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arXiv:2104.09553

Interpolating between symmetric and asymmetric hypothesis testing

Salzmann, Robert, Nilanjana Datta

Published Apr 19, 2021• 2 citations
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Abstract

The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say $ρ$ and $σ$. This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call $s$-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter $s$. In particular, $s$-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of $s$-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as $ξ_s(ρ\|σ)$, and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for $s = 1$, and asymmetric hypothesis testing (the Umegaki relative entropy) for $s = 0$.

DOI
10.48550/arxiv.2104.09553

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The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say ρρρ and σσσ. This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call sss-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter sss. In particular, sss-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of sss-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as ξs(ρ∥σ)ξ_s(ρ\|σ)ξs​(ρ∥σ), and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for s=1s = 1s=1, and asymmetric hypothesis testing (the Umegaki relative entropy) for s=0s = 0s=0.
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing. The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support. The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.
Given two families of quantum states AAA and BBB, called the null and the alternative hypotheses, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to AAA or BBB. Mistaking AAA for BBB is a type I error, and vice versa for the type II error. In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the asymptotic rate of decay of the type II error probability for a given threshold on the type I error probability. Stein exponents have been thoroughly investigated -- and, sometimes, calculated. However, most currently available solutions apply to settings where the hypotheses simple (i.e. composed of a single state), or else the families AAA and BBB need to satisfy stringent constraints that exclude physically important sets of states, such as separable states or stabiliser states. In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite: the alternative hypothesis BBB is assumed to be either composite i.i.d. or arbitrarily varying, with components taken from a known base set, while the null hypothesis AAA is fully general, and required to satisfy only weak compatibility assumptions that are met in most physically relevant cases -- for instance, by the sets of separable or stabiliser states. Our result extends and subsumes the findings of [BBH, CMP 385:55, 2021] (that we also simplify), as well as the 'generalised quantum Sanov theorem' of [LBR, arXiv:2408.07067]. The proof relies on a careful quantum-to-classical reduction via measurements, followed by an application of the results on classical Stein exponents obtained in [Lami, arXiv:today]. We also devise new purely quantum techniques to analyse the resulting asymptotic expressions.
Hypothesis testing is an important task in mathematics and physics. Hypothesis testing of two random variables is related to the Kullback-Leibler divergence of the two corresponding distributions. Similarly, quantum hypothesis testing of two quantum states is characterized by the quantum relative entropy. While a quantum state can be abstracted as a device that only has outputs but no inputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend hypothesis testing to general quantum channels. In both the one-shot and asymptotic scenario, we study several quantifiers for hypothesis testing under different assumptions of how the channels are used. As the quantifiers are analogs to the quantum relative entropy of states, we call them the quantum relative entropy of channels. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarizing channel. We investigate the properties that the quantum relative entropy of channels should satisfy, and we study its interplay with entanglement. With the broad applications of the quantum relative entropy of states, our results can be useful for understanding general properties of quantum channels.
Quantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states (ρ or σ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n) identical copies of the state (ρ⊗n or σ⊗n) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability αn of erroneously inferring the state to be σ, the probability βn of erroneously inferring the state to be ρ decays exponentially in n, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures, and quasi-free states of fermionic lattice gases.
Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, we develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry (TTT symmetry) and diagonal symmetry (ZZZ symmetry) with limited queries. Contrasting with the advantages of indefinite causal-order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal-order strategies have equal power for our tasks. We establish optimal protocols for TTT symmetry testing and ZZZ symmetry testing for six and five queries, respectively, from which we infer that the type-II error exhibits a decay rate of $O({m}^{\ensuremath{-}2})$ with respect to the number of queries mmm. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of $O({m}^{\ensuremath{-}1})$.
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.