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arXiv:1812.05142

From asymptotic hypothesis testing to entropy inequalities

Christoph Hirche

Published Dec 12, 2018• 3 citations
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Abstract

This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-$2$ entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.

DOI
10.48550/arxiv.1812.05142

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This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-222 entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.
Abstract We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ⊗n\rho ^{\otimes n}ρ⊗n ρ ⊗ n against convex combinations of quantum states σ⊗n\sigma ^{\otimes n}σ⊗n σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.
The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras. After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.
A central yet mysterious problem in quantum information is the asymptotic distinguishability of quantum states under restricted measurements. Here we look at asymptotic asymmetric hypothesis testing for general multi-mode bosonic Gaussian states restricted to arbitrary collective Gaussian measurements. We establish a single-letter formula for the Stein exponent for many pairs of states, featuring two distinct contributions, corresponding to first and second moments, respectively. Remarkably, the Gaussian-measured relative entropy exhibits genuine non-additivity, but the only effect of additivity violations is to asymptotically decouple first and second moments. We prove this by establishing sufficient conditions for additivity of the second-moment term via novel log-determinant matrix inequalities. Our results not only identify a single-letter expression for the Gaussian Stein exponent in a wide range of practical situations, but also rule out any possible advantage from collective Gaussian measurements when considering centred Gaussian states.
We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ( ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.
The hypothesis testing problem of two quantum states is treated. We show a new inequality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the first kind error when the power exponent for the second kind error exceeds the quantum relative entropy, and the bound yields the strong converse in the quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in the classical hypothesis testing.
In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein’s lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein’s lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.