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

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