Entanglement and output entropy of the diagonal map

TL;DR

This paper explores the properties of the convex roof extension in quantum information, focusing on the entanglement entropy of the diagonal map, revealing a bifurcation phenomenon at dimension six.

Contribution

It provides a detailed analysis of the entanglement entropy of the diagonal map for symmetric states and uncovers a surprising bifurcation at dimension six.

Findings

Entanglement entropy of the diagonal map for permutation symmetric states

Solution for the case z<0 in N=3 states

Bifurcation in output entropy behavior at N=6

Abstract

We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real N=3 states $\omega(z)$ and solve the case $z<0$ where $z$ is the non-diagonal entry in the density matrix. We also report a surprising result about the behaviour of the output entropy of the diagonal map for arbitrary dimensions $N$; showing a bifurcation at N=6.