A semidefinite programming upper bound of quantum capacity

TL;DR

This paper introduces a new semidefinite programming upper bound for quantum capacity, demonstrating its effectiveness and superiority over previous bounds, and explores its implications for quantum communication and super-activation phenomena.

Contribution

The paper presents a novel SDP-based upper bound for quantum capacity that is additive, computationally efficient, and improves upon existing bounds, with applications to super-activation.

Findings

The SDP upper bound $Q_\Gamma$ is always less than or equal to the Partial transposition bound.

$Q_\Gamma$ is strictly better than several known bounds for certain channels.

$Q_\Gamma$ can be used to analyze super-activation of quantum capacity.

Abstract

Recently the power of positive partial transpose preserving (PPTp) and no-signalling (NS) codes in quantum communication has been studied. We continue with this line of research and show that the NS/PPTp/NS$\cap$PPTp codes assisted zero-error quantum capacity depends only on the non-commutative bipartite graph of the channel and the one-shot case can be computed efficiently by semidefinite programming (SDP). As an example, the activated PPTp codes assisted zero-error quantum capacity is carefully studied. We then present a general SDP upper bound $Q_\Gamma$ of quantum capacity and show it is always smaller than or equal to the "Partial transposition bound" introduced by Holevo and Werner, and the inequality could be strict. This upper bound is found to be additive, and thus is an upper bound of the potential PPTp assisted quantum capacity as well. We further demonstrate that $Q_\Gamma$ is strictly better than several previously known upper bounds for an explicit class of quantum channels. Finally, we show that $Q_\Gamma$ can be used to bound the super-activation of quantum capacity.