TITLE: Doubly stochastic matrices as squares of entires of unitary matrices
QUESTION [8 upvotes]: 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 $|\alpha|^2_{ii} = \beta_{ii}$?

REPLY [3 votes]: 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]"