Additivity rates and PPT property for random quantum channels

TL;DR

This paper introduces the concept of additivity rates for quantum channels, analyzes their properties using random matrix models, and explores the PPT property and entropy additivity violations in random quantum channels.

Contribution

It defines additivity rates for quantum channels, provides bounds using operator norms, and analyzes random channels to determine PPT thresholds and entropy violation conditions.

Findings

Additivity rates give the linear term of minimal output p-Renyi entropies.

Upper bounds for classical capacity are derived from operator norm bounds.

Random quantum channels can violate entropy additivity for p ≥ 30.95.

Abstract

Inspired by Montanaro's work, we introduce the concept of additivity rates of a quantum channel $L$, which give the first order (linear) term of the minimum output $p$-R\'enyi entropies of $L^{\otimes r}$ as functions of $r$. We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is showed. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the $p$-R\'enyi entropy for all $p\geq30.95$.