Perfectly Discriminable States

Miroslav Ježek

Scott M. Cohen

Quantum state discrimination involves identifying a given state out of a set of possible states. When the states are mutually orthogonal, perfect state discrimination is always possible using a global measurement. In the case of multipartite systems when the parties are constrained to use multiple rounds of local operations and classical communication (LOCC), perfect state discrimination is often impossible even with the use of asymptotic LOCC, wherein an error is allowed but must vanish in the limit of an infinite number of rounds. Utilizing our recent results on asymptotic LOCC, we derive a lower bound on the error probability for LOCC discrimination of any given set of mutually orthogonal pure states. Informed by the insights gained from this lower bound, we are able to prove necessary conditions for perfect state discrimination by asymptotic LOCC. We then illustrate by example the power of these necessary conditions in significantly simplifying the determination of whether perfect discrimination of a given set of states can be accomplished arbitrarily closely using LOCC. The latter examples include a proof that perfect discrimination by asymptotic LOCC is impossible for any minimal unextendible product basis (UPB), where minimal means that for the given multipartite system no UPB with a smaller number of states can exist. We also give a simple proof that what has been called strong nonlocality without entanglement is considerably stronger than had previously been demonstrated.

Hanwool Lee, Teiko Heinosaari

A set of pairwise non-orthogonal quantum states cannot be perfectly discriminated, and this remains true even when one bit of classical partial information is available prior to the measurement. Contrary to the usual intuition that earlier information is at least as valuable as later information, we show that the same bit can be more useful when it arrives after the measurement. We present a framework for state discrimination in which the sender provides classical information in response to a request from the receiver, which we refer to as a query. In some cases, pairwise non-orthogonal states can be perfectly discriminated when the query is allowed to depend on the measurement outcome. We give a general method for finding the optimal strategy in this setting, which turns out to be the standard minimum-error discrimination problem for an auxiliary ensemble.

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Adam Bílek, Paulina Lewandowska, Ryszard Kukulski

Quantum channel discrimination plays a crucial role in quantum information theory. Of particular interest is the case in which the channels can be discriminated perfectly. In this work, we focus on the perfect quantum channel discrimination task in a parallel scheme. We develop an SDP formulation combined with a bisection procedure to compute a quantum state for perfect discrimination in time linear in the number of copies. In addition, we obtain the minimal number of copies of quantum channels to achieve perfect discrimination. Thanks to that, we settle in the affirmative Conjecture 1 of Duan, Guo, Li and Li [arXiv:1605.02294, IEEE ISIT 2016], which characterizes the number of parallel uses needed to discriminate perfectly a distinguished family of operator subspaces. All our results are obtained using the notion and basic properties of the numerical range. In particular, the key fact that we prove and use is that the minimal angle of the numerical range of a tensor product of matrix subspaces equals the sum of the minimal angles of the numerical ranges of the individual subspaces.

Hayato Arai, Yuuya Yoshida, Masahito Hayashi

Abstract We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states perfectly, and find that some non-orthogonal states are perfectly distinguishable. However, the above framework does not improve the capacity , namely, the maximum number of states that are simultaneously and perfectly distinguishable.