On the quantum no-signalling assisted zero-error classical simulation cost of non-commutative bipartite graphs

TL;DR

This paper investigates the quantum no-signalling assisted zero-error classical simulation cost of non-commutative bipartite graphs, providing conditions for multiplicativity and additivity, and demonstrating cases where these properties do not hold.

Contribution

It introduces a general sufficient condition for the multiplicativity and additivity of simulation costs and constructs examples disproving multiplicativity.

Findings

A sufficient condition for cost multiplicativity and additivity.

Identification of a large class of graphs with additive asymptotic cost.

Explicit construction showing non-multiplicativity of one-shot simulation cost.

Abstract

Using one channel to simulate another exactly with the aid of quantum no-signalling correlations has been studied recently. The one-shot no-signalling assisted classical zero-error simulation cost of non-commutative bipartite graphs has been formulated as semidefinite programms [Duan and Winter, IEEE Trans. Inf. Theory 62, 891 (2016)]. Before our work, it was unknown whether the one-shot (or asymptotic) no-signalling assisted zero-error classical simulation cost for general non-commutative graphs is multiplicative (resp. additive) or not. In this paper we address these issues and give a general sufficient condition for the multiplicativity of the one-shot simulation cost and the additivity of the asymptotic simulation cost of non-commutative bipartite graphs, which include all known cases such as extremal graphs and classical-quantum graphs. Applying this condition, we exhibit a large class of so-called \emph{cheapest-full-rank graphs} whose asymptotic zero-error simulation cost is given by the one-shot simulation cost. Finally, we disprove the multiplicativity of one-shot simulation cost by explicitly constructing a special class of qubit-qutrit non-commutative bipartite graphs.