On the additivity conjecture for channels with arbitrary constraints

TL;DR

This paper explores the additivity conjecture for quantum channels with arbitrary constraints, establishing equivalences with other additivity problems and providing new characterizations and proof techniques.

Contribution

It generalizes the additivity conjecture to channels with arbitrary constraints and links it to subadditivity and superadditivity properties, offering new proof methods.

Findings

Additivity conjecture for constrained channels is equivalent to unconstrained case.

Characterization of channels with subadditive $\, ext{chi}$-function.

Global additivity implies superadditivity of entanglement of formation.

Abstract

Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the $\chi$-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the $\chi $-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the $\chi $-function and strong superadditivity of the entanglement of formation.