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 $\alpha^2_{ii} = \beta_{ii}$?

A brief googling yielded another interesting paper: http://www.sciencedirect.com/science/article/pii/0024379578900228 Topological properties of orthostochastic matrices ☆
"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]" 


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 nonzero volume, and which cannot be convex because it visits all of the vertices. 


Interesting topic. Today we do not have a clear picture about the relationship between being unistochastic (or orthostochastic, if you restrict your attention to orthogonal matrices) and doublystochastic. 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) 


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, 307324 (2005). 

