On Hastings' counterexamples to the minimum output entropy additivity conjecture

TL;DR

This paper simplifies Hastings' original proof of the violation of the minimum output entropy additivity conjecture, extending the result to a broader class of channels using concentration of measure techniques.

Contribution

It provides a simplified proof of non-additivity and broadens the class of channels known to violate the additivity conjecture.

Findings

Non-additivity holds for most channels with Haar random isometry and large environment.

The proof avoids complex probability distributions, using concentration of measure instead.

The result extends Hastings' original findings to a wider set of channels.

Abstract

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.