Question

Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a doubly stochastic matrix $B'$ with entries $\beta_{ii}$, does there exist a unitary matrix with entries $\alpha_{ii}$ such that $|\alpha|^2_{ii} = \beta_{ii}$?

Answer 1

A brief googling yielded another interesting paper:

http://www.sciencedirect.com/science/article/pii/0024379578900228

Topological properties of orthostochastic matrices ☆

Tony F. Heinz

"In this paper, it is shown that for n⩾3 the orthostochastic matrices are not everywhere dense in the set of doubly stochastic matrices, thus answering a question of L. Mirsky in his survey article on doubly stochastic matrices [2]"

Answer 2

Counterexample: $$\begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0 \end{pmatrix}.$$ The question is of interest in quantum probability. Your map from unitary matrices to doubly stochastic matrices defines an interesting region which has non-zero volume, and which cannot be convex because it visits all of the vertices.

Answer 3

Interesting topic. Today we do not have a clear picture about the relationship between being uni-stochastic (or ortho-stochastic, if you restrict your attention to orthogonal matrices) and doubly-stochastic.

A couple of references:

Defect of a unitary matrix, Wojciech Tadej, Karol Zyczkowski http://arxiv.org/abs/math/0702510

Recent work, it contains a number of references on the discussion.

On the digraph of a unitary matrix, Simone Severini http://arxiv.org/abs/math/0205187

A more combinatorial perspective (but superficial)

Answer 4

Yet another reference

I. Bengtsson, A. Ericsson, M. Kus, W. Tadej, and K. Zyczkowski, Birkhoff's polytope and unistochastic matrices, N=3 and N=4, Comm. Math. Phys. 259, 307-324 (2005).