A hierarchy of efficient bounds on quantum capacities exploiting symmetry

TL;DR

This paper introduces a hierarchy of semidefinite programming bounds leveraging symmetry to efficiently approximate regularized quantum information measures, improving capacity estimates for certain quantum channels.

Contribution

It develops a novel hierarchy of bounds based on symmetry for regularized quantum information quantities, enabling efficient approximations and slight capacity improvements.

Findings

Provides a general method for bounding regularized Umegaki divergence.

Achieves improved capacity bounds for the amplitude damping channel.

Shows polynomial-time approximation of sandwiched Rényi divergence for fixed dimensions.

Abstract

Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introduced $D^{\#}$ in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched R\'enyi divergence between any two quantum channels can be approximated up to an $\epsilon$ accuracy in time that is polynomial in $1/\epsilon$.