Concurrence and Entanglement Entropy of Stochastic 1-Qubit Maps

TL;DR

This paper derives explicit formulas for the concurrence of all stochastic 1-qubit maps, introduces a new convex roof construction method, and explores entanglement entropy and capacity in symmetric cases.

Contribution

It provides the first explicit expressions for concurrence of all stochastic 1-qubit maps and introduces a novel method for convex roof construction.

Findings

Existence of two-component optimal decompositions.

Explicit concurrence formulas for 1-qubit stochastic maps.

Analysis of entanglement entropy and capacity in symmetric cases.

Abstract

Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. We construct the relevant convex roof patterns by a new method. We conclude that two component optimal decompositions always exist. Our results can be transferred to 2xN-quantumsystems providing the concurrence for all rank two density operators as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of 1-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we analyze the bifurcation patterns appearing in the convex roof of optimal decompositions and give results for the one-shot (Holevo-Schumacher-Westmoreland) capacity of those maps.