Free Pressure, Free Entropy and Hypothesis Testing

Home , Free entropy

Free pressure, free entropy and hypothesis testing

Fumio Hiai (Tohoku University)

2008, January (at Banﬀ)

1 Plan

1. Hypothesis testing: conventional framework

2. Free pressure and free entropy: microstate approach

3. Free analog of hypothesis testing – free Stein’s lemma

4. The single variable case

2 1. Hypothesis testing: conventional framework

• (Hn): a sequence of ﬁnite-dimensional Hilbert spaces

• ρn, σn: states on Hn

• Null-hypothesis (H0): the true state of the nth system is ρn

• Counter-hypothesis (H1): the true state of the nth system is σn

• Test: binary measurement 0 ≤ Tn ≤ I on Hn

Tn corresponds to outcome 0, I − Tn corresponds to outcome 1

outcome = 0: (H0) is accepted, outcome = 1: (H1) is accepted

• Error probabilities of the ﬁrst/second kinds:

αn(Tn) := ρn(In − Tn), βn(Tn) := σn(Tn)

3 Bayesian error probabilities

• ρn and σn have a priori probabilities πn and 1 − πn

• Optimal Bayesian probability of an erroneous decision:

Pmin(ρn : σn|πn) := min πnαn(Tn) + (1 − πn)βn(Tn) 0≤Tn≤I n o

Results for i.i.d. case

⊗n ⊗n ⊗n • I.i.d. setting: Hn = H , ρn = ρ1 , σn = σ1 • Rate function: ψ(t) := log Tr ρt σ1−t, ϕ(a) := max {at − ψ(t)} 1 1 0≤t≤1

Stein’s lemma (H-Petz, 1991; Ogawa-Nagaoka, 2000) 1 lim log min{βn(Tn) : αn(Tn) ≤ ε} = −S(ρ1, σ1) for any 0 <ε< 1. n→∞ n

4 Chernoﬀ bound (Audenaert-Calsamiglia-et al., Nussbaum-Szko la, 2006) 1 lim log Pmin(ρn : σn|π)= min ψ(t)= −ϕ(0) n→∞ n 0≤t≤1

Hoeﬀding bound (Hayashi, Nagaoka, 2006) For any r ∈ R, 1 1 −tr − ψ(t) inf lim sup log βn(Tn) : lim sup log αn(Tn) < −r = − max . (Tn)( n n n n ) 0≤t<1 1 − t

Results for non-i.i.d. case

H-Mosonyi-Ogawa

• Large deviations and Chernoﬀ bound for certain correlated states on the spin chain, J. Math. Phys.

• Error exponents in hypothesis testing for correlated states on a spin chain, J. Math. Phys.

5 2. Free pressure and free entropy: microstate approach sa C ∗ • For R> 0, (MN )R := A ∈ MN ( ) : A = A , kAk≤ R n o sa ∼ RN 2 • ΛN : the “Lebesgue” measure on MN =

(n) ⋆n ∗ • AR := C([−R, R]) : the n-fold universal free product C -algebra, ∗ i.e., the C -completion of ChX1,...,Xni w.r.t. the norm sa N kpkR := sup kp(A1,...,An)k : A1,...,An ∈ (MN )R, N ∈ n o (n) (n) • TS(AR ): the set of tracial states on AR

6 (n) • Free entropy: for µ ∈ TS(AR ), 1 n ( ):= lim lim sup logΛ⊗n(Γ ( ; )) + log χR µ 2 N R µ N,m,δ N m→∞,δց0 N→∞ "N 2 #

(n) sa • Free pressure (free energy): for h ∈ (AR ) (considered as a free probabilistic potential), 1 π (h) := lim sup log exp −N 2tr (h(A ,...,A )) R 2 sa n N 1 n N→∞ "N (M ) Z N R ⊗n n dΛN (A1,...,An)+ log N 2 #

(n) • η-version of free entropy: for µ ∈ TS(AR ), (n) sa ηR(µ) := inf µ(h)+ πR(h) : h ∈ (AR ) , n o the (minus) Legendre transform of πR.

7 (n) sa • For every h0 ∈ (AR ) , (n) πR(h0) = max −µ(h0)+ ηR(µ) : µ ∈ TS(AR ) . (n) n o µ0 ∈ TS(AR ) is called an equilibrium tracial state associated with h0 if

πR(h0)= −µ0(h0)+ ηR(µ0).

(n) sa Note An equilibrium tracial state exists for every h ∈ (AR ) , and (n) sa is unique for almost all h ∈ (AR ) (i.e., in a dense Gδ subset) by the Baire category theorem. But it is not easy to prove the uniqueness for a given h.

Fact

ηR(µ) ≥ χR(µ) and equality holds if X1,...,Xn are free w.r.t. µ.

8 3. Free analog of hypothesis testing – free Stein’s lemma (n) sa N • Micro Gibbs measure: for h ∈ (AR ) and N ∈ , h 1 2 dλ (A1,...,An) := exp −N trN (h(A1,...,An)) R,N Zh R,N ⊗n × χ sa n (A1,...,An) dΛ (A1,...,An) (MN )R N h with normalization constant ZR,N .

(n) sa • Micro pressure: for h ∈ (AR ) , h PR,N (h) := log ZR,N 2 ⊗n = log exp −N trN (h(A1,...,An)) dΛ (A1,...,An) (M sa)n N Z N R

(n) sa h0 (n) • N-level tracial state: for each h0 ∈ (AR ) , µR,N ∈ TS(AR ) is deﬁned by

h0 h0 (n) µ (h) := trN (h(A1,...,An)) dλ (A1,...,An) for h ∈ A . R,N (M sa)n R,N R Z N R 9 Fact If limit exists in the deﬁnition of πR(h0), i.e., 1 n πR(h0)= lim PR,N (h0)+ log N , N→∞ N 2 2 !

h0 then any limit point of (µR,N )N∈N is an equilibrium tracial state associated with h0.

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(n) sa Let h0, h1 ∈ (AR ) and consider the hypothesis testing for

h0 h1 (λR,N )N∈N (null-hypothesis) vs. (λR,N )N∈N (counter-hypothesis). sa n For a Borel subset (test) T ⊂ (MN )R,

h0 c h1 αN (T ) := λR,N (T ), βN (T ) := λR,N (T ).

10 For the free Stein’s lemma, deﬁne for 0 <ε< 1

h1 h0 h0 sa n h1 c N βε(λR,N kλR,N ) := min λR,N (T ) : T ⊂ (MN )R, λR,N (T ) ≤ ε , N ∈ , n 1 o B((λh1 ) (λh0 )) := inf lim inf log λh0 (T ): lim λh1 (T c ) = 0 , R,N k R,N 2 R,N N R,N N (TN )( N→∞ N N→∞ ) 1 B((λh1 ) (λh0 )) := inf lim sup log λh0 (T ): lim λh1 (T c ) = 0 , R,N k R,N 2 R,N N R,N N (TN )( N→∞ N N→∞ ) 1 (( h1 ) ( h0 )) := inf lim log h0 ( ): lim h1 ( c ) = 0 B λR,N k λR,N 2 λR,N TN λR,N TN . (TN )(N→∞ N N→∞ )

1 h1 h0 h1 h0 sup lim inf log βε(λR,N kλR,N ) = B((λR,N )k(λR,N )) ε>0 N→∞ N 2 ! 1 (( h1 ) ( h0 )) = sup lim sup log ( h1 h0 ) ≤ B λR,N k λR,N 2 βε λR,N kλR,N ε>0 N→∞ N !

11 Theorem Assume that there is a unique equilibrium tracial state µh1 associated with h1. Then for every 0 <ε< 1, 1 lim sup log β (λh1 λh0 ) η (µ ) µ (h ) π (h ) 2 ε R,N k R,N ≥ R h1 − h1 0 − R 0 N→∞ N

≥ χR(µh1) − µh1(h0) − πR(h0). If, moreover, limit exists in the deﬁnition of πR(h1), then for every 0 <ε< 1,

1 h1 h0 lim inf log βε(λ kλ ) ≥ ηR(µh ) − µh (h0) − πR(h0). N→∞ N 2 R,N R,N 1 1

Theorem Assume that limit exists in the deﬁnition of πR(h1). Then h1 for any limit point µ of (µR,N )N∈N,

h1 h0 B((λR,N )k(λR,N )) ≥ ηR(µ) − µ(h0) − πR(h0).

h1 Moreover, there exists a limit point µ1 of (µR,N )N∈N such that

h1 h0 B((λR,N )k(λR,N )) ≥ ηR(µ1) − µ1(h0) − πR(h0).

12 In particular when h0 = 0, the theorems give

(n) sa Cor. Let h ∈ (AR ) and assume that there is a unique equilibrium tracial state µh associated with h. Then

χR(µh) ≤ ηR(µh) 1 n lim sup log min Λ⊗n(T ) : T (M sa)n , λh (T c) ε + log N ≤ 2 N ⊂ N R R,N ≤ N→∞ "N 2 # n o for every 0 <ε< 1. If, moreover, limit exists in the deﬁnition of πR(h), then for every 0 <ε< 1,

ηR(µh) 1 ⊗n sa n h c n ≤ lim inf log min ΛN (T ) : T ⊂ (MN )R, λR,N (T ) ≤ ε + log N . N→∞ "N 2 2 # n o

13 (n) sa Cor. Let h ∈ (AR ) and assume that limit exists in the deﬁnition of h πR(h). Then for any limit point µ of (µR,N )N∈N,

ηR(µ) 1 n inf lim sup logΛ⊗n(T )+ log N : lim λh (T c ) = 0 . ≤ 2 N N R,N N (TN )( N→∞ N 2 ! N→∞ ) h Moreover, for some limit point µ1 of (µR,N )N∈N,

ηR(µ1) 1 n inf lim inf logΛ⊗n( )+ log : lim h ( c ) = 0 ≤ 2 N TN N λR,N TN . (TN )( N→∞ N 2 ! N→∞ )

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(n) sa sa n Let h0 ∈ (AR ) . For each (A, ...,An) ∈ (MN )R deﬁne (n) µ (h) := tr (h(A ,...,A )), h ∈ A , N,(A1,...,An) N 1 n R which is a random tracial state when (A1,...,An) is distributed under h0 λR,N . 14 Fact

(1) If the random tracial state µ satisﬁes LDP in the scale N,(A1,...,An) −2 N with a good rate function having a unique minimizer µ0, then µ weakly* converges to µ almost surely and so N,(A1,...,An) 0 h0 N λR,N (ΓR(µ0; N,m,δ)) → 1 as N →∞ for every m ∈ and δ > 0.

h0 N (2) If λR,N (ΓR(µ0; N,m,δ)) → 1 as N → ∞ for every m ∈ and δ > 0, h0 then µR,N → µ0 weakly* as N →∞.

Cor In addition to the assumption of (2), assume

(i) µ0 is a unique equilibrium tracial state associated with h0, or

(ii) limit exists in the deﬁnition of πR(h0).

Then ηR(µ0)= χR(µ0). Moreover, in the case (ii), µ0 is regular.

15 Guionnet – Maurel-Segala, 2006 k C sa Consider a potential Vt = i=1 tiqi ∈ hX1,...,Xni with monomials qi. t There exists an ε> 0 suchP that when | | < ε the following hold:

(i) The Schwinger-Dyson equation µ ⊗ µ(∂ih) = µ((DiVt + Xi)h) has a (n) unique solution µt (∈ TS(AR )).

(ii) Limit exists in the deﬁnition of πR(Vt).

Vt (iii) µ , under λ , weakly* converges to µt almost surely. N,(A1,...,AN ) R,N

(iv) µt is regular.

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Consequently, the above µt is an equilibrium (unique?) tracial state associated with Vt and ηR(µt)= χR(µt).

16 4. The single variable case (1) (1) AR = C([−R, R]), TS(AR ) = Prob([−R, R]).

Fact For every h ∈ CR([−R, R]),

1 2 1 • πR(h)= lim exp −N trN (h(A)) dΛN (A)+ log N , N→∞N 2 (M sa) 2  Z N R   1 3 • ηR(µ)= χR(µ)= log |x − y| dµ(x) dµ(y)+ log 2π + , ZZ 2 4

• there exists a unique equilibrium measure µh ∈ Prob([−R, R]):

πR(µh)= −µh(h)+ χR(µh),

• πR is Gˆateaux diﬀerentiable and ′ πR(h + th ) − πR(h) ′ ′ lim = −µh(h ), h ∈ CR([−R, R]). t→0 t

17 Let h0, h1 ∈ CR([−R, R]).

Theorem (Chernoﬀ bound) Deﬁne

ψ(s) := πR(sh1 + (1 − s)h0) − sπR(h1) − (1 − s)πR(h0), 0 ≤ s ≤ 1, ϕ(a) := sup {as − ψ(s)}, a ∈ R. 0≤s≤1 Then for every a ∈ R with a 6= ψ′(0),ψ′(1),

2 1 −N a h1 c h0 lim log min e λR,N (T )+ λR,N (T ) = −ϕ(a). N→∞ N 2 T ⊂(M sa) N Rn o

Fact ′ ψ (0) = χR(µh0) − µh0(h1) − πR(h1), ′ ψ (1) = −χR(µh1)+ µh1(h0)+ πR(h0), ′ ψ (1) is the relative free entropy of µh1 with respect to h0 (or µh0) introduced by Biane-Speicher, 2001. 18 For r ∈ R introduce

B(r|µh1kµh0) 1 1 := inf lim inf log λh0 (T ) : lim sup log λh1 (T c ) < r , 2 R,N N 2 R,N N − (TN )( N→∞ N N→∞ N )

B(r|µh1kµh0) 1 1 := inf lim sup log h0 ( ) : lim sup log h1 ( c ) 2 λR,N TN 2 λR,N TN < −r , (TN )( N→∞ N N→∞ N )

B(r|µh1kµh0) 1 1 := inf lim log h0 ( ) : lim sup log h1 ( c ) 2 λR,N TN 2 λR,N TN < −r . (TN )(N→∞ N N→∞ N )

Theorem (Hoeﬀding bound) For any r ∈ R, −sr − ψ(s) B(r|µh1kµh0)= B(r|µh1kµh0)= B(r|µh1kµh0)= − sup . 0≤s≤1 1 − s

19 Theorem (free Stein’s lemma) For every 0 <ε< 1,

1 h1 h0 lim log βε(λ kλ ) N→∞ N 2 R,N R,N

= B(µh1kµh0)= B(µh1kµh0)= B(µh1kµh0)

= B(0|µh1kµh0)= B(0|µh1kµh0)= B(0|µh1kµh0) = −ψ′(1)

= χR(µh1) − µh1(h0) − πR(h0).

20 In particular when h0 = 0, the above theorem gives

Cor. For every h ∈ CR([−R, R]) and 0 <ε< 1,

χR(µh) 1 sa h c 1 = lim log min ΛN (T ) : T ⊂ (MN )R, λR,N (T ) ≤ ε + log N N→∞"N 2 2 # n o 1 1 = inf lim inf logΛ (T )+ log N : lim λh (T c ) = 0 2 N N R,N N (TN )( N→∞ N 2 ! N→∞ ) 1 1 = inf lim sup logΛ (T )+ log N : lim λh (T c ) = 0 2 N N R,N N (TN )( N→∞ N 2 ! N→∞ ) 1 1 = inf lim logΛ (T )+ log N : lim λh (T c ) = 0 . 2 N N R,N N (TN )(N→∞ N 2 ! N→∞ )

21 Thank you for attention!