# Can a unitary matrix be constructed from any doubly stochastic matrix?

Here is a question that came up while I was thinking about the foundations of quantum mechanics:

Consider a unitary $n\times n$ complex matrix $U$, with elements $u_{ij}$. We know that the rows and columns of such a matrix must form orthonormal bases of $\mathbb{C}^n$. It follows that the matrix $P$, with elements $p_{ij} = |u_{ij}|^2$, must be doubly stochastic, i.e. $\sum_i p_{ij} = \sum_j p_{ij} = 1$, and every $p_{ij}\ge 0$.

The question is, is the reverse also true? That is, for any arbitrary doubly stochastic matrix $P = (p_{ij})$, does there exist a unitary matrix $U$ with elements $u_{ij}$ such that $|u_{ij}|^2 = p_{ij}$?

If so, is there a systematic way to construct such a unitary matrix, given $P$? (I realise the solution will not be unique.)

If not, is there some other property, besides being doubly stochastic, that $P$ must have in order to make this conjecture true?

• According to Wikipedia, the following is an example of a doubly stochastic matrix that is not unistochastic: $$\frac{1}{2} \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}.$$ Commented Nov 27, 2017 at 3:22