Unistochastic matrix

In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the square of the absolute value of some unitary matrix.

A square matrix B of size n is doubly stochastic (or bistochastic) if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix U such that

$B_{ij}=|U_{ij}|^2 \text{ for } i,j=1,\dots,n. \,$

All 2-by-2 doubly stochastic matrices are unistochastic and orthostochastic, but for larger n it is not the case. Already for $n=3$ there exist a bistochastic matrix B which is not unistochastic:

$B= \frac{1}{2} \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}$

since any two vectors with moduli equal to the square root of the entries of two columns (rows) of B cannot be made orthogonal by a suitable choice of phases.

Properties

the set of unistochastic matrices contains all permutation matrices
for $n \ge 3$ this set is not convex
for $n =3$ the set of unistochastic matrices is star shaped.
for $n =3$ the relative volume of the set of unistochastic matrices with respect to the Birkhoff polytope of bistochastic matrices is $8\pi^2/105 \approx 75.2 \%$

References

Bengtsson, Ingemar; Ericsson, Åsa; Kuś, Marek; Tadej, Wojciech; Życzkowski, Karol (2005), "Birkhoff's Polytope and Unistochastic Matrices, N = 3 and N = 4", Communications in Mathematical Physics 259 (2): 307–324, arXiv:math/0402325, doi:10.1007/s00220-005-1392-8 .