1):50, ⇤, ok, ok) 2
/ sW
8 t 2 [n:00, n:50] : (t, ⇤, ⇤, ⇤) 2 s
W
(8.11)
(8.12)
holds for all n 2 N.
8.3
Proof
We split the proof of Theorem 3 in three parts. In the first we apply the compatibility
condition (C). Then, we analyse the extended Wigner’s Friend Experiment, as described
in the previous section, separately from the viewpoints of F1, F2, A, and W. For this we
only use property (QT), which means that the analysis is based on the rules of standard
quantum theory. Then, in the last part of the proof, we use properties (SW) and (SC) to
combine the results obtained in the first, which then leads to a contradiction.
For the following, let s be any story that is not forbidden by T.
8.3.1
Applying the compatibility assumption
The experiments F1, F2, A, and W have certain overlapping elements – after all, they
are all part of one big experiment. From the assumption (C) the following compatibility
conditions follow. First, the requirement
(t, ⇤, ⇤, w) 2 sF1
()
115
(t, ⇤, ⇤, w) 2 sW
(8.13)
8. SINGLE-WORLD THEORIES ARE NOT SELF-CONSISTENT
models that the quantity w defined in the experiment W is the same as the one of F1.
Similarly, the compatibility constraints which model that r, z, and x denote the same
quantities in each experiment can be written as
(t, r, ⇤, ⇤) 2 sF1
(t, ⇤, ⇤, z) 2 sF2
(t, ⇤, x, ⇤) 2 s
8.3.2
(t, r, ⇤, ⇤) 2 sF2
()
(t, ⇤, z, ⇤) 2 sA
()
A
()
(t, ⇤, ⇤, x) 2 s
W
(8.14)
(8.15)
.
(8.16)
Analysis of individual views
Analysis of Experiment F1
By linearity, the vector
U |!iE = U
p
1/2|#i
E
+
p
1/2|"i
E
= |failiF2
is orthogonal to |okiF2 . This is equivalent to say that |!iE has no overlap with the projector
H . Property (QT), which we can apply via (8.3), thus implies that
⇡ok
(n:40, ⇤, ⇤, w = ok) 2 sF1
=)
9
E
: (n:10, ⇤,
E , ⇤)
2 sF1 and
E
6 k |!iE .
Using furthermore the constraint (8.2), we conclude that
(n:40, ⇤, ⇤, w = ok) 2 sF1
(n:10, r = head, ⇤, ⇤) 2 sF1 .
=)
(8.17)
Analysis of Experiment F2
Property (QT), applied via (8.6), implies that
(n:20, ⇤, ⇤, z = + 12 ) 2 sF2
=)
9
E
: (n:10, ⇤,
E , ⇤)
2 sF2 and
E
6 k |#iE .
By the constraint (8.5), we conclude that
(n:20, ⇤, ⇤, z = + 12 ) 2 sF2
=)
(n:10, r = tail, ⇤, ⇤) 2 sF2 .
(8.18)
Analysis of Experiment A
Using the explicit form (8.1) of C0 we find that
p
p
V C0 = 1/3|headiF1 ⌦ |#iE + 2/3|tailiF1 ⌦ |!iE
p
p
p
= 1/3|headiF1 ⌦ |#iE + 1/3|tailiF1 ⌦ |#iE + 1/3|tailiF1 ⌦ |"iE
p
p
= 2/3|failiF1 ⌦ |#iE + 1/3|tailiF1 ⌦ |"iE .
(8.19)
This vector is obviously orthogonal to |okiF1 ⌦ |#iE . This means that C0 is orthogonal
H
⇡ok,
1 . It follows from Property (QT), applied via (8.8), as well as (8.7) that
2
(n:30, ⇤, z =
1
2, x
116
= ok) 2
/ sA .
(8.20)
8.3 Proof
Analysis of Experiment W
Using (8.19) we find
UV
0
C
=
p
2/3
⌦ |failiF1 ⌦ |
1
2 iF2
+
p
1/3
⌦ |tailiF1 ⌦ |+ 12 iF2 .
The overlap between this state and the state |okiF1 ⌦ |okiF2 equals
p
p
1/3hok|tailiF1 hok|+ 1 iF2 =
1/12 .
2
H
In other words, the state C0 has a non-zero overlap with the projector ⇡ok,ok
. Furthermore,
(8.9) ensures that the system is always prepared in this state, unless no state was prepared
at all. Property (QT), which applies due to (8.10), hence implies that there must exist a
round n where the outcomes y = w = ok occur, i.e.,
(n:00, ⇤, ⇤, ⇤) 2 sW 8 n
8.3.3
=)
9 n, t 2 [n:40, n:50] : (t, ⇤, x = ok, w = ok) 2 sW . (8.21)
Combining the views
Assume now by contradiction that T satisfies all four properties, i.e., (QT), (C), (SW),
and (SC). Property (SC) implies that there must exist a story s that is not forbidden by
T such that all of sF1 , sF2 , sA , and sW are defined.
Let n be such that (8.12) holds. Due to the compatibility constraint (8.13) this also
implies that
8 t 2 [n:00, n:50] : (t, ⇤, ⇤, ⇤) 2 sF1 .
Furthermore, because of (8.4), there exists Exp 2 O{head,tail} such that
8 t 2 [n:00, n:50] :
(t, ⇤, ⇤, ⇤) 2 sF1
()
(t0 , ⇤) 2 sExp
(with an appropriate mapping t 7! t0 ). We can thus apply property (SW) which implies
that the value r must remain the same, in the sense that
{r : (t, r, ⇤, ⇤) 2 sF1 for t 2 [n:00, n:30]} = 1 .
(8.22)
A similar argument can be applied to find that
{z : (t, ⇤, ⇤, z) 2 sF2 for t 2 [n:20, n:40]} = 1
(8.23)
{(x, w) : (t, ⇤, x, w) 2 sW for t 2 [n:40, n:50]} = 1
(8.24)
{(z, x) : (t, ⇤, z, x) 2 sA for t 2 [n:30, n:50]} = 1 .
(8.25)
and
and
117
8. SINGLE-WORLD THEORIES ARE NOT SELF-CONSISTENT
Condition (8.24) implies that (8.21) is equivalent to
(n:50, ⇤, x = ok, w = ok) 2 sW .
(8.26)
It follows that there must exist some n such that (8.26) holds. Indeed, if this was not
the case condition (8.24), together with the repetition criterion (i.e., (8.11) =) (8.12))
would imply via induction that
(n:00, ⇤, ⇤, ⇤) 2 sW 8 n .
From this it would follow from the quantum-mechanical analysis from W’s viewpoint
(8.21), that there must still exist some n such that
t 2 [n:40, n:50] : (t, ⇤, x = ok, w = ok) 2 sW
holds and using condition (8.24) again (8.26) follows. We can therefore continue our analysis for a choice of n such that (8.26) is valid.
The compatibility condition (8.16), which asserts that W and A see the same value x,
implies
(n:50, ⇤, ⇤, x = ok) 2 sA .
From (8.25), we conclude that
(n:30, ⇤, ⇤, x = fail) 2
/ sA .
Also, because of (5.6), the measurement outcome x cannot admit the value ? at time
t = n:30. Using once again (8.25), we thus obtain
(n:30, ⇤, ⇤, x = ok) 2 sA .
Similarly, the measurement outcome z cannot admit the value ? at time t = n:30. Constraint (8.20), which resulted from the quantum-mechanical analysis from A’s viewpoint,
hence implies that the z value must be equal to + 12 , i.e.,
(n:30, ⇤, z = + 12 , x = ok) 2 sA .
The compatibility condition (8.15), according to which A and F2 see the same value z,
implies that
(n:30, ⇤, ⇤, z = + 12 ) 2 sF2 .
We now use (8.23) to infer that
(n:20, ⇤, ⇤, z = + 12 ) 2 sF2 .
118
8.4 Related work
Applying (8.18), which resulted from the quantum-mechanical analysis from F2’s viewpoint, we obtain
(n:10, r = tail, ⇤, ⇤) 2 sF2 .
The compatibility condition (8.14), which asserts that F2 and F1 see the same value r,
implies
(n:10, r = tail, ⇤, ⇤) 2 sF1 .
Using (8.22) we find that this is equivalent to
(n:10, r = head, ⇤, ⇤) 2
/ sF1 .
(8.27)
Conversely, starting again from (8.26), we can employ (8.24) to obtain
(n:40, ⇤, x = ok, w = ok) 2 sW .
Then we can use the compatibility condition (8.13), which ensures that F1 and W talk
about the same value w, to obtain
(n:40, ⇤, ⇤, w = ok) 2 sF1 .
However, this and (8.27), taken together, are in contradiction to (8.17), which was the
result of the quantum-mechanical analysis from F1’s viewpoint. This concludes the proof
of Theorem 3.
8.4
Related work
Bell’s theorem [75] is probably the best known work showing that outcomes of quantum
measurements cannot be real and one may wonder why our result does not simply follow
from it. The main di↵erence is that in contrast to Bell we do not assume that the outcomes
are real for microscopic systems but only for conscious observers such as Wigner’s friend.
Also our result does not involve any assumption about locality.
David Deutsch proposed an actual experiment testing between collapse theories and
the multiverse interpretation which is also based on Wigner’s friend Gedankenexperiment
[76]. The idea is that the friend actually reveals to Wigner whether he has seen a definite
outcome – but crucially not what the outcome was. This guarantees that from Wigner’s
point of view the joint state of the system and the friend remains entangled and he could
test the validity of his unitary description by performing interference experiments. Similar experiments have been proposed later by Michael Lockwood [77] and Lev Vaidman [28].
Only recently Časlav Brukner provided a proof based on Deutsch’s idea showing that
the assumption that the measurement outcome of the friend coexists for Wigner leads to
a problem for hidden variable theories [33]. More precisely, he extended the experiment
119
8. SINGLE-WORLD THEORIES ARE NOT SELF-CONSISTENT
introducing additional systems and observers and constructed an experimental setup violating a Bell inequality. Similar to our result Brukner views his result as a restriction
on hidden variable theories. However, the assumptions that enter the proof are di↵erent
from ours. Also, the conclusions Brukner derives from his result di↵er from our point
of view. While we see it as an indication against the assumption that there is only one
definite measurement outcome Brukner argues that one may still assume that there is a
single outcome – however, this outcome only exists for the friend. Hence, he takes a point
of view similar to the QBist approach according to which there are no facts of the world
per se but only relative to observers. However, he also stresses that in contrast to the
general QBist subjective interpretation of the wavefunction one may also assume that the
wavefunction is indeed objective in the sense that it represents a maximal possible degree
of belief.
Another argument supporting the multiverse interpretation was given by Gilles Brassard and Paul Raymond-Robichaud [78]. They show that there is another possibility than
the generally drawn conclusions from Bell’s theorem. While it usually assumed that the
theorem shows that the world does not have a local realistic description they argue that
one may also drop the assumption that measurements have single outcomes. Assuming
that all possible outcomes occur in parallel they construct a deterministic, local toy theory
in which the correlations between individual observers are those given by quantum theory.
Nicolas Gisin constructed a Bell inequality based on position measurements that is violated within Bohmian mechanics [79]. Arguing that Bohmian mechanics holds for single
particles as well as for “elephants” he concludes that the assumption that the Bohmian
hidden parameter models where the elephant actually is must be wrong. One may also
view this result in the context of a Wigner’s friend scenario. Assume that the elephant
was originally in a superposition between “right” and “left” and the friend measures his
position. Then Gisin’s result implies that even after the friend perceived that the elephant
is on the right (or left) this does not describe reality from Wigner’s point of view. Our
result is more general as it is not restricted to Bohmian mechanics. Rather, it follows from
our result that Bohmian mechanics cannot be self-consistent because it is a single-world
theory that is compatible with unitary quantum theory.
In contrast to most of the proposed experiments to test the multiverse interpretation
which are not practical with current techniques Don Page proposed one which seems to be
realisable within the near future [80]. His idea is to consider the case in which the number
of observations, or the number of observers, may di↵er for di↵erent branches of the wavefunction. He shows that this would lead to observable di↵erences between the expectation
values of measurements within many-world and single-world interpretations leading to the
conclusion that an expanding universe hints towards a many-world interpretation.
120
Chapter 9
Conclusions
The goal of this thesis was to introduce a novel framework for scientific theories. Our hope
is that the general structure of the framework will allow us to gain new insights in some
open questions in research. In particular, we hope that its application will force us to
question the structure of existing formalisms and to become aware of implicit assumptions
within them.
9.1
Applications of the framework
In this thesis we discussed two applications of the framework. First we derived a nonprobabilistic formulation of the Born rule as an objective law of nature which we termed
(BornObj). While the Born rule is generally stated as the probabilistic axiom of quantum
theory we derived it from the two more natural assumptions (i) that measurement outcomes corresponding to projectors that are orthogonal to the quantum state are forbidden
and (ii) that the statements of quantum theory are robust under small perturbations of
the formal description. We think that this result is interesting for various reasons. First,
motivated by the rather unnatural formulation of the probabilistic axiom there have been
previous approaches to derive the Born rule from more operational assumptions. However,
because these results are all formulated in terms of probabilities they also su↵er from the
well known problems when one tries to define probabilities objectively. Thus, in order to
be consistent, the standard probabilistic Born rule can only have a subjective meaning.
In contrast to this, our law (BornObj) can be seen as an objective law of nature without
problems regarding its consistency.
We also showed how the standard subjective probabilistic interpretation of the Born
rule can be derived from the objective law (BornObj). In order to give the probabilities an
operational meaning we introduced an agent and followed the Bayesian approach according to which the probability P (x) of an outcome x corresponds to the maximum amount
of money the agent would be willing to pay to enter the bet in which he obtains $P (x)
in case x occurs and nothing otherwise. Based on this idea we showed that under certain
assumptions about the agent’s rational beliefs he would assign probability P (x) = |hx| i|2
121
9. CONCLUSIONS
to the outcome x of a measurement described by a quantum state with respect to a set
of projectors {⇡x }x2X . This separation of the standard Born rule in a part expressing a
law of nature as well as in a part expressing an agent’s beliefs has not been discussed in
the literature before.
As a second application we showed that single-world interpretations of quantum theory
cannot be self-consistent. The general structure of our framework allowed us to formally
express a set of assumptions that are so natural that they are usually made implicitly
in existing formalisms. For instance, the compatibility assumption (C) captures the idea
that experiments may be analysed from di↵erent point of views without running into
contradictions and the self-consistency requirement (SC) demands that a theory should
allow that something happens in an experiment. Finally, we showed that single-world
theories cannot satisfy a set of such natural assumptions. Apart from the main result
itself this application also demonstrates how general properties of theories can be expressed
naturally within the framework.
9.2
Outlook and open questions
Obviously one of the most interesting research topics is to gain a better understanding of
time. A possible application of the framework would be to formulate scenarios such as the
alternate tic game introduced in [81] within the framework. Here the question was how
accurately a clock can be modelled without the assumption of a pre-existing time. The
scenario may be extended by additionally dropping the assumption of a pre-existing order
of events.
Another idea would be to use the framework to investigate the symmetry between
position and time. For example it is interesting that while we do generally not find it
problematic that there are several copies of us at di↵erent times at the same position
many of us do find it rather unacceptable that there might be several copies of us at
di↵erent positions at the same time.
A question that has been discussed in the literature before is whether quantum theory
can be derived from more natural assumptions. Several interesting results have been
derived within the framework of generalised probabilistic theories [65, 66, 67]. Because
these approaches are based on the notion of probabilities they involve implicit assumptions
that are related to their operational interpretation. Another open question is whether
it is possible to formalise these results within our framework without any reference to
probabilities.
122
Appendix A
Bell’s Theorem
Bell’s theorem is generally understood to show that there is no local, realistic extension of
quantum theory. Sometimes it is also said to show that quantum theory is non-local. This
is a bit misleading especially because it is also said for example that Bohmian mechanics
is non-local and this is commonly used as an argument to deem Bohmian mechanics a
“bad” theory. This confusion originates from the problem that while the proof of Bell’s
theorem is part of a standard quantum mechanics lecture the assumptions are generally
not made explicit. Therefore, we will review here the formalisation of these assumptions
and discuss which of them is violated within quantum theory and which of them is violated
within Bohmian mechanics. We hope that this will help to clarify that the non-locality
within Bohmian mechanics is fundamentally di↵erent from quantum theory. In particular,
it follows that Bohmian mechanics allows for faster than light signalling. Note that we
will not restate the proof of the theorem, which can be found for example in [82]. Most of
the statements of this chapter can be found in [83].
Bell’s theorem is formulated for a bipartite scenario with measurement settings A and
B and outputs X and Y . The settings, the outputs as well as possible additional information ⇤ an be modelled as space-time variables [27]. In the following we will call a theory
with ⇤ as a parameter an extension.
An assumption that usually enters the theorem implicitly is that the choice of the
measurement settings is free, i.e., that an experimenter carrying out the measurement has
a free choice. This assumption is sometimes criticised to be unphysical. However, what
is meant can be made mathematically precise – in contrast some kind of meta-physical or
psychological concept of free will.1
In John Bell’s word variables are free if “such variables have implications only in their
future light cones.” [84] Note that it does not suffice to demand that the choice is uncor-
1
Free choice and free will are not the same and we do not know how to define the latter.
123
A. BELL’S THEOREM
related to anything in the past-light cone within a specific reference frame. The reason is
that in the case that the choice is correlated to a space-like separated variable there exists
another reference frame such that this variable is in its past light cone.
The disadvantage of the informal definition is that it assumes the validity of special
relativity. Although many scientists would agree that this assumption is justified, there are
also theories, such as Bohmian mechanics, that have a preferred reference frame. Therefore, within such a theory a choice is free if it is only correlated to future events w.r.t. that
frame. Fortunately, the assumption of the validity of special relativity is not necessary to
express what is meant by a free choice. All that is needed is that the theory has a notion of future and past, thereby introducing a causal order (see Figure A.1) [85]. A choice
is then said to be free if it is only correlated with events in its future w.r.t. that causal order.
X
Y
A
B
⇤
Figure A.1: Causal structure for Bell type measurement.
Formally, the condition that the choice of the measurement settings A and B is free
can be expressed as the condition
PA|BY ⇤ = PA
and
PB|AX⇤ = PB .
(A.1)
Besides free choice the less controversial assumptions of determinism and compatibility
with quantum theory enter the original statement of Bell’s theorem. Formally, they are
expressed as follows and it follows from the theorem that at least one of them cannot hold.
1. Free choice w.r.t. causal order:
PA|BY ⇤ = PA
and
PB|AX⇤ = PB .
2. Local determinism:
PXY |AB⇤ = PX|A⇤ · PY |B⇤
and
PX|A⇤ , PY |B⇤ 2 {0, 1}.
3. Compatibility with quantum theory:
PXY |AB⇤ obeys quantum statistics.
124
It was shown that this set of conditions can be relaxed [27]. In order to see this we first
note that the second assumption can be divided into two di↵erent assumptions as follows.
2a. Parameter independence
PX|AB⇤ = PX|A⇤ .
2b. Outcome independence
PX|ABY ⇤ = PX|AB⇤ .
Note that 2a. is generally understood as “locality” or non-signalling condition.
First, we observe that determinism follows from 2a., 2b. and 3., because the quantum
correlations used in the proof of the theorem are perfect:
PXY |AB =
x,y .
Therefore we get
x,y
2b
2a
= PX|ABY ⇤ = PX|AB⇤ = PX|A⇤ .
Parameter independence follows from free choice as follows.
PX|AB⇤ = PXAB⇤ /PAB⇤
= PB|XA⇤ · PXA⇤ /(PB|A⇤ · PA⇤ )
= PB · PXA⇤ /(PB · PA⇤ )
= PXA⇤ /PA⇤
= PX|A⇤
Therefore, Bell’s theorem actually shows that the following three conditions cannot be
satisfied together.
1. Free choice w.r.t. causal order.
2. Outcome independence.
3. Compatibility with quantum theory.
Which of the assumptions do we want to give up? Most physicists are somewhat attached to the idea that nature is compatible with free choice. Because we are explicitly
asking about the existence of a possible extension of quantum theory we cannot really
give up 3. Therefore, in general the conclusion is that nature does not satisfy outcome
independence. This is what is meant when we say that “quantum theory is non-local”.
125
A. BELL’S THEOREM
Note it does not correspond to give up the non-signalling condition 2a.
If we drop outcome independence it follows that there cannot exist a hidden variable
⇤ that determines the outcome X. The reason is that the condition
PX|ABY ⇤ 6= PX|AB⇤
tells us that X = f (A, B, ⇤) cannot hold for any function f . If there is such a function on
the other hand, as it is for example the case within Bohmian mechanics, it follows directly
that outcome independence is satisfied. This is simply the case because Y does not reveal
any more information about X than A, B and ⇤. Hence, it necessarily follows that such
a theory cannot satisfy free choice w.r.t. a causal order.
126
Appendix B
Typicality Lemma
Here we show Eq. (6.10). The proof is technically equivalent to the law of large numbers.
However, whereas the latter is generally understood as a probabilistic statement we use it
in a probability-free context. Because this might be a bit unfamiliar, we also give a proof
of the lemma.
We show the following Lemma stated in (6.10).
Lemma 4. For a frequency distribution f 2 [0, 1]d and the the distribution Q = |hx| i|2
the following relations holds
kf
Qk
"+µ
=)
h
⌦n
µ
|⇡f,n
|
⌦n
i < nd
1
·2
n 2
"
2
8n
µ
where the set of projectors {⇡f,n
} is given in (6.5) and d is the dimension of the
Hilbert space H of .
Proof. For a given sequence s 2 {0, 1}n we write
Qn (s) :=
Y
Q(si ).
i
For f 0 2 [f
µ
2,f
+ µ2 ] we can make use of the steps analogous to the derivation of the
127
B. TYPICALITY LEMMA
law of large numbers
h
⌦n
|⇡f 0 ,n |
⌦n
⌦n
i=h
|
X
=
X
s: Psn =f 0 si
Y
s: Psn =f 0 si
X
=
O
Y
⇡s i |
⌦n
i
h |⇡si | i
Q(si )
s: Psn =f 0 si
X
=
Qn (s)
s: Psn =f 0
 |{s : Psn = f 0 }|
 2nH(f
0)
 2nH(f
0)
max Qn (s)
s: Psn =f 0
n(H(f 0 )+D(f 0 kQ))
max
2
n
0
s: Ps =f
nH(f 0 )
2
max Qn (s)
s: Psn =f 0
2
n(H(f )+D(f 0 kQ))
nD(f 0 kQ)
=2
where H is the Shannon entropy and D the relative entropy.
Here we used Cover [86] Theorem 11.1.2.
Qn (s) = 2
n(H(f 0 )+D(f 0 kQ))
and Theorem 11.1.3
0
|{s : Psn = f 0 }|  2nH(f ) .
Using the triangle inequality for kf 0
bound
kf 0
f k < µ and kf
|kf 0
Qk
= |µ
µ
fk
"|
kf
=".
From this and Theorem 11.6.1. in [86]
D(f 0 kQ)
1 0
kf
2
1 2
"
2
128
Qk2
Qk < " + µ we find the following
Qk|
follows and hence
kf 0
f k < µ and kf
holds for each individual f 0 2 [f
Qk < " + µ
µ
2,f
h |⇡f 0 ,n | i  2
=)
n 2
"
2
+ µ2 ].
Finally we find
h
⌦n
µ
|⇡f,n
|
⌦n
i=

f 0 2[f
X
X
µ
,f + µ
]
2
2
2
h
⌦n
|⇡f 0 ,n |
⌦n
i
n 2
"
2
f 0 ⇡µ f
< |f 2 Fn | · 2
< nd
1
2
n 2
"
2
n 2
"
2
,
where Fn is the set of frequencies on strings with length n. This concludes the proof.
129
B. TYPICALITY LEMMA
130
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Daniela Frauchiger
Greifenseestrasse 29
8050 Zürich
[email protected]
Telefon: +41 78 707 5865
31.07.1985
Swiss and German
Education
10.2012 – 12.2015 ETH Zürich, Doctoral Studies in Theoretical Physics, Quantum Information
defense planned for December 2015
09.2011
09.2009
09.2009
09.2006
–
–
–
–
08.2012
08.2011
12.2009
08.2009
USI Lugano, Doctoral Studies in Computer Science, Quantum Information
ETH Zürich, MSc in Physics, Thesis in Theoretical Physics, (5.72/6.0)
ENS Paris, Erasmus Semester
ETH Zürich, BSc in Physics, (5.27/6.0)
08.2002 – 12.2005 Muttenz, Switzerland, Matura, Major subject: Mathematics
Maturandenpreis
07.2001 – 06.2002 High School, Arizona (USA), Exchange year with EF
Further Education
09.2006 – 12.2015 Schweizerische Studienstiftung
Participation in workshops, intellectual tools, reading group, summer academy
July 2005 SPHAIR, two week aviation training by the Swiss Air Force
completion with recommendation for application as military pilot
Practical Experience
2009–2015 Teaching, ETH Zürich
Exercises classes and substitute teaching in lectures (around one hundred students)
-
thermodynamics
classical mechanics
linear algebra
numerics
2011–2015 Talks and Poster Presentations at Conferences
attendance at around ten different conferences during Doctoral studies
1/2
2011–2015 Publications
- Daniela Frauchiger, Renato Renner, Matthias Troyer, True randomness from realistic
quantum devices, arXiv:311.4547, November 2013
- Daniela Frauchiger, Renato Renner, Truly Random Number Generation, SPIE
Security+Defense Conference Proceedings Vol 8899, 2013
2011-2015 Organisation of Conferences
- one out of two organisers of a conference about “Time in Physics” with approx. 80
international participants and a budget of ≥ CHF 30 000
- one out of two organisers of the “Junior Meeting” (conference for young scientists)
with around 40 participants and a budget of ≥ CHF 10 000
January 2006 Teaching, Realschule Pratteln
four weeks substitute teaching Niveau A
Computer skills
Advanced LATEX
Basic python, Microsoft Windows
Worked with html, css, C++
Projects - lecture homepage using Silva Web Content Management System
- conference homepage using html and css (time-workshop.phys.ethz.ch)
Languages
German Mothertongue
English Fluent
French Intermediate
Cambridge Certificate in Advanced English
Interests
Running regular training and participation in races
PB in Marathon 2:58
participation in Comrades Marathon
Sports swimming, cycling, hiking, yoga
Effective Altruism movement with the goal to reduce as much unnecessary suffering in the world
as possible
2/2

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