Self-Dual Maps and Symmetric Bistochastic Matrices

TL;DR

This paper explores the geometric relationship between H-unistochastic matrices and symmetric bistochastic matrices, providing bounds on line segments within the convex set and introducing self-dual doubly stochastic maps.

Contribution

It establishes geometric bounds for H-unistochastic matrices within symmetric bistochastic matrices and introduces self-dual doubly stochastic maps with new theoretical results.

Findings

Line segments through the centroid spend at least two-thirds in the H-unistochastic convex hull for n=3,4

Partial results for higher n regarding convex hulls

Self-dual doubly stochastic maps extend the Laudau-Streater theorem

Abstract

The H-unistochastic matrices are a special class of symmetric bistochastic matrices obtained by taking the square of the absolute value of each entry of a Hermitian unitary matrix. We examine the geometric relationship of the convex hull of the n by n H-unistochastic matrices relative to the larger convex set of n by n symmetric bistochastic matrices. We show that any line segment in the convex set of the n by n symmetric bistochastic matrices which passes through the centroid of this convex set must spend at least two-thirds of its length in the convex hull of the n by n H-unistochastic matrices when n is either three or four and we prove a partial result for higher n. A class of completely positive linear maps called the self-dual doubly stochastic maps is useful for studying this problem. Some results on self-dual doubly stochastic maps are given including a self-dual version of the Laudau-Streater theorem.