Different approaches to mutual information in free
probability theory
Fumio Hiai
Tohoku University
2014, July (at Krakow)
(Based on joint work with Yoshimich Ueda)
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
1 / 32
Plan
Introduction
Free pressure π R and η-entropy
Orbital free entropy χorb
Orbital free pressure πorb,R and ηorb -entropy
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
2 / 32
Introduction
Classical / quantum entropies
Shannon entropy: S(p) := −
∑
pi log pi
∫
Boltzmann-Gibbs entropy: H(X) := − p(x) log p(x) dx
( p(x) = dµ X /dx)
von Neumann entropy: S(ρ) := −Tr ρ log ρ
Classical / quantum mutual informations
classical:
I(X; Y) := −H(X, Y) + H(X) + H(Y) = D(µ X,Y ∥µ X ⊗ µY )
quantum:
IρAB (A; B) := −S(ρAB ) + S(ρA ) + S(ρB ) = D(ρAB ∥ρA ⊗ ρB )
( D(· ∥ ·) is the relative entropy.)
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
3 / 32
Expression of classical H(X1 , . . . , X n) via microstates
Let m ∈ N, δ > 0 and R ≥ max1≤i≤n ∥Xi ∥∞ . Define
∆ R (X1 , . . . , X n; N, m, δ)
:= {(x1 , . . . , x n) ∈ ([−R, R] N ) n :
|tr N (xi1 · · · xi k ) − E(Xi1 · · · Xi k )| < δ, 1 ≤ i j ≤ k, k ≤ m}.
∑N
where tr N (x) := N1 i=1
xi for x = (x1 , . . . , x N ). Then
1
log λ⊗n
(∆ R (X1 , . . . , X n; N, m, δ)),
N
N→∞ N
δ↘0
where λ N is the Lebesgue measure on R N .
H(X1 , . . . , X n) = lim
lim
m→∞
Pressure and variational expression
P(H) := log Tr e−H = max{−Tr ρH + S(ρ) : ρ ∈ S(A)}
S(ρ) = inf{Tr ρH + P(H) : H ∈ A sa }
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
4 / 32
Free analogs
free entropy
microstate
approach
microstate-free
approach
microstate/pressure
approach
mutual free information
χ Voiculescu 1994
χ˜
microstate/orbital
approach
Biane-Dabrowski 2013
χ∗ Voiculescu 1998
η H. 2005
Fumio Hiai (Tohoku University)
microstate-free/
liberation approach
microstate/orbital/
pressure approach
mutual information in free probability theory
χorb
χ˜ orb
H.-Miyamoto-Ueda 2009
Ueda 2014
Biane-Dabrowski 2013
i∗ Voiculescu 1999
ηorb H.-Ueda 2014
2014, July (at Krakow)
5 / 32
Definition: Voiculescu’s free entropy χ
Let (M, τ) be a tracial W ∗ -probability space, and
X1 , . . . , X n ∈ M sa . Let R > 0, N, m ∈ N and δ > 0. Define
Γ R (X1 , . . . , X n; N, m, δ)
:= {( A1 , . . . , A n) ∈ (M N (C) sa
)n :
R
|tr N ( Ai1 · · · Ai k ) − τ(Xi1 · · · Xi k )| < δ, 1 ≤ i j ≤ k, k ≤ m},
χ R (X1 , . . . , X n)
[ 1
]
n
⊗n
lim sup
:= lim
log Λ N (Γ R (X1 , . . . , X n; N, m, δ)) + log N ,
m→∞
2
N→∞
N2
δ↘0
χ(X1 , . . . , X n) := sup χ R (X1 , . . . , X n),
R>0
:= {A ∈ M N (C) : A = A∗ , ∥ A∥∞ ≤ R} and Λ N is
where M N (C) sa
R
the “Lebesgue” measure on M N (C) sa R N .
2
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
6 / 32
Free pressure π R and η-entropy
Setting and notations
n
x = (xi )i=1
is an n-tuple of non-commutative indeterminates.
n
For R > 0, C R (x) := ⋆i=1
C[−R, R] is the universal C∗ -free
product with the identification xi (t) = t in the ith copy of
C[−R, R].
TS(C R (x)) is the set of tracial states on C R (x).
n
For any unital C∗ -algebra A and a = (ai )i=1
in A sa with
∥ai ∥ ≤ R we have a canonical ∗-homomorphism
h ∈ C R (x) 7−→ h(a) ∈ A, uniquely determined by xi 7−→ ai .
In particular, we have a ∗-homomorphism
n
) n.
h ∈ C R (x) 7−→ h(A) ∈ M N (C) for A = ( Ai )i=1
∈ (M N (C) sa
R
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
7 / 32
Definition: Free pressure π R
For every h ∈ C R (x) sa define the free pressure of h by
π R (h) := lim sup
N→∞
1
N2
∫
log
(M N (C) sa ) n
( 2
)
dΛ⊗n
(A)
exp
−N
tr
(h(A))
N
N
R
(in analogy of log Tr e−H ).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
8 / 32
Definition: Free pressure π R
For every h ∈ C R (x) sa define the free pressure of h by
π R (h) := lim sup
N→∞
1
N2
∫
log
(M N (C) sa ) n
( 2
)
dΛ⊗n
(A)
exp
−N
tr
(h(A))
N
N
R
(in analogy of log Tr e−H ).
π R is convex on C R (x) sa and Lipschitz continuous, i.e.,
|π R (h1 ) − π R (h2 )| ≤ ∥h1 − h2 ∥ R for h1 , h2 ∈ C R (x) sa .
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
8 / 32
Definition: Free pressure π R
For every h ∈ C R (x) sa define the free pressure of h by
π R (h) := lim sup
N→∞
1
N2
∫
log
(M N (C) sa ) n
( 2
)
dΛ⊗n
(A)
exp
−N
tr
(h(A))
N
N
R
(in analogy of log Tr e−H ).
π R is convex on C R (x) sa and Lipschitz continuous, i.e.,
|π R (h1 ) − π R (h2 )| ≤ ∥h1 − h2 ∥ R for h1 , h2 ∈ C R (x) sa .
Definition
For every φ ∈ C R (x)∗,sa define
η R (φ) := inf{φ(h) + π R (h) : h ∈ C R (x) sa }.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
8 / 32
Proposition
If φ ∈ C R (x)∗,sa and η R (φ) > −∞, then φ ∈ TS(C R (x)).
For every h ∈ C R (x) sa ,
π R (h) = max{−τ(h) + η R (τ) : τ ∈ TS(C R (x))}.
Therefore, π R and η R are the Legendre transforms of each other,
similarly to the case of P(H) and S(ρ).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
9 / 32
Proposition
If φ ∈ C R (x)∗,sa and η R (φ) > −∞, then φ ∈ TS(C R (x)).
For every h ∈ C R (x) sa ,
π R (h) = max{−τ(h) + η R (τ) : τ ∈ TS(C R (x))}.
Therefore, π R and η R are the Legendre transforms of each other,
similarly to the case of P(H) and S(ρ).
Definition
n
Let X = (Xi )i=1
∈ (M sa ) n with ∥Xi ∥∞ ≤ R for all i. We have
τX(R) ∈ TS(C R (x)) determined by τX(R) (h) := τ(h(X)) for h ∈ C R (x).
Define
η R (X1 , . . . , X n) := η R (τ(R)
),
X
η(X1 , . . . , X n) := sup η R (X1 , . . . , X n)
R>0
called the η-entropy of (X1 , . . . , X n).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
9 / 32
Theorem: H. 2005
η(X) = η R (X) = χ(X) for a single X ∈ M sa and R ≥ ∥X∥∞ .
η(X1 , . . . , X n) is upper semicontinuous in the strong topology
on (M sa ) n.
η(X1 , . . . , X n) ≤ η(X1 , . . . , X k ) + η(X k+1 , . . . , X n) for 1 ≤ k < n.
χ(X1 , . . . , X n) ≤ η(X1 , . . . , X n). (χ = η does not hold in
general.)
If X1 , . . . , X n are free and R ≥ maxi ∥Xi ∥∞ , then
η(X1 , . . . , X n) = η R (X1 , . . . , X n) = χ(X1 , . . . , X n)
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
10 / 32
Remark
For each h0 ∈ C R (x) sa there exists a τ0 ∈ TS(C R (x)) such that
π R (h0 ) = −τ0 (h0 ) + η R (τ0 ), a variational principle.
Call τ0 an equilibrium tracial state associated with h0 .
From the general theory of Legendre transforms, the
uniqueness (no phase transition) of an equilibrium tracial state
associated with h0 is equivalent to the differentiability of π R (h)
at h0 , i.e.,
π R (h0 + t h) − π R (h0 )
exists.
t→0
t
π R is differentiable at points in a dense Gδ set of C R (x) sa .
However, no effective (sufficient) condition for h0 ∈ C R (x) sa at
which π R is differentiable is known so far.
lim
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
11 / 32
Orbital free entropy χorb
xi = (xi j ) r(i)
, 1 ≤ i ≤ n, are non-commutative
j=1
multi-indeterminates.
x := x1 ⊔ · · · ⊔ x n.
Consider the universal C∗ -free product
n
n
C R (x) = ⋆i=1
C R (xi ) = ⋆i=1
(C[−R, R]⋆r(i) ).
)
(∏ n
P i=1
(M N (C) sa ) r(i) is the set of Borel probability measures
∏n
on i=1
(M N (C) sa ) r(i) .
Consider the map
n
n
∏
∏
sa r(i)
Φ N : U(N) ×
(M N (C) ) −→
(M N (C) sa ) r(i) ,
n
i=1
i=1
n
n
n )
(Ui )i=1 , (Ai )i=1 7−→ (Ui Ai Ui∗ )i=1
(Ui Ai1 u∗i , . . . , Ui Air(i) Ui∗ ) for Ai = ( Ai j ) r(i)
.
j=1
(
where Ui Ai Ui∗ :=
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
12 / 32
Definition: Ueda
Let Xi = (Xi j ) j=1 ∈ (M sa ) r(i) , 1 ≤ i ≤ n. For each N, m ∈ N and
δ > 0 define
r(i)
χorb,R (X1 , . . . , X n; N, m, δ)
(
)
:= sup log (γ⊗n
⊗ µ) ◦ Φ−1
(Γ(X1 ⊔ · · · ⊔ X n; N, m, δ))
N
U(N)
µ
(∏ n
( µ is taken over P
χorb,R (X1 , . . . , X n) := lim
lim sup
m→∞
i=1
)
(M N (C) sa ) r(i) ),
1
χorb,R (X1 , . . . , X n; N, m, δ),
N2
χorb (X1 , . . . , X n) := sup χorb,R (X1 , . . . , X n)
δ↘0
N→∞
R>0
called the orbital free entropy of X1 , . . . , X n.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
13 / 32
When W ∗ (Xi ) is hyperfinite for each i, one can choose microstates
Ξi (N) = (ξi j (N)) r(i)
∈ (M N (C) sa ) r(i) such that ∥ξi j (N)∥∞ ≤ ∥Xi j ∥∞
j=1
and Ξi (N) −→ Xi as N → ∞ in the sense of multi-moments.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
14 / 32
When W ∗ (Xi ) is hyperfinite for each i, one can choose microstates
Ξi (N) = (ξi j (N)) r(i)
∈ (M N (C) sa ) r(i) such that ∥ξi j (N)∥∞ ≤ ∥Xi j ∥∞
j=1
and Ξi (N) −→ Xi as N → ∞ in the sense of multi-moments.
Proposition: Ueda
Assume that W ∗ (Xi ) is hyperfinite for every i, and let Ξi (N) be
chosen as above. Then
χorb,R (X1 , . . . , X n)
= lim
lim sup
m→∞
δ↘0
N→∞
1
N2
n
log γ⊗n
(Γ(X1 , . . . , X n; (Ξi (N))i=1
; N, m, δ)
U(N)
for any R ≥ max{∥Xi j ∥∞ : 1 ≤ i ≤ n, 1 ≤ j ≤ r(i)}, where
n
n
Γ(X1 , . . . , X n; (Ξi (N))i=1
; N, m, δ) is the set of (Ui )i=1
∈ U(N) n such
n
that (Ui Ξi (N)Ui∗ )i=1
∈ Γ(X1 ⊔ · · · ⊔ X n; N, m, δ).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
14 / 32
Returning to general X1 , . . . , X n,
Theorem: Ueda
χorb (X1 , . . . , X n) ≤ χorb (X1 , . . . , X k ) + χorb (X k+1 , . . . , X n) for
1 ≤ k < n.
(k)
(k)
If X1 ⊔ · · · ⊔ X n → X1 ⊔ · · · ⊔ X n in multi-moments as
k → ∞, then
χorb (X1 , . . . , X n) ≥ lim sup χorb (X(k)
, . . . , X(k)
n ).
1
k→∞
If Yi = (Yi j ) j=1 ⊂ W ∗ (Xi ) for 1 ≤ i ≤ n, then
χorb (X1 , . . . , X n) ≤ χorb (Y1 , . . . , Y n). In particular,
χorb (X1 , . . . , X n) depends only on the von Neumann
subalgebras W ∗ (Xi ).
s(i)
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
15 / 32
Theorem: Ueda (continued)
χorb (X1 , . . . , X n) = 0 if and only if each Xi has f.d.a. (i.e., for
every m ∈ N, δ > 0 and R > max j ∥Xi j ∥∞ , Γ R (Xi ; N, m, δ) , ∅
for some N ∈ N) and X1 , . . . , X n are free.
χorb (X1 , . . . , X n) = χorb,R (X1 , . . . , X n) for every
R ≥ max{∥Xi j ∥∞ : 1 ≤ i ≤ n, 1 ≤ j ≤ r(i)}.
∑n
χ(X1 ⊔ · · · ⊔ X n) ≤ χorb (X1 , . . . , X n) + i=1
χ(Xi ).
If Xi consists of a single variable Xi for every i, then
n
∑
χ(X1 , . . . , X n) = χorb (X1 , . . . , X n) +
χ(Xi ).
i=1
Remarks
Very recently, Ueda found an example of
χ(X ⊔ Y) = −∞ < χorb (X, Y) + χ(X) + χ(Y).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
16 / 32
Remarks (continued)
The last identity in the above theorem suggests that
−χorb (X1 , . . . , X n) is a variant of the mutual free information (at
least when the Xi ’s are single variables).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
17 / 32
Remarks (continued)
The last identity in the above theorem suggests that
−χorb (X1 , . . . , X n) is a variant of the mutual free information (at
least when the Xi ’s are single variables).
Voiculescu (1999) introduced the mutual free information
i∗ (A1 ; . . . ; A n) for ∗-subalgebras Ai , 1 ≤ i ≤ n, of M, so it is
very interesting (but perhaps quite difficult) to see whether the
equality
−χorb (X1 , . . . , X n) = i∗ (W ∗ (X1 ); . . . ; W ∗ (X n))
holds true or not.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
17 / 32
Remark (continued)
The problem is not at all easy even for two projections. Indeed,
Izumi and Ueda (independently, Collins and Kemp) recently
proved that
−χorb (p, q) = i∗ (C p + C(1 − p); Cq + C(1 − q)
for two projections p, q with τ( p) = τ(q) = 1/2.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
18 / 32
Remark (continued)
The problem is not at all easy even for two projections. Indeed,
Izumi and Ueda (independently, Collins and Kemp) recently
proved that
−χorb (p, q) = i∗ (C p + C(1 − p); Cq + C(1 − q)
for two projections p, q with τ( p) = τ(q) = 1/2.
Another variant of the orbital entropy is χ˜ orb proposed by Biane
and Dabrowski (2013). χorb is not greater than χ˜ orb , and both
coincides at least when W ∗ (X1 ⊔ · · · ⊔ X n) is a factor.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
18 / 32
Orbital free pressure πorb,R and ηorb -entropy
Definition: H.-Ueda
Let τi ∈ TS(C R (xi )), 1 ≤ i ≤ n, and h ∈ C R (x) sa . For every N, m ∈ N
and δ > 0 define
n
πorb,R (h : (τi )i=1
; N, m, δ) :=
∫
log
U(N) n
sup
∏
(Ai ) n ∈ n Γ R (τi ;N,m,δ)
i=1
i=1
(
)
n
dγ⊗n
(Vi ) exp −N2 tr N (h((Vi Ai Vi∗ )i=1
)) ,
U(N)
where γU(N) is the Haar probability measure on U(N). Then, define
n
πorb,R (h : (τi )i=1
) := lim
lim sup
m→∞
δ↘0
N→∞
1
N2
n
πorb,R (h : (τi )i=1
; N, m, δ)
n
called the orbital free pressure of h relative to (τi )i=1
.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
19 / 32
n
πorb,R (h; (τi )i=1
) is a convex and Lipschitz continuous function of
sa
h ∈ C R (x) .
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
20 / 32
n
πorb,R (h; (τi )i=1
) is a convex and Lipschitz continuous function of
sa
h ∈ C R (x) .
Proposition
Assume that the von Neumann algebra generated by xi via the
GNS representation associated with τi is hyperfinite for every i, and
choose microstates Ξi (N) ∈ (M N (C) sa ) r(i) with Ξi (N) −→ xi (under
τi ) as N → ∞. Then for every h ∈ C R (x) sa ,
n
πorb,R (h : (τi )i=1
)
∫
1
( 2
)
∗ n
= lim sup
log
dγ⊗n
(V
)
exp
−N
tr
(h((V
Ξ
(N)V
)
))
.
i
N
i
i
i i=1
U(n)
N→∞ N 2
U(N) n
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
20 / 32
Definition
For φ ∈ C R (x)∗,sa ,
n
n
ηorb,R (φ : (τi )i=1
) := inf{φ(h) + πorb,R (h : (τi )i=1
) : h ∈ C R (x) sa }.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
21 / 32
Definition
For φ ∈ C R (x)∗,sa ,
n
n
ηorb,R (φ : (τi )i=1
) := inf{φ(h) + πorb,R (h : (τi )i=1
) : h ∈ C R (x) sa }.
Proposition
n
If φ ∈ C R (x)∗,sa and ηorb,R (φ : (τi )i=1
) > −∞, then
φ ∈ TS(C R (x)) and φ|CR (xi ) = τi for every i.
For every h ∈ C R (x) sa ,
n
n
):
πorb,R (h : (τi )i=1
) = max{−τ(h) + ηorb,R (τ : (τi )i=1
τ ∈ TS(C R (x)), τ|CR (xi ) = τi , 1 ≤ i ≤ n}.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
21 / 32
Definition
For φ ∈ C R (x)∗,sa ,
n
n
ηorb,R (φ : (τi )i=1
) := inf{φ(h) + πorb,R (h : (τi )i=1
) : h ∈ C R (x) sa }.
Proposition
n
If φ ∈ C R (x)∗,sa and ηorb,R (φ : (τi )i=1
) > −∞, then
φ ∈ TS(C R (x)) and φ|CR (xi ) = τi for every i.
For every h ∈ C R (x) sa ,
n
n
):
πorb,R (h : (τi )i=1
) = max{−τ(h) + ηorb,R (τ : (τi )i=1
τ ∈ TS(C R (x)), τ|CR (xi ) = τi , 1 ≤ i ≤ n}.
For each τ ∈ TS(C R (x)), letting τi := τ|CR (xi ) , 1 ≤ i ≤ n, we write
n
ηorb,R (τ) := ηorb,R (τ : (τi )i=1
).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
21 / 32
Theorem: H.-Ueda
For 1 ≤ i ≤ n let Xi = (Xi j ) j=1 and Yi = (Yi j ) j=1 be in M sa such that
∥Xi j ∥∞ ≤ R and ∥Yi j ∥∞ ≤ S for all i, j. Let X := X1 ⊔ · · · ⊔ X n and
Y := Y1 ⊔ · · · ⊔ Y n, and take τ(R)
∈ TS(C R (x)) and τ(S)
∈ TS(CS (y)),
X
Y
r(i)
s(i)
where y := y1 ⊔ · · · ⊔ y n with multi-indeterminates yi = (yi j ) j=1 ,
1 ≤ i ≤ n. If Yi ⊂ W ∗ (Xi ) for every i, then
s(i)
) ≤ ηorb,S (τ(S)
).
ηorb,R (τ(R)
Y
X
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
22 / 32
Theorem: H.-Ueda
For 1 ≤ i ≤ n let Xi = (Xi j ) j=1 and Yi = (Yi j ) j=1 be in M sa such that
∥Xi j ∥∞ ≤ R and ∥Yi j ∥∞ ≤ S for all i, j. Let X := X1 ⊔ · · · ⊔ X n and
Y := Y1 ⊔ · · · ⊔ Y n, and take τ(R)
∈ TS(C R (x)) and τ(S)
∈ TS(CS (y)),
X
Y
r(i)
s(i)
where y := y1 ⊔ · · · ⊔ y n with multi-indeterminates yi = (yi j ) j=1 ,
1 ≤ i ≤ n. If Yi ⊂ W ∗ (Xi ) for every i, then
s(i)
) ≤ ηorb,S (τ(S)
).
ηorb,R (τ(R)
Y
X
In particular, when Xi = Yi for every i, ηorb,R (τX ) is independent of
the choice of R ≥ maxi, j ∥Xi j ∥∞ , and thus the next definition is
justified.
(R)
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
22 / 32
Definition
For every Xi = (Xi j ) j=1 ∈ (M sa ) r(i) , 1 ≤ i ≤ n, define
r(i)
ηorb (X1 , . . . , X n) := ηorb,R (τ(R)
)
X
with X := X1 ⊔ · · · ⊔ X n and R ≥ maxi, j ∥Xi j ∥∞ , called the orbital
η-entropy of X1 , . . . , X n.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
23 / 32
Theorem: H.-Ueda
ηorb (X1 , . . . , X n) ≤ ηorb (X1 , . . . , X k ) + ηorb (X k+1 , . . . , X n) for
1 ≤ k < n.
r(i)
(k)
(k) r(i)
If Xi = (Xi j ) j=1 and Xi = (Xi j ) j=1 are in M sa for 1 ≤ i ≤ n
and k ∈ N and Xi j −→ Xi j strongly as k → ∞ for every i, j,
then
(k)
ηorb (X1 , . . . , X n) ≥ lim sup ηorb (X1(k) , . . . , X(k)
n ).
k→∞
If Yi ⊂ W ∗ (Xi ) for every i, then
ηorb (X1 , . . . , X n) ≤ ηorb (Y1 , . . . , Y n). In particular,
ηorb (X1 , . . . , X n) depends only on the von Neumann
subalgebras W ∗ (Xi ).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
24 / 32
Theorem (continued)
χorb (X1 , . . . , X n) ≤ ηorb (X1 , . . . , X n). (χorb = ηorb does not hold
in general.)
ηorb (X1 , . . . , X n) = 0 if and only if each Xi has f.d.a. and
X1 , . . . , X n are free.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
25 / 32
Theorem (continued)
χorb (X1 , . . . , X n) ≤ ηorb (X1 , . . . , X n). (χorb = ηorb does not hold
in general.)
ηorb (X1 , . . . , X n) = 0 if and only if each Xi has f.d.a. and
X1 , . . . , X n are free.
Definition
Let τ ∈ TS(C R (x)) and h ∈ C R (x) sa . We say that τ is an orbital
equilibrium tracial state associated with h if
n
ηorb,R (τ) = τ(h) + πorb,R (h : (τi )i=1
)
holds with finite value, where τi := τ|CR (xi ) , 1 ≤ i ≤ n.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
25 / 32
We need the next transportation cost inequality to prove the last
assertion (characterizing freeness) of the above theorem.
Lemma
Let τ ∈ TS(C R (x)) and τi := τ|CR (xi ) , 1 ≤ i ≤ n. If τ is an orbital
equilibrium tracial state associated with some h ∈ C R (x) sa , then
√ √
n
W2 (τ, ⋆i=1
τi ) ≤ 4R r −ηorb (τ),
where r := maxi r(i) and W2 is the free 2-Wasserstein distance
introduced by Biane and Voiculescu (2001).
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
26 / 32
n
Let τ ∈ TS(C R (x)) and Ξ(N) = (Ξi (N))i=1
with
Ξi (N) = (ξi j (N)) r(i)
∈ (M N (C) sa
) r(i) be a sequence of microstates
R
j=1
such that the tracial state g ∈ C R (xi ) 7−→ tr N (g(Ξi (N)) converges to
τi := τ|CR (xi ) in the weak* topology as N → ∞ for every i. For given
h ∈ C R (x) sa , define the orbital Gibbs micro-ensemble on U(N) n as a
probability measure
dµ(h,Ξ(N))
(Vi ) :=
N
1
Z (h,Ξ(N))
N
n
exp(−N2 tr N (h((Vi Ξi (N)Vi∗ )i=1
))) dγ⊗n
(Vi ).
U(N)
The next result gives a sufficient criterion for orbital equilibrium
tracial states.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
27 / 32
Proposition
Assume that the von Neumann algebra generated by C R (xi ) with
respect τi is hyperfinite for every i. If
lim
N→∞
1
N2
(Γorb (τ : Ξ(N); N, m, δ)) = 0,
log µ(h,Ξ(N))
N
(1)
then τ is orbital equilibrium associated with h, and moreover
χorb (τ) = ηorb (τ) holds.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
28 / 32
Remark
Assumption (1) is satisfied when the empirical orbital tracial state
n
f ∈ C R (x) 7−→ tr N ( f ((Vi Ξi (N)Vi∗ )i=1
)) converges to τ in the weak*
topology as N → ∞, almost surely when (Vi ) ∈ U(N) n is distributed
(h,Ξ(N))
under µ N
. In fact, this implies a much stronger convergence
lim µ(h,Ξ(N))
(Γorb (τ : Ξ(N); N, m, δ)) = 1
N
N→∞
for every m ∈ N large enough and every δ > 0 small enough.
A random matrix model studied by Collins, Guionnet and
Maurel-Segala (2009) produces an example of orbital equilibrium
states with suitable self-adjoint polynomials h in x. Indeed, their
matrix model realizes the situations described in the above remark.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
29 / 32
References
Ph. Biane and Y. Dabrowski, Concavification of free entropy,
Adv. Math. 234 (2013), 667–696.
F. Hiai, Free analog of pressure and its Legendre transform, Comm.
Math. Phys. 255 (2005), 229–252.
F. Hiai, T. Miyamoto and Y. Ueda, Orbital approach to microstate free
entropy, Internat. J. Math. 20 (2009), 227–273.
F. Hiai and Y. Ueda, Orbital free pressure and its Legendre transform,
Comm. Math. Phys., to appear. arXiv:1310.3877.
Y. Ueda, Orbital free entropy, revisited, Indiana Univ. Math. J., to
appear.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
30 / 32
References (continued)
D. Voiculescu, The analogues of entropy and of Fisher’s information
measure in free probability theory, II, Invent. Math. 118 (1994),
411–440.
D. Voiculescu, The analogues of entropy and of Fisher’s information
measure in free probability theory, V, Noncommutative Hilbert
transforms, Invent. Math. 132 (1998), 189–227.
D. Voiculescu, The analogue of entropy and of Fisher’s information
measure in free probability theory VI: Liberation and mutual free
information, Adv. Math. 146 (1999), 101–166.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
31 / 32
Thank you for your attention.
Fumio Hiai (Tohoku University)
mutual information in free probability theory
2014, July (at Krakow)
32 / 32
© Copyright 2024 ExpyDoc
