Unistochastic matrix

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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.

The detailed definition is as follows. A square matrix B of size n is doubly stochastic (or bistochastic) if all its rows and columns sum to 1 and all its entries are nonnegative real numbers, each of whose rows and columns sums 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.

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