On complementary channels and the additivity problem

TL;DR

This paper investigates the relationship between quantum channels and their complements, showing that certain properties like additivity are preserved between them, and introduces new classes of channels where these properties hold.

Contribution

It demonstrates the equivalence of output purity characteristics for complementary CP maps and identifies new classes of channels satisfying additivity.

Findings

Complementary CP maps share output purity characteristics.

Additivity conjecture validity for one class implies it for the complement.

New class of channels where multiplicativity/additivity holds, including diagonal and covariant channels.

Abstract

We explore complementarity between output and environment of a quantum channel (or, more generally, CP map), making an observation that the output purity characteristics for complementary CP maps coincide. Hence, validity of the mutiplicativity/additivity conjecture for a class of CP maps implies its validity for complementary maps. The class of CP maps complementary to entanglement-breaking ones is described and is shown to contain diagonal CP maps as a proper subclass, resulting in new class of CP maps (channels) for which the multiplicativity/additivity holds. Covariant and Gaussian channels are discussed briefly in this context.