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arXiv:1709.07268 · Communications in Mathematical Physics

On Composite Quantum Hypothesis Testing

Mario Berta, Fernando G. S. L. Brandão, Christoph Hirche

Published Jun 10, 2021• 34 citations
arXivHTMLPDF

Abstract

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 $$\rho ^{\otimes n}$$ ρ ⊗ n against convex combinations of quantum states $$\sigma ^{\otimes n}$$ 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.

DOI
10.1007/s00220-021-04133-8

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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 generalized quantum Stein's lemma characterizes the optimal asymptotic exponent of the type-II error in quantum hypothesis testing for an independent and identically distributed (IID) null hypothesis against a composite alternative hypothesis. Classically, a probabilistic mixture of IID sources arises as a natural generalization of IID sources, and, in the non-composite setting, the optimal type-II error exponent in hypothesis testing for such classical mixed sources is known to be characterized concisely by the worst-case component of the mixture. In this work, we extend these foundational results to composite quantum hypothesis testing where the null hypothesis is a mixed source, i.e., a probabilistic mixture of IID quantum states, and the alternative hypothesis is composite as in the generalized quantum Stein's lemma. When the type-I error vanishes asymptotically, we characterize the optimal type-II error exponent of this composite quantum hypothesis testing problem in terms of the worst-case component of the mixture, by developing techniques for the non-commutative quantum setting inspired by the classical information-spectrum analysis. We also show that the analogous characterization does not hold in general for a fixed nonzero type-I error threshold, by providing a counterexample beyond the vanishing type-I error regime. These results clarify the applicability of the generalized quantum Stein's lemma to highly non-IID null hypotheses arising from arbitrary finite probabilistic mixtures of IID quantum states.
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.
We establish duality relations for conditional entropies relative to subalgebras of arbitrary von Neumann algebras, including the Umegaki relative entropy as well as the sandwiched- and Petz-Rényi divergences. Even in finite dimensions, these results extend the familiar dualities for conditional entropies from tensor-product subsystems to general subalgebras, while also lifting the framework to the fully infinite-dimensional setting. As an application, we establish an infinite-dimensional extension of quantum Stein's lemma when the alternative hypothesis is the state space of a finite-index subalgebra, which gives an operational interpretation of the relative entropy to a subalgebra as the optimal asymptotic error exponent in composite quantum hypothesis testing-the first beyond finite dimensions.
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.
Given a sequence of random variables Xn=X1,…,XnX^n=X_1,\ldots, X_nXn=X1​,…,Xn​, discriminating between two hypotheses on the underlying probability distribution is a key task in statistics and information theory. Of interest here is the Stein exponent, i.e. the largest rate of decay (in nnn) of the type II error probability for a vanishingly small type I error probability. When the hypotheses are simple and i.i.d., the Chernoff-Stein lemma states that this is given by the relative entropy between the single-copy probability distributions. Generalisations of this result exist in the case of composite hypotheses, but mostly to settings where the probability distribution of XnX^nXn is not genuinely correlated, but rather, e.g., a convex combination of product distributions with components taken from a base set. Here, we establish a general Chernoff-Stein lemma that applies to the setting where both hypotheses are composite and genuinely correlated, satisfying only generic assumptions such as convexity (on both hypotheses) and some weak form of permutational symmetry (on either hypothesis). Our result, which strictly subsumes most prior work, is proved using a refinement of the blurring technique developed in the context of the generalised quantum Stein's lemma [Lami, IEEE Trans. Inf. Theory 2025]. In this refined form, blurring is applied symbol by symbol, which makes it both stronger and applicable also in the absence of permutational symmetry. The second part of the work is devoted to applications: we provide a single-letter formula for the Stein exponent characterising the discrimination of broad families of null hypotheses vs a composite i.i.d. or an arbitrarily varying alternative hypothesis, and establish a 'constrained de Finetti reduction' statement that covers a wide family of convex constraints. Applications to quantum hypothesis testing are explored in a related paper [Lami, arXiv:today].
We prove a generalized quantum Stein lemma for i.i.d. normal states against convex, tensor-stable families on arbitrary von Neumann algebras. Assuming the existence of an alternative state with finite relative entropy from the null state, we show that, at every type-I error tolerance ε∈(0,1)\varepsilon\in (0,1)ε∈(0,1), the worst case type-II error exponent is achieved with the regularized relative entropy with a strong converse. The proof combines the integral representation of relative entropy by hockey-stick divergences, a modular testing bound, and convex minimax argument.