On asymmetric quantum hypothesis testing

Transcription

1 On asymmetric quantum hypothesis testing JMP, Vol 57, 6, / arxiv: Cambyse Rouzé (Cambridge) Joint with Nilanjana Datta (University of Cambridge) and Yan Pautrat (Paris-Saclay) Workshop: Quantum trajectories, parameter and state estimation Jan 2017, Toulouse January 25, / 31

2 Overview 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 2 / 31

3 Motivation: the quantum Stein lemma and its refinements Framework Let H finite dimensional Hilbert space, ρ, σ D(H) +, set of states on H Goal is to distinguish between: ρ (null hypothesis) and σ (alternative hypothesis). A test is a POVM {T, 1 T }, T B(H), 0 T 1 Errors when guessing are therefore given by: α(t ) = Tr(ρ(1 T )) β(t ) = Tr(σT ) type I error type II error Asymmetric case, minimize the type II error while controlling the type I error: β(ɛ) = inf {β(t ) α(t ) ɛ}. 0 T 1 3 / 31

4 Motivation: the quantum Stein lemma and its refinements Size-dependent hypothesis testing Let H n a sequence of Hilbert spaces, {ρ n} n N, {σ n} n N D(H n) +. Examples: the i.i.d. case: ρ n = ρ n vs. σ n = σ n. Gibbs states on a lattice Λ n of size Λ n = n d : e βρhρ Λn e βσ Hσ Λn ρ n := ( ) vs. σ n := ( ). Tr e βρhρ Λn Tr e βσ Hσ Λn Quasi-free fermions on a lattice etc. Question: Given a uniform bound ɛ on the type I errors, how quickly does β n(ɛ) decrease with n? 4 / 31

5 Motivation: the quantum Stein lemma and its refinements First-order asymptotics First proved in the i.i.d. setting Theorem (Quantum Stein s lemma Hiai&Petz91, Ogawa&Nagaoka00) When ρ n = ρ n, and σ n = σ n, 1 log βn(ɛ) D(ρ σ), ɛ (0, 1), where n D(ρ σ) = Tr(ρ(log ρ log σ)) Umegaki s relative entropy. r < D(ρ σ) lim 1 n n log min{α(tn) : β(tn) e nr } := sup 0<α<1 r > D(ρ σ) lim 1 n n log max{(1 α(tn)) : β(tn) e nr } = sup 1<α D α(ρ σ) : = 1 α 1 log Tr(ρα σ 1 α ) α 1 [r Dα(ρ σ)], α α 1 α [r D α(ρ σ)], where Rényi-α divergence, D α (ρ σ) : = 1 α 1 log Tr(ρ1/2 σ 1 α α ρ 1/2 ) α sandwiched Rényi-α divergence. 5 / 31

6 Motivation: the quantum Stein lemma and its refinements Interpretation of Stein s lemma α (1) 1 0 D(ρ σ) t 1 α (1) = inf {lim sup α(t n) : 0 T n 1 n lim inf n 1 n log β(tn) t 1}. Discontinuity of α (1) : manifestation of coarse-grained analysis. Question: can we better quantify the behaviour of the asymptotic type I error by adding a sub-exponential term for the type II error? 1 1 Lots of generalisations to the non i.i.d. framework: Bjelakovic&Siegmund-Schultze04, Bjelakovic,&Deuschel&Krueger&Seiler&Siegmund-Schultze&Szkola08, Hiai&Mosonyi&Ogawa08, Mosonyi&Hiai&Ogawa&Fannes08, Brandao&Plenio10, Jaksic&Ogata&Pillet&Seiringer12, etc. 6 / 31

7 Second-order asymptotics in Stein s lemma 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 7 / 31

8 Second-order asymptotics in Stein s lemma Second-order asymptotics in the i.i.d. case Tomamichel&Hayashi13 and Li14 proved the following second-order result: Theorem (Second-order asymptotics: the i.i.d. case Tomamichel&Hayashi13, Li14 ) When ρ n = ρ n and σ n = σ n, log β n(ɛ) = nd(ρ σ) + ns 1 (ɛ) + O(log n), s 1 (ɛ) := V (ρ σ)φ 1 (ɛ), where V (ρ σ) = Tr(ρ(log ρ log σ) 2 ) D(ρ σ) 2 quantum information variance, and Φ the cumulative distribution function of law N (0, 1). α (2) α (2) (t 2) = inf 0 Tn 1 { lim sup α(t n) lim inf n n 0 1 ( log βn(tn)+n D(ρ σ) ) } t 2 = Φ(t2 / n t 2 V (ρ σ)). 8 / 31

9 Second-order asymptotics in Stein s lemma Main result 1: Second-order asymptotics in the non-i.i.d. scenario For a given strictly increasing sequence of weights w n, under some conditions (see later), the quantum information variance rate is well-defined, and: Theorem (DPR16) Fix ɛ (0, 1). Define Then 1 v({ρ n}, {σ n}) = lim V (ρ n n σ n) w n t 2 (ɛ) = v({ρ n}, {σ n}) Φ 1 (ɛ). Optimality: t 2 > t 2 (ɛ) there exists a function ft 2 (x) = + o( x) such that n N and any sequence of tests T n: log β(t n) D(ρ n σ n) + w n t 2 + f t2 (w n), (1) Achievability: t 2 < t 2 (ɛ), f (x) = o( x), there exists a sequence of test T n such that + for n large enough: log β n(t n) D(ρ n σ n) + w n t 2 + f (w n). (2) The i.i.d. case of Tomamichel&Hayashi13 and Li14 follows (Bryc CLT). The Berry-Esseen theorem even provides information on third order: log β n(ɛ) = nd(ρ σ) + nv (ρ σ)φ 1 (ɛ) + O(log n), 9 / 31

10 Second-order asymptotics in Stein s lemma Key tools 1: the relative modular operator The relative modular operator is a generalisation of the Radon-Nikodym derivative for general von Neuman algebras. In finite dimensions, for two faithful states, it reduces to: ρ σ : B(H) B(H), A ρaσ 1. ρ σ is a positive operator, admits a spectral decomposition. Quantities of relevance can be rewritten in terms of ρ σ : D(ρ σ) = ρ 1/2, log( ρ σ )(ρ 1/2 ), Ψ s(ρ σ) := log Tr(ρ 1+s σ s ) = log ρ 1/2, s ρ σ (ρ1/2 ) where A, B = Tr(A B). Define X to be the classical random variable taking values on spec(log ρ σ ) such that for any measurable f : spec(log ρ σ ) R, ρ 1/2, f (log ρ σ )(ρ 1/2 ) E[f (X )]. f : x x E[X ] = D(ρ σ), f : x e sx log E[e sx ] = log ρ 1/2, e s log ρ σ (ρ 1/2 ) Ψ s(ρ σ), the cumulant-generating function. 10 / 31

11 Second-order asymptotics in Stein s lemma Key tools 2: Bryc s theorem The following theorem is a non i.i.d. generalization of the Central limit theorem: Theorem (Bryc s theorem Bryc93) Let (X n) n N be a sequence of random variables, and (w n) n N an increasing sequence of weights. If there exists r > 0 such that: ] n N, H n(z) := log E [e zxn is analytic in the complex open ball B C (0, r), 1 H(x) = lim n Hn(x) exists x ( r, r), wn 1 sup n N sup z BC (0,r) Hn(z) < +, wn then H is analytic on B C (0, r) and X n H n(0) wn d n N ( 0, H (0) ), 11 / 31

12 Second-order asymptotics in Stein s lemma The working condition Take X n associated to log ρn σn, ρn 1/2 H n(z) = Ψ z (ρ n σ n). The conditions of Bryc s theorem translate into: Condition Let {w n} an increasing sequence of weights. Assume there exists r > 0 such that n N, z Ψ z (ρ n σ n) is analytic in the complex open ball B C (0, r), 1 H(x) = lim n Ψz (ρn σn) exists x ( r, r), wn sup n N sup z BC (0,r) 1 Ψz (ρn σn) < +, wn Here H n (0) = D(ρn σn), By Bryc s theorem, the quantum information variance rate is well-defined. 1 v({ρ n}, {σ n}) = lim V (ρ n n σ n) H (0) w n 12 / 31

13 Second-order asymptotics in Stein s lemma Proof of achievability (2) We will need the following crucial technical lemma: Lemma (Li14, DPR16) Let ρ, σ D +(H). For all L > 0 there exists a test T such that Tr(ρ(1 T )) ρ 1/2, P (0,L) ( ρ σ )(ρ 1/2 ) and Tr(σT ) L 1. (3) Let L n := exp(d(ρ n σ n) + w nt 2 + f (w n)), f (x) = ( x). (3) log β(t n) D(ρ n σ n) + w n t 2 + f (w n) α(t n) ρ n 1/2, P (0,Ln)( ρn σ n ) ρ n 1/2, = ρ 1/2 1/2 n, P (,log Ln)(log ρn σn ) ρ n. ( Xn D(ρ n σ n) = P(X n log L n) = P t 2 + f (wn) ). wn wn Bryc s theorem RHS converges to Φ(t 2 / v({ρ n}, {σ n})) < ε for n large enough, α(t n) ε. Optimality (1) follows similarly from a lower bound on the total error provided by Jaksic&Ogata&Pillet&Seiringer / 31

14 Some examples 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 14 / 31

15 Some examples Quantum spin systems Lattice Zd. System prepared at each site: x, Hx = H. Example: particle (spin ± 21 ): Hx = C2. 15 / 31

16 Some examples X Zd, HX := x X Hx, Interaction between sites: Φ : X 7 ΦX Bsa (HX ). Φ Translation invariant: invariant on sets of same shape. Φ Finite range: R > 0 : diam(x ) > R ΦX = 0. Φ induces dynamics, with equilibrium states (Gibbs states): X HXΦ := ΦY Y X Φ ρφ,β := X e βhx Φ Tr(e βhx ). 16 / 31

17 Some examples Λ n := { n,..., n} d. Translation invariant, of finite range interactions Φ, Ψ. Suppose given one of two states ρ n := ρ Φ,β 1 Λ n or σ n := ρ Ψ,β 2 Λ n Tests performed on H Λn Proposition (DPR16) For β 1, β 2 small enough, ρ n and σ n satisfy the conditions for second order asymptotics w.r.t. w n = Λ n. 17 / 31

18 Some examples Free lattice fermions One-particle Hilbert space h := l 2 (Z d ), standard basis {e i : i Z d } (particle localized at site i). Construct the Fock space F(h) = n N n h, where x 1... x n = 1 n! σ S n sgn(σ)x σ(1)... x σ(n). Algebra of observables: C algebra generated by creation and anihilation operators on the Fock space, obeying the canonical anti-commutation relations: a(x)a(y) + a(y)a(x) = 0, a(x)a (y) + a (y)a(x) = x, y 1, x, y h. a (y): unique linear extension a (y) : F(h) F(h) of a (y)x 1... x n = y x 1... x n, a(y) = (a (y)) 18 / 31

19 Some examples Let H B(h) be a one-particle Hamiltonian, define Ĥ := dγ(h) the differential second quantization on F(h), closure of ( n ) H n(x 1... x n) = P x 1... Hx k... x n, k=1 where P is the projection onto the antisymmetric subspace. Define the Gibbs state: ρ H β = e βĥ ( ) Tr e βĥ Gibbs states are expressed as linear forms on CAR(h) as follows: Tr(ρ H β a (x 1 )...a (x n)a(y m)...a(y 1 )) = δ mn det( y i, Qx i ) i,j, Q := e βh 1 + e βh. ρ H β is called the quasi-free state with symbol Q. Define the shift operators T j : e i e i+j, i, j Z d. A symbol which commutes with all shift operators is said to be shift invariant. 19 / 31

20 Some examples Assume now that we only have access to part of the lattice: denote Λ n := {0,..., n 1} d Z d, h n = l 2 (Λ n) h. Assume given two shift invariant symbols δ < Q, R < 1 δ. Denote Q n = P nqp n, R n = P nrp n, where P n orthogonal projection onto h n. The sequences of states that we want to distinguish between are then: {ρ n} associated with symbols {Q n} vs. {σ n} associated with symbols {R n}. Dierckx&Fannes&Pogorzelska08: ρ n, σ n can be written as: ρ n = det(1 Q n) dim h n k=0 k dim Qn h n σ n = det(1 R n) 1 Q n k=0 k Rn 1 R n. Proposition (DPR16) ρ n, and σ n satisfy the conditions for second order asymptotics w.r.t. w n = Λ n. 20 / 31

21 Finite sample size quantum hypothesis testing: the i.i.d. case 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 21 / 31

22 Finite sample size quantum hypothesis testing: the i.i.d. case Finite sample size bounds of Audenaert&Mosonyi&Verstraete12 Practical situations: finitely many copies available. Theorem (Audenaert&Mosonyi&Verstraete12) For ρ n := ρ n and σ n := σ n, D(ρ σ) f (ɛ) 1 g(ɛ) log βn(ɛ) D(ρ σ) +, n n n where f (ɛ) = 4 2 log η log(1 ɛ) 1, g(ɛ) = 4 2 log η log ɛ 1, and η := 1 + e 1/2D 3/2 (ρ σ) + e 1/2D 1/2 (ρ σ). The second order parts of the bounds scale as log ɛ 1, to compare with Φ 1 (ɛ) in the asymptotic case not tight. A better upper bound can be found by means of (non-commutative) martingale concentration inequalities. 22 / 31

23 Finite sample size quantum hypothesis testing: the i.i.d. case Non-commutative martingales A noncommutative probability space is a couple (M, τ), where M is a von Neumann algebra, and τ a normal tracial state on M. Let N be a von Neumann subalgebra of M. Then there exists a unique map E[. N ] : M N, called conditional expectation, such that E[1 N ] = 1, E[AXB N ] = AE[X N ]B, A, B N, X M, E [τ N ] = τ, A noncommutative filtration of M is an increasing sequence {M j } 1 j n of von Neumann subalgebras of M. A martingale is a sequence of noncommutative random variables {X j } 1 j n M n such that for each j, X j M j, τ( X j ) < E[X j+1 M j ] = X j (integrability), (martingale property). 23 / 31

24 Finite sample size quantum hypothesis testing: the i.i.d. case A noncommutative martingale concentration inequality Theorem (Noncommutative Azuma inequality, Sadeghi&Moslehian14) Let {X j, M j } 1 j n be a self-adjoint martingale. If d j X j+1 X j d j (d j > 0), ( ) α 2 τ(1 [α, ) (X n)) exp 2 n, α > 0. j=1 d2 j Idea (borrowed from Sason11): Take ρ n = ρ 1... ρ n, and σ n = σ 1... σ n. Then ρn σn = n n 1 ρj σ j log ρn σn = id j log ρj σ j id n j 1 j=1 Define algebras M k generated by id i log ρi σ i id n i 1, i k. ρ 1/2 n, (.) ρ 1/2 n is a tracial state on M n, and (log ρk σ k D(ρ k σ k ), M k ) is a martingale, where E[. M k ] := ρ k+1... ρ n, (.) ρ k+1... ρ n. j=0 24 / 31

25 Finite sample size quantum hypothesis testing: the i.i.d. case {log ρk σ k D(ρ k σ k )} 1 k n is a self-adjoint martingale, hence by NC Azuma: Recall: n, L n, T n: Hence, ρ 1/2 n, 1 (r, ) (log( ρn σn ) D(ρ n σ n))(ρn 1/2 r2 2 j ) e d2 j. α(t n) ρ 1/2 n, P (0,Ln)( ρn σ n )(ρ 1/2 n ) and β(t n) L 1 n. (log Ln D(ρn σn))2 α(t n) ρ 1/2 n, P (,log Ln)(log ρn σn )(ρn 1/2 2 j ) e d2 j ɛ log L n 2 dj 2 ln ɛ 1 + D(ρ n σ n) j β n(ɛ) β(t n) e D(ρn σn) 2 j d2 log ɛ j / 31

26 Finite sample size quantum hypothesis testing: the i.i.d. case Remember Audenaert&Mosonyi&Verstraete12 bound: log β n(ɛ) nd(ρ σ) + ng(ɛ), g(ɛ) := K ρ,σ log ɛ 1. Theorem (DR16) Suppose given states of the form n n ρ n = ρ j and σ n = σ j, j=1 j=1 where for each j, ρ j, σ j D(H j ) +. Then for each n N: n n log β n(ɛ) D( ρ j σ j ) + 2 log ɛ 1 dk 2, d k := log ρk σ k D( ρ k σ k ). j=1 k=1 In the i.i.d. case (ρ n = ρ n vs. σ n = σ n ), log β n(ɛ) nd(ρ σ) + n h(ɛ), h(ɛ) := 2 log ɛ 1 d / 31

27 Finite sample size quantum hypothesis testing: the i.i.d. case Comparison with Audenaert&Mosonyi&Verstraete12 Our error scales as: log ɛ 1, as opposed to log ɛ 1. We are getting closer to the asymptotic behavior in Φ 1 (ɛ). Rates ϵ g(ϵ) (ϵ) h s 1 (ϵ) -f(ϵ) / 31

28 Conclusion Summary and open questions We studied second order asymptotics as well as finite sample size hypothesis testing in the asymmetric scenario. We proved a theorem providing second order asymptotics for a large class of quantum systems, including the i.i.d. scenario of Tomamichel&Hayashi13 and Li14, as well as states of quantum spin systems and free fermions on a lattice. We found finite sample size bounds on the optimal type II error in the i.i.d. scenario using noncommutative martingale concentration inequalities. Open question: Does the martingale approach give bounds in physically relevant non i.i.d. examples? 28 / 31

29 Conclusion Thank you for your attention! 29 / 31

30 References References Bjelakovic&Siegmund-Schultze04: An ergodic theorem for quantum relative entropy. Comm. Math. Phys. 247, 697 (2004) Bjelakovic&Deuschel,&Krueger&Seiler&Siegmund-Schultze&Szkola08: Typical support and Sanov large deviations of correlated states. Comm. Math. Phys. 279, 559 (2008). Brandão&Plenio: A Generalization of Quantum Stein s Lemma. Commun. Math. Phys. 295: 791 (2010). Datta&Pautrat&Rouze16: Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. quantum lattice systems and more. Journal of Mathematical Physics, 57, 6, (2016) Audenaert&Koenraad&Verstraete12: Quantum state discrimination bounds for finite sample size. Journal of Mathematical Physics, Vol. 53, (2012). Hiai&Mosonyi&Ogawa08: Error exponents in hypothesis testing for correlated states on a spin chain. J. Math. Phys. 49, (2008). Hiai&Petz91: The proper formula for the relative entropy an its asymptotics in quantum probability. Comm. Math. Phys. 143, 99 (1991). 30 / 31

31 References Jaksic&Ogata&Piller&Seiringer12: Quantum hypothesis testing and non-equilibrium statistical mechanics. Rev. Math. Phys. 24, (2012). Li14: Second-order asymptotics for quantum hypothesis testing. Ann. Statist. Volume 42, Number 1, (2014). Mosonyi&Hiai&Ogawa&Fannes08: Asymptotic distinguishability measures for shiftinvariant quasi-free states of fermionic lattice systems. J. Math. Phys. 49, (2008). Ogawa&Nagaoka00: Strong Converse and Stein s Lemma in the Quantum Hypothesis Testing. IEEE Trans. Inf. Theo. 46, 2428 (2000). Sadghi&Moslehian14: Noncommutative martingale concentration inequalities. Illinois Journal of Mathematics, Volume 58, Number 2, (2014). Sason11: Moderate deviations analysis of binary hypothesis testing IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, pp (2012). Tomamichel&Hayashi13: A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks. IEEE Transactions on Information Theory, vol. 59, no. 11, pp (2013). 31 / 31