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IEEE International Symposium on Information Theory (ISIT)

Tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state

Masahito Hayashi, Masaki Owari

Published Jun 1, 2015• 2 citations
Publisher

Abstract

We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state |ψ〉 and the completely mixed state under one-way LOCC (local operations and classical communications), two-way LOCC, and separable POVMs. As a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable POVMs. Further, we derive a Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC, and separable POVMs up to the third order, which clarifies the difference between one-way and two-way LOCC POVM. Our study gives a very rare example in which the optimal performance under the infinite-round two-way LOCC is also equal to that under separable operations and can be attained with two-round communication, but not attained with the oneway LOCC.

DOI
10.1109/isit.2015.7282543

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We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state |ψ〉 and the completely mixed state under one-way LOCC (local operations and classical communications), two-way LOCC, and separable POVMs. As a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable POVMs. Further, we derive a Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC, and separable POVMs up to the third order, which clarifies the difference between one-way and two-way LOCC POVM. Our study gives a very rare example in which the optimal performance under the infinite-round two-way LOCC is also equal to that under separable operations and can be attained with two-round communication, but not attained with the oneway LOCC.
We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state |Ψ| and the white noise state (the completely mixed state) under one-way local operations and classical communications (LOCC), two-way LOCC, and separable Positive Operator Valued Measures (POVMs). As a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable POVMs. Further, we derive Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC, and separable POVMs up to the third order, which clarifies the difference between one-way and two-way LOCC POVM. Our results clarify the relationship between the entanglement of Renyi entropy and the hypothesis testing under LOCC, since the entanglement of Renyi entropy appears in the formula of both the Hoeffding bounds and Stein's lemma type of error exponents. This paper gives a very rare example in which the optimal performance under the infinite-round two-way LOCC is also equal to that under separable operations and can be attained with two-round communication, but not with the one-way LOCC.
In this paper, we treat an asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state $|\ensuremath{\Psi}\ensuremath{\rangle}$ and the completely mixed state by one-way local operation and classical communication (LOCC), two-way LOCC, and separable positive operator-valued measures (POVMs). As a result, we derive single-letterized formulas for the Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC, and separable POVMs, the Chernoff bounds under one-way LOCC POVMs and separable POVMs, and the Hoeffding bounds under one-way LOCC POVMs in the whole region of a parameter and under separable POVMs on a restricted region of a parameter. We also numerically calculate the Chernoff and the Hoeffding bounds under a class of three-step LOCC protocols in low-dimensional systems and show that these bounds not only outperform the bounds for one-way LOCC POVMs but also almost approximate the bounds for separable POVMs in the parameter region where analytical bounds for separable POVMs are derived.
In this paper, we treat a local discrimination problem in the framework of asymmetric hypothesis testing. We choose a known bipartite pure state |ψ) as an alternative hypothesis and the completely mixed state as a null hypothesis. As a result, we analytically derive an optimal type-2 error and an optimal positive operator valued measure (POVM) for one-way local operations and classical communication (LOCC) POVM and separable POVM. For two-way LOCC POVM, we study a family of simple three-step LOCC protocols, and show that the best protocol in this family has strictly better performance than any one-way LOCC protocol in low-dimensional systems when there may exist differences between two-way LOCC POVM and one-way LOCC POVM.
We analyze the difference in the local distinguishability among the following three restrictions: (i) local operations and only one-way classical communications (one-way LOCC) are permitted; (ii) local operations and two-way classical communications (two-way LOCC) are permitted; and (iii) all separable operations are permitted. We obtain two main results concerning the discrimination between a given bipartite pure state and the completely mixed state with the condition that the given state should be detected perfectly. As the first result, we derive the optimal discrimination protocol for a bipartite pure state in cases (i) and (iii). As the second result, by constructing a concrete two-way local discrimination protocol, it is proven that case (ii) is much better than case (i), i.e. two-way classical communication remarkably improves the local distinguishability in comparison with one-way classical communication at least for a low-dimensional bipartite pure state.
Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d times d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations. This is a shortened version of a paper [1] recently submitted to Communications in Mathematical Physics; here the proofs have been omitted.
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.