Asymptotic local hypothesis testing between a pure bipartite state and the completely mixed state

TL;DR

This paper analyzes the asymptotic hypothesis testing between a bipartite pure state and the completely mixed state using various LOCC and separable measurements, deriving formulas for error exponents and bounds.

Contribution

It provides single-letter formulas for error exponents and bounds under different measurement classes, and numerically compares their performance in low-dimensional systems.

Findings

Derived formulas for Stein's lemma, Chernoff, and Hoeffding bounds.

Numerical results show three-step LOCC bounds approach separable POVM bounds.

Bounds outperform one-way LOCC and closely match separable POVM bounds.

Abstract

In this paper, we treat an asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state and the completely mixed state by one-way LOCC, two-way LOCC, and separable 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 approximates the bounds for separable POVMs in the parameter region where analytical bounds for separable POVMs are derived.