When do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination?

TL;DR

This paper investigates when local operations and classical communication (LOCC) are sufficient for optimal two-qubit state discrimination, revealing surprising limitations and conditions, especially for ensembles of pure, mixed, and product states, and exploring multi-party scenarios.

Contribution

It provides necessary and sufficient conditions for perfect discrimination of two-qubit states, and shows that almost all three pure state ensembles cannot be optimally distinguished by LOCC.

Findings

Almost all three pure states ensembles cannot be optimally discriminated by LOCC.

A sufficient condition is given for when three product states cannot be distinguished by LOCC.

LOCC can asymptotically discriminate states in the N-copy scenario as N approaches infinity.

Abstract

In this paper we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the \emph{perfect} distinguishability problem of two-qubit ensembles - both for separable operations and LOCC - by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then for the well-known task of minimum error discrimination, it is shown that \textit{almost all} two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that \textit{any} two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as "non-locality without entanglement". We next consider an example of $N$ parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant "trine ensemble", and we establish a tight connection between the $N$-copy ensemble and Shor's "lifted" single-copy ensemble. For any finite $N$, we prove that optimal identification of the states cannot be achieved by LOCC; however as $N\to\infty$, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum-error probability is considered.