On Composite Quantum Hypothesis Testing

Transcription

1 University of York 7 November 207 On Composite Quantum Hypothesis Testing Mario Berta Department of Computing with Fernando Brandão and Christoph Hirche arxiv:

2 Overview Introduction 2 Composite Hypothesis Testing 3 Proof Idea 4 Examples 5 Conclusion

3 Introduction: Hypothesis Testing Discriminate between two sequences of quantum states ρ n, σ n on H n null and alternative hypothesis with two outcome POVM {M n, M n}. M n is associated with accepting ρ n and M n with accepting σ n. This leads to two types of errors α nm n := Tr [ ρ n M n ] Type error β nm n := Tr [ σ nm n ] Type 2 error. Symmetric setting for ρ n = ρ n, σ n = σ n with leads to ξ nρ, σ := inf 0 M n α nm n 2 + βnmn 2 Quantum Chernoff bound [Audenaert et al., PRL 07] log ξnρ, σ ξρ, σ := lim = log min n n Tr [ ρ s σ s]. 0 s

4 Introduction: Asymmetric Hypothesis Testing Same two type of errors α nm n, β nm n and ρ n = ρ n, σ n = σ n but asymmetric setting with βε n ρ, σ := inf { βnmn α nm n ε }. 0 M n leads to asymptotic error exponent Quantum Stein s lemma [Hiai and Petz, CMP 9] βρ, σ := lim log βn ερ, σ n n ε 0 = Dρ σ := Tr [ ρ log ρ log σ ]. Note: this led to the definition of the quantum relative entropy Dρ σ. Motivation: fundamental task in quantum statistics + underlying technical core problem for many applications in QIT as, e.g., quantum channel coding, quantum illumination, quantum reading, etc. [very many references].

5 Composite Hypothesis Testing: Setup Composite null and alternative hypotheses { } { } S n := ρ n dν ρ S vs. T n := σ n dµ σ T } {{ } } {{ } =:ρ nν =:σ nµ with S, T sets of quantum states and ν, µ measures on S, T, resp. For the asymmetric setting we define { βε n S, T := inf sup Tr [M nσ nµ] 0 M n sup Tr [ M nρ nν] ν S }{{}}{{} =:β nm n =:α nm n This leads to the definition of the composite asymptotic error exponent βs, T := lim log βn εs, T. n n ε 0 } ε.

6 Composite Hypothesis Testing: Classical Case If all involved quantum states pairwise commute classical setting probability distributions P, Q we have Composite Stein s lemma [Levitan and Merhav, IEEE 02] βs, T = inf P S Q T βp, Q = inf DP Q with Kulback-Leibler divergence. P S Q T Question: does this hold in the general non-commutative case as well? Yes, if T = {σ}, i.e., only composite null hypothesis [Hayashi, JPA 02]. Some related cases are understood as well [Brandão and Plenio, CMP 0] + [Hayashi and Tomamichel, JMP 6]. However, the general case remained open see also [Bjelaković et al., CMP 05]. Motivation: fundamental task in quantum statistics, composite version of applications in QIT e.g., network quantum Shannon theory.

7 Composite Hypothesis Testing: Quantum Case Our main result is regularized formula Composite quantum Stein s lemma [this talk] βs, T = lim n n inf ρ S D ρ n σ n dµσ inf ρ S σ T Dρ σ in general. Hence, in general D ρ n σ n dµσ n inf σ T Dρ σ. Converse: βs, T RHS based on MONO of quantum relative entropy under quantum channels [Hiai and Petz, CMP 9]. Achievability: βs, T RHS via measure: post-measurement probability distributions 2 apply classical composite Stein s lemma 3 mathematical properties of quantum entropy Regularization: examples + novel quantum entropy inequalities.

8 Proof Idea: Classical Strategy βs, T := lim lim log βn εs, T ε 0 n n For n N, ε 0,, and POVM N n with P n := N nρ n, Q n := N nσ n composite Stein s lemma for probability distributions gives achievability bound log β n εs, T inf ρ S σ T D N nρ n Nnσ n inf ν S D N nρ nν N nσ nµ. Optimizing over all POVM N n we find the measured relative entropy D N ρ σ as introduced by [Donald, CMP 86] log βn ε S, T n n sup inf N n ν S minimax = n inf ν S D N nρ nν N nσ nµ sup D N nρ nν N nσ nµ. N n }{{} =:D N ρ nν σ nµ

9 Proof Idea: Properties of Quantum Entropy Hence, so far we have βs, T lim n n inf ν S and it remains to prove that asymptotically n inf ν S D N ρ nν σ nµ i n inf ν S D N ρ nν σ nµ Dρ ii nν σ nµ n inf D ρ n σ nµ. ρ S Using asymptotic spectral pinching [Hayashi, JPA 02] + [Sutter et al., CMP 7] it can be shown D N ρ σ Dρ σ log specσ MONO: D N ρ σ Dρ σ. However, since σ nµ = σ n dµσ is permutation invariant, we have by Schur-Weyl duality specσ nµ polyn and step i follows. Step ii is deduced from the quasi-convexity of the von Neumann entropy.

10 Examples Composite quantum Stein s lemma [this talk] βs, T = lim n n inf ρ S D ρ n σ n dµσ inf ρ S σ T Dρ σ in general. When do we get single-letter formula? From [Hayashi, JPA 02] we have β S, T = {σ} = inf ρ S Dρ σ. An example for composite alternative hypotheses: relative entropy of coherence [Baumgratz et al., PRL 4] D C ρ := inf Dρ σ for set of states C diagonal in a fixed basis { c }. σ C

11 Examples: Relative Entropy of Coherence Goal: discrimination problem with asymptotic error exponent given by the relative entropy of coherence D C ρ := inf Dρ σ for set of states C diagonal in a fixed basis { c }. σ C Null hypothesis: the fixed states ρ n Alternative hypothesis: convex sets of iid coherent states C n := { σ n dµσ σ C } β {ρ}, C = lim ρ n n inf D n µ C σ n dµσ = D C ρ. More examples possible, e.g., quantum mutual information for product state testing cf. [Hayashi and Tomamichel, JMP 6].

12 Examples: Regularization and Entropy Inequalities I Goal: give discrimination problem such that lim n n inf D ρ n σ n dµσ inf Dρ σ ρ S ρ S σ T Quantum Markov testing see also [Cooney et al., PRA 6] Null hypothesis: the fixed state ρ n ABC Alternative hypothesis: the { convex sets of quantum Markov iid states } R n := IA R C BC ρ AC n dµr with R C BC local quantum channels For this example we claim that our formula does not become single-letter β {ρ ABC }, R = lim n n inf D ρ n IA µ R ABC R C BC ρ AC n dµr inf R D ρ ABC I A R C BC ρ AC.

13 Examples: Regularization and Entropy Inequalities II lim n n inf D ρ n IA µ R ABC R C BC ρ AC n dµr inf R D ρ ABC I A R C BC ρ AC. We show improved lower bound on quantum conditional mutual information CQMI [Sutter et al., CMP 7], relaxed to see also [Brandão et al., PRL 5] I A : B C ρ := Dρ ABC ρ A ρ BC Dρ AC ρ A ρ C lim n n inf D ρ n IA µ R ABC R C BC ρ AC n dµr. However, [Fazwi and Fawzi, arxiv 7] give explicit quantum state ρ ABC with I A : B C ρ inf R D ρ ABC I A R C BC ρ AC. Note: use of additive CQMI nicely allows to circumvent asymptotics.

14 Conclusion Composite quantum Stein s lemma [this talk] βs, T = lim n n inf ρ S D ρ n σ n dµσ inf ρ S σ T Dρ σ in general. Single-letter examples possible, even with refinements: Hoeffding bound, strong converse exponent, second-order expansion as in [Hayashi and Tomamichel, JMP 6] + [Tomamichel and Hayashi, arxiv 5]. Symmetric setting: open question about composite quantum Chernoff bound ξρ, σ = log min Tr [ ρ s σ s] ξs, T =? inf ξρ, σ 0 s ρ S σ T only known up to a factor of two [Audenaert and Mosonyi, JMP 4]. Applications in QIT, e.g., network quantum Shannon theory [Qi et al., arxiv 7]?

15 Extra: Entropy inequalities CQMI bounds [Junge et al., arxiv 5], [Sutter et al., CMP 7], [this talk] For any quantum state ρ ABC the CQMI is lower bounded by the incomparable bounds I A : B C ρ I A : B C ρ D N ρ ABC I A : B C ρ lim sup n β 0 t log ρabc σ [t] 2 ABC dt β 0 tσ [t] ABC dt n D ρ n ABC β 0 t σ [t] n ABC dt, where β 0 t := π 2 coshπt + is a universal probability distribution and σ [t] ABC := I A R [t] C BC ρ AC with R [t] C BC := ρ +it 2 BC are rotated Petz local recovery quantum channels. ρc it 2 ρc +it 2 it 2 ρbc