Conditions for uniqueness of product representations for separable quantum channels and separable quantum states

TL;DR

This paper establishes a sufficient condition for the uniqueness of product operator representations of separable quantum channels and states, with implications for local implementation and state ensemble representations.

Contribution

It introduces a new criterion for the uniqueness of product representations in separable quantum channels and states, expanding understanding of their structure and implementation.

Findings

Provides a sufficient condition for unique product representations.

Constructs examples of such channels for any number of parties.

Connects the condition to the structure of subspaces spanned by product operators.

Abstract

We give a sufficient condition that an operator sum representation of a separable quantum channel in terms of product operators is the unique product representation for that channel, and then provide examples of such channels for any number of parties. This result has implications for efforts to determine whether or not a given separable channel can be exactly implemented by local operations and classical communication. By the Choi-Jamiolkowski isomorphism, it also translates to a condition for the uniqueness of product state ensembles representing a given quantum state. These ideas follow from considerations concerning whether or not a subspace spanned by a given set of product operators contains at least one additional product operator.