The Convex Closure of the Output Entropy of Infinite Dimensional Channels and the Additivity Problem

TL;DR

This paper explores the properties of the convex closure of output entropy in infinite dimensional quantum channels, establishing that superadditivity in finite dimensions implies the same in infinite dimensions, thus advancing understanding of the additivity problem.

Contribution

It proves that superadditivity of the convex closure of output entropy in finite channels implies the same in infinite channels, extending key additivity results to infinite dimensions.

Findings

Superadditivity in finite channels implies superadditivity in infinite channels.

Infinite dimensional generalization of Shor's theorem on additivity properties.

Superadditivity results for channels combining noiseless and entanglement-breaking components.

Abstract

The continuity properties of the convex closure of the output entropy of infinite dimensional channels and their applications to the additivity problem are considered. The main result of this paper is the statement that the superadditivity of the convex closure of the output entropy for all finite dimensional channels implies the superadditivity of the convex closure of the output entropy for all infinite dimensional channels, which provides the analogous statements for the strong superadditivity of the EoF and for the additivity of the minimal output entropy. The above result also provides infinite dimensional generalization of Shor's theorem stated equivalence of different additivity properties. The superadditivity of the convex closure of the output entropy (and hence the additivity of the minimal output entropy) for two infinite dimensional channels with one of them a direct sum of noiseless and entanglement-breaking channels are derived from the corresponding finite dimensional results. In the context of the additivity problem some observations concerning complementary infinite dimensional channels are considered.