On the structure of optimal sets for tensor product channel

TL;DR

This paper investigates the structure of optimal state sets related to the Holevo capacity and minimal output entropy in quantum channels, exploring conditions for their coincidence and implications for additivity properties in tensor product channels.

Contribution

It provides necessary and sufficient conditions for the coincidence of optimal sets and links their structure to additivity properties of quantum channels.

Findings

Optimal sets exhibit hereditary properties under certain assumptions.

Additivity of Holevo capacity implies strong hereditary properties.

Structural relations between optimal sets of tensor product channels and individual channels.

Abstract

For a given quantum channel we consider two optimal sets of states, related to the Holevo capacity and to the minimal output entropy of this channel. Some properties of these sets as well as the necessary and sufficient condition for their coincidence are obtained. The relations between additivity properties for two quantum channels and the structure of the optimal sets for tensor product of these channels are considered. It turns out that these additivity properties are connected with the special hereditary property of the optimal sets for tensor product channel. We explore the structural properties of these optimal sets under two different assumptions. The first assumption means that for tensor product of two channels with arbitrary constraints there exists optimal ensemble with the product state average. It turns out that exactly this assumption guarantees hereditary property of the both optimal sets for tensor product channel and provides some observations concerning the additivity problems. The second assumption means additivity of the Holevo capacity for two channels with arbitrary constraints. It turns out that exactly this assumption guarantees strong hereditary property of the both optimal sets for tensor product channel and provides the natural "projective" relations between these sets and the optimal sets for single channels.