Institute of Information Science, Academia Sinica

In this paper, we revisit hypothesis testing between two bipartite quantum states by local operations and classical communication (LOCC), in the scenario of many copies.
Two hypotheses correspond to distinct states of a quantum system are considered. The aim of the testing is to decide which of the two hypotheses is true by performing a measurement on multiple identical copies of the state.  When the underlying quantum system is multipartite, a class of measurements, LOCC, is of fundamental and practical importance to quantum computation. Our goal is thus to analyze the asymptotic  behavior of the incurred average error and the trade-off between two types of errors under LOCC.

We firstly determine an expression for the optimal average error probability for testing maximally entangled state against its orthogonal complement in the symmetric setting. This leads to the so-called Chernoff exponent for distinguishing the pair of states. Secondly, we establish the optimal trade-off between the type-I and type-II errors in the asymmetric setting. As a result, we calculate the optimal exponential error rates such as the corresponding Stein, Hoeffding, and the strong converse exponents. Lastly, we extend the above studies to the case of testing pure state with even non-zero Schmidt coefficients against its orthogonal complement, and testing completely symmetric state against completely anti-symmetric state as well. We show that the mentioned operational exponent quantities are faithful in the sense that the errors could not decay faster than exponential for the considered pairs of orthogonal states. Surprisingly, our results of using LOCC demonstrate a fundamental difference on the second-order asymptotics from that of the global measurements.