Revisiting Additivity Violation of Quantum Channels

Abstract

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of ǫ-net argument and Lévy'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.

Quantum Channels With Polytopic Images and Image Additivity

“…On the other hand, in such an asymptotic regime, the violation of additivity was proven to be typical in [2], [7], [9], and [10] (see also [1], [5], [12]). That is, there exists some channel T 2 : M d 2 (C) → M n (C) that violates the additivity equation (11) for some ε > 0:…”

Random matrix techniques in quantum information theory

“…An improved version of Dvoretzky's theorem was used in [ASW11] to show violations at p = 1. Later, Fukuda provided a simpler proof of violation [Fuk14], using this time ε-net arguments and Levy's lemma, the techniques used also in the pioneering work [HLW06]. In [BCN12,BCN13], the authors use free probability theory to compute exactly the minimum output entropy of a random quantum channel [BCN13, Theorem 5.2].…”

On the Minimum Output Entropy of Random Orthogonal Quantum Channels

“…The interest in the study of random quantum channels comes mainly from the fact that, to date, violation of additivity is proved only through random techniques (typically with random unitary quantum channels generated by random unitary matrices), see [Has09,FKM10,FK10,ASW11,BCN12,Fuk14,BCN16,Col16]. Non-random counter-examples have been obtained only for p-Rényi minimum output entropies, see [WH02,GHP10].…”