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 $|\alpha|^2\_{ii} = \beta\_{ii}$?



Doubly stochastic matrices as squares of entires of unitary matrices
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]"


