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
All 2-by-2 doubly stochastic matrices are unistochastic and orthostochastic, but for larger n it is not the case. Already for there exists a bistochastic matrix B which is not unistochastic:
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 this set is not convex
- for the set of unistochastic matrices is star shaped.
- for the relative volume of the set of unistochastic matrices with respect to the Birkhoff polytope of bistochastic matrices is
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.