Free pressure, free entropy and hypothesis testing Fumio Hiai - PowerPoint PPT Presentation

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 frame- work • ( H n ): a sequence of finite-dimensional Hilbert spaces • ρ n , σ n : states on H n • Null-hypothesis (H0): the true state of the n th system is ρ n • Counter-hypothesis (H1): the true state of the n th system is σ n • Test: binary measurement 0 ≤ T n ≤ I on H n T n corresponds to outcome 0, I − T n corresponds to outcome 1 outcome = 0: (H0) is accepted, outcome = 1: (H1) is accepted • Error probabilities of the first/second kinds: α n ( T n ) := ρ n ( I n − T n ) , β n ( T n ) := σ n ( T n ) 3

Bayesian error probabilities • ρ n and σ n have a priori probabilities π n and 1 − π n • Optimal Bayesian probability of an erroneous decision: � � P min ( ρ n : σ n | π n ) := min π n α n ( T n ) + (1 − π n ) β n ( T n ) 0 ≤ T n ≤ I Results for i.i.d. case ρ n = ρ ⊗ n σ n = σ ⊗ n • I.i.d. setting: H n = H ⊗ n , 1 , 1 1 σ 1 − t • Rate function: ψ ( t ) := log Tr ρ t , ϕ ( a ) := max 0 ≤ t ≤ 1 { at − ψ ( t ) } 1 Stein’s lemma (H-Petz, 1991; Ogawa-Nagaoka, 2000) 1 lim n log min { β n ( T n ) : α n ( T n ) ≤ ε } = − S ( ρ 1 , σ 1 ) for any 0 < ε < 1 . n →∞ 4

Chernoff bound (Audenaert-Calsamiglia-et al., Nussbaum-Szko� la, 2006) 1 lim n log P min ( ρ n : σ n | π ) = min 0 ≤ t ≤ 1 ψ ( t ) = − ϕ (0) n →∞ Hoeffding bound (Hayashi, Nagaoka, 2006) For any r ∈ R , � � 1 1 − tr − ψ ( t ) inf lim sup n log β n ( T n ) : lim sup n log α n ( T n ) < − r = − max . 1 − t n n 0 ≤ t< 1 ( T n ) 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 � � A ∈ M N ( C ) : A = A ∗ , � A � ≤ R • For R > 0, ( M sa N ) R := = R N 2 N ∼ • Λ N : the “Lebesgue” measure on M sa • A ( n ) := C ([ − R, R ]) ⋆n : the n -fold universal free product C ∗ -algebra, R i.e., the C ∗ -completion of C � X 1 , . . . , X n � w.r.t. the norm � � � p ( A 1 , . . . , A n ) � : A 1 , . . . , A n ∈ ( M sa � p � R := sup N ) R , N ∈ N • TS ( A ( n ) R ): the set of tracial states on A ( n ) R 6

• Free entropy: for µ ∈ TS ( A ( n ) R ), � � 1 N (Γ R ( µ ; N, m, δ )) + n N 2 log Λ ⊗ n χ R ( µ ) := m →∞ ,δ ց 0 lim sup lim 2 log N N →∞ for h ∈ ( A ( n ) R ) sa (considered as a free • Free pressure (free energy): probabilistic potential), � 1 � � − N 2 tr N ( h ( A 1 , . . . , A n )) � π R ( h ) := lim sup N 2 log exp ( M sa N ) n N →∞ R � N ( A 1 , . . . , A n ) + n d Λ ⊗ n 2 log N • η -version of free entropy: for µ ∈ TS ( A ( n ) R ), µ ( h ) + π R ( h ) : h ∈ ( A ( n ) � R ) sa � η R ( µ ) := inf , the (minus) Legendre transform of π R . 7

• For every h 0 ∈ ( A ( n ) R ) sa , − µ ( h 0 ) + η R ( µ ) : µ ∈ TS ( A ( n ) � � π R ( h 0 ) = max R ) . µ 0 ∈ TS ( A ( n ) R ) is called an equilibrium tracial state associated with h 0 if π R ( h 0 ) = − µ 0 ( h 0 ) + η R ( µ 0 ) . An equilibrium tracial state exists for every h ∈ ( A ( n ) Note R ) sa , and is unique for almost all h ∈ ( A ( n ) R ) sa (i.e., in a dense G δ subset) by the Baire category theorem. But it is not easy to prove the uniqueness for a given h . Fact η R ( µ ) ≥ χ R ( µ ) and equality holds if X 1 , . . . , X n are free w.r.t. µ . 8

3. Free analog of hypothesis testing – free Stein’s lemma • Micro Gibbs measure: for h ∈ ( A ( n ) R ) sa and N ∈ N , 1 − N 2 tr N ( h ( A 1 , . . . , A n )) dλ h � � R,N ( A 1 , . . . , A n ) := exp Z h R,N R ( A 1 , . . . , A n ) d Λ ⊗ n × χ ( M sa N ( A 1 , . . . , A n ) N ) n with normalization constant Z h R,N . • Micro pressure: for h ∈ ( A ( n ) R ) sa , P R,N ( h ) := log Z h R,N � � − N 2 tr N ( h ( A 1 , . . . , A n )) � d Λ ⊗ n = log exp N ( A 1 , . . . , A n ) ( M sa N ) n R • N -level tracial state: for each h 0 ∈ ( A ( n ) R ) sa , µ h 0 R,N ∈ TS ( A ( n ) R ) is defined by � for h ∈ A ( n ) µ h 0 tr N ( h ( A 1 , . . . , A n )) dλ h 0 R,N ( h ) := R,N ( A 1 , . . . , A n ) R . ( M sa N ) n R 9

Fact If limit exists in the definition of π R ( h 0 ), i.e., � � 1 N 2 P R,N ( h 0 ) + n π R ( h 0 ) = lim 2 log N , N →∞ then any limit point of ( µ h 0 R,N ) N ∈ N is an equilibrium tracial state associ- ated with h 0 . ————————— Let h 0 , h 1 ∈ ( A ( n ) R ) sa and consider the hypothesis testing for ( λ h 0 ( λ h 1 R,N ) N ∈ N (null-hypothesis) vs. R,N ) N ∈ N (counter-hypothesis) . For a Borel subset (test) T ⊂ ( M sa N ) n R , α N ( T ) := λ h 0 β N ( T ) := λ h 1 R,N ( T c ) , R,N ( T ) . 10

For the free Stein’s lemma, define for 0 < ε < 1 β ε ( λ h 1 R,N � λ h 0 λ h 0 R , λ h 1 � R,N ( T ) : T ⊂ ( M sa N ) n R,N ( T c ) ≤ ε � R,N ) := min N ∈ N , , � � 1 B (( λ h 1 R,N ) � ( λ h 0 N 2 log λ h 0 N →∞ λ h 1 R,N ( T c R,N )) := inf lim inf R,N ( T N ) : lim N ) = 0 , N →∞ ( T N ) � � 1 B (( λ h 1 R,N ) � ( λ h 0 N 2 log λ h 0 N →∞ λ h 1 R,N ( T c R,N )) := inf lim sup R,N ( T N ) : lim N ) = 0 , ( T N ) N →∞ � � 1 B (( λ h 1 R,N ) � ( λ h 0 N 2 log λ h 0 N →∞ λ h 1 R,N ( T c R,N )) := inf lim R,N ( T N ) : lim N ) = 0 . N →∞ ( T N ) � � 1 N 2 log β ε ( λ h 1 R,N � λ h 0 = B (( λ h 1 R,N ) � ( λ h 0 sup lim inf R,N ) R,N )) N →∞ ε> 0 � � 1 ≤ B (( λ h 1 R,N ) � ( λ h 0 N 2 log β ε ( λ h 1 R,N � λ h 0 R,N )) = sup lim sup R,N ) ε> 0 N →∞ 11

Theorem Assume that there is a unique equilibrium tracial state µ h 1 associated with h 1 . Then for every 0 < ε < 1, 1 N 2 log β ε ( λ h 1 R,N � λ h 0 lim sup R,N ) ≥ η R ( µ h 1 ) − µ h 1 ( h 0 ) − π R ( h 0 ) N →∞ ≥ χ R ( µ h 1 ) − µ h 1 ( h 0 ) − π R ( h 0 ) . If, moreover, limit exists in the definition of π R ( h 1 ), then for every 0 < ε < 1, 1 N 2 log β ε ( λ h 1 R,N � λ h 0 lim inf R,N ) ≥ η R ( µ h 1 ) − µ h 1 ( h 0 ) − π R ( h 0 ) . N →∞ Theorem Assume that limit exists in the definition of π R ( h 1 ). Then for any limit point µ of ( µ h 1 R,N ) N ∈ N , B (( λ h 1 R,N ) � ( λ h 0 R,N )) ≥ η R ( µ ) − µ ( h 0 ) − π R ( h 0 ) . Moreover, there exists a limit point µ 1 of ( µ h 1 R,N ) N ∈ N such that B (( λ h 1 R,N ) � ( λ h 0 R,N )) ≥ η R ( µ 1 ) − µ 1 ( h 0 ) − π R ( h 0 ) . 12

In particular when h 0 = 0, the theorems give Let h ∈ ( A ( n ) R ) sa and assume that there is a unique equilibrium Cor. tracial state µ h associated with h . Then χ R ( µ h ) ≤ η R ( µ h ) � � 1 + n � �� Λ ⊗ n � N ( T ) : T ⊂ ( M sa N ) n R , λ h R,N ( T c ) ≤ ε ≤ lim sup N 2 log min 2 log N N →∞ for every 0 < ε < 1. If, moreover, limit exists in the definition of π R ( h ), then for every 0 < ε < 1, η R ( µ h ) � � 1 + n � �� � Λ ⊗ n N ( T ) : T ⊂ ( M sa N ) n R , λ h R,N ( T c ) ≤ ε ≤ lim inf N 2 log min 2 log N . N →∞ 13

Let h ∈ ( A ( n ) R ) sa and assume that limit exists in the definition of Cor. π R ( h ). Then for any limit point µ of ( µ h R,N ) N ∈ N , η R ( µ ) � � � � 1 N ( T N ) + n N 2 log Λ ⊗ n N →∞ λ h R,N ( T c ≤ inf lim sup 2 log N : lim N ) = 0 . ( T N ) N →∞ Moreover, for some limit point µ 1 of ( µ h R,N ) N ∈ N , η R ( µ 1 ) � � � � 1 N ( T N ) + n N 2 log Λ ⊗ n N →∞ λ h R,N ( T c ≤ inf lim inf 2 log N : lim N ) = 0 . N →∞ ( T N ) ————————— Let h 0 ∈ ( A ( n ) R ) sa . For each ( A , . . . , A n ) ∈ ( M sa N ) n R define h ∈ A ( n ) µ N, ( A 1 ,...,A n ) ( h ) := tr N ( h ( A 1 , . . . , A n )) , R , which is a random tracial state when ( A 1 , . . . , A n ) is distributed under λ h 0 R,N . 14

Fact (1) If the random tracial state µ N, ( A 1 ,...,A n ) satisfies LDP in the scale N − 2 with a good rate function having a unique minimizer µ 0 , then µ N, ( A 1 ,...,A n ) weakly* converges to µ 0 almost surely and so λ h 0 R,N (Γ R ( µ 0 ; N, m, δ )) → 1 as N → ∞ for every m ∈ N and δ > 0. (2) If λ h 0 R,N (Γ R ( µ 0 ; N, m, δ )) → 1 as N → ∞ for every m ∈ N and δ > 0, then µ h 0 R,N → µ 0 weakly* as N → ∞ . Cor In addition to the assumption of (2), assume (i) µ 0 is a unique equilibrium tracial state associated with h 0 , or (ii) limit exists in the definition of π R ( h 0 ). Then η R ( µ 0 ) = χ R ( µ 0 ). Moreover, in the case (ii), µ 0 is regular. 15