Revisiting additivity violation of quantum channels

TL;DR

This paper presents a new proof of additivity violation of quantum channels' minimum output entropy using Dvoretzky's theorem, extending the regimes where violation occurs and connecting to previous methods.

Contribution

The authors develop an alternative proof of additivity violation based on Milman's view of Dvoretzky's theorem, broadening the known regimes of violation and linking to Hastings' approach.

Findings

Additivity violation proven in broader regimes

Extension of Dvoretzky's theorem techniques to norm-like functions

Discussion on relation between regularized entropy and classical capacity

Abstract

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of \epsilon-net argument and L\'evy's lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley's bound on Gaussian process (or Dvoretzky's theorem with Schechtman's improvement). In this paper, we develop another proof along Dvoretzky's theorem in Milman's view showing additivity violation in broader regimes than the existing proofs. Importantly, Dvoretzky's theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm-like structures - positive homogeneity and triangle inequality. Then, a connection between Hastings' method and ours is also discussed. Besides, we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels.