General approach to quantum channel impossibility by local operations and classical communication

TL;DR

This paper introduces a general linear-equation-based method to prove the impossibility of implementing certain quantum channels via LOCC, and to design LOCC protocols when possible, revealing new insights into the structure of LOCC channels.

Contribution

The authors develop a unified approach to determine LOCC implementability of quantum channels, including cases with infinite rounds, and demonstrate its effectiveness through several examples.

Findings

Some quantum channels are not LOCC and the set is not closed.

Channels implementable in one or three rounds can lie on the boundary of LOCC channels.

Analyzing LOCC channels is computationally comparable to analyzing LOCC measurements.

Abstract

We describe a general approach to proving the impossibility of implementing a quantum channel by local operations and classical communication (LOCC), even with an infinite number of rounds, and find that this can often be demonstrated by solving a set of linear equations. The method also allows one to design an LOCC protocol to implement the channel whenever such a protocol exists in any finite number of rounds. Perhaps surprisingly, the computational expense for analyzing LOCC channels is not much greater than that for LOCC measurements. We apply the method to several examples, two of which provide numerical evidence that the set of quantum channels that are not LOCC is not closed and that there exist channels that can be implemented by LOCC either in one round or in three rounds that are on the boundary of the set of all LOCC channels. Although every LOCC protocol must implement a separable quantum channel, it is a very difficult task to determine whether or not a given channel is separable. Fortunately, prior knowledge that the channel is separable is not required for application of our method.