Unistochastic matrix

In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries 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 exists 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 $n\geq 3$ 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.
• Bengtsson, Ingemar (2004-03-11). "The importance of being unistochastic". arXiv:quant-ph/0403088.