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

TL;DR

This paper establishes tight asymptotic bounds for local hypothesis testing between a pure bipartite state and white noise, revealing differences in optimal error exponents under various LOCC and separable measurement protocols.

Contribution

It derives the Hoeffding bounds and Stein's lemma type error exponents for different measurement classes, highlighting cases where two-way LOCC matches separable operations but differs from one-way LOCC.

Findings

Two-way LOCC achieves optimal performance equal to separable operations.

Explicit Hoeffding bounds are derived for two-way LOCC and separable POVMs.

Optimal error exponents under one-way, two-way LOCC, and separable POVMs are characterized.

Abstract

We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state $\ket{\Psi}$ 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 one-way LOCC.