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.

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 ext{ 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} egin{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.

References

* | year=2005 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=259 | issue=2 | pages=307–324.