Extended necessary condition for local operations and classical communication: Tight bound for all measurements

TL;DR

This paper establishes a new tight necessary condition for implementing separable measurements via LOCC, extending previous bounds to all measurement operators and applying it to demonstrate the nonlocality of domino states.

Contribution

It generalizes and strengthens existing bounds on LOCC implementability, providing a universal necessary condition applicable to any finite or infinite measurement operators.

Findings

New necessary condition tight for any finite measurement operators

Applied to domino states to show they cannot be perfectly distinguished by LOCC

Extended the condition to cases with infinite measurement operators

Abstract

We give a necessary condition that a separable measurement can be implemented by local quantum operations and classical communication (LOCC) in any finite number of rounds of communication, generalizing and strengthening a result obtained previously. That earlier result involved a bound that is tight when the number of measurement operators defining the measurement is relatively small. The present results generalize that bound to one that is tight for any finite number of measurement operators, and we also provide an extension which holds when that number is infinite. We apply these results to the famous example on a $3\times3$ system known as "domino states", which were the first demonstration of nonlocality without entanglement. Our new necessary condition provides an additional way of showing that these states cannot be perfectly distinguished by (finite-round) LOCC. It directly shows that this conclusion also holds for their cousins, the rotated domino states. This illustrates the usefulness of the present results, since our earlier necessary condition, which these results generalize, is not strong enough to reach a conclusion about the domino states.