Almost one bit violation for the additivity of the minimum output entropy

TL;DR

This paper determines the limit of the minimum output entropy for random quantum channels, demonstrating a violation of additivity in dimensions as low as 183, with potential violations approaching log 2.

Contribution

It precisely computes the limit of the minimum output entropy for random quantum channels and establishes the minimal dimension for observing additivity violations.

Findings

Violation of additivity can occur in dimensions as low as 183.

Maximum l^p norms on the set K_{k,t} are attained on specific vectors.

The violation size can approach log 2 - epsilon.

Abstract

In a previous paper, we proved that the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_{k,t}. We also showed that the set K_{k,t} is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of l^p norms on the set K_{k,t} and prove that the maximum is attained on a vector of shape (a,b,...,b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any eps > 0, it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as log 2 - eps. Conversely, our result implies that, with probability one, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.