Unistochastic matrix - Alchetron, The Free Social Encyclopedia

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 i j = | U i j | 2 for i , j = 1 , … , 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 = 1 2 [ 1 1 0 0 1 1 1 0 1 ]

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 ≥ 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 π 2 / 105 ≈ 75.2 %

References

Unistochastic matrix Wikipedia

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