On some additivity problems in quantum information theory

TL;DR

This paper discusses the challenging additivity problems in quantum information theory related to channel capacities and output purity, presenting new partial results that support the conjecture of additivity.

Contribution

It formulates key additivity problems, explores their interrelations, and provides new partial proofs that bolster the conjecture of additivity in quantum channels.

Findings

Partial results supporting the additivity conjecture

Relations established between different additivity problems

Numerical evidence consistent with the conjecture

Abstract

A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", and the "maximal output purity". All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture.