Self-Dual Maps and Symmetric Bistochastic Matrices

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.