Relative Entropy and Single Qubit Holevo-Schumacher-Westmoreland Channel Capacity

TL;DR

This paper applies relative entropy to analyze the classical capacity of single qubit channels, deriving formulas and proofs that clarify optimal signaling states and their properties.

Contribution

It introduces a simple formula for qubit relative entropy and proves the optimality and uniqueness of certain signaling states for single qubit channels.

Findings

Derived a simple formula for qubit relative entropy.

Proved optimal HSW signaling states minimize channel output entropy.

Showed the average output density matrix of optimal states is unique.

Abstract

The relative entropy description of Holevo-Schumacher-Westmoreland (HSW) classical channel capacity is applied to single qubit channels. A simple formula for the relative entropy of qubit density matrices in the Bloch sphere representation is derived. This formula is combined with the King-Ruskai-Szarek-Werner qubit channel ellipsoid picture to analyze several unital and non-unital qubit channels in detail. An alternate proof that the optimal HSW signalling states for single qubit unital channels are those states with the minimum channel output entropy is presented. The derivation is based on the symmetries of the qubit relative entropy formula and the King-Ruskai-Szarek-Werner qubit channel ellipsoid picture. A proof is given that the average output density matrix of any set of optimal HSW signalling states for a (qubit or non-qubit) quantum channel is unique.