In mathematics, a unistochastic matrix (also called ''unitary-stochastic'') is a

doubly stochastic matrix In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1 ...

whose entries are the squares of the absolute values of the entries of some

unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...

. A square matrix ''B'' of size ''n'' is doubly stochastic (or ''bistochastic'') if all its entries are non-negative

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix ''U'' such that :

B_=, U_, ^2 \text i,j=1,\dots,n. \,

This definition is analogous to that for an

orthostochastic matrix In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix. The detailed definition is as follows. A square matrix ''B'' of size ''n'' is ...

, which is a doubly stochastic matrix whose entries are the squares of the entries in some

orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity m ...

. Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger ''n'' this is not the case. For example, take

n=3

and consider the following doubly stochastic matrix: :

B= \frac 
\begin
1  & 1 & 0  \\
0  & 1 & 1  \\
1  & 0 & 1   \end.

This matrix is not unistochastic, since any two vectors with moduli equal to the square root of the entries of two columns (or rows) of ''B'' cannot be made orthogonal by a suitable choice of phases. For

n > 2

, the set of orthostochastic matrices is a

proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

of the set of unistochastic matrices. * the set of unistochastic matrices contains all

permutation matrices In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...

and its convex hull is the

Birkhoff polytope The Birkhoff polytope ''B'n'' (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph K_) is the convex polytope in R''N'' (where ''N'' = ''n''2) who ...

of all doubly stochastic matrices * for

n \ge 3

this set is not convex * for

n = 3

the set of triangle inequality on the moduli of the raw is a sufficient and necessary condition for the unistocasticity * for

n =3

the set of unistochastic matrices is star shaped and unistochasticity of any bistochastic matrix ''B'' is implied by a non-negative value of its Jarlskog invariant * for

n =3

the relative volume of the set of unistochastic matrices with respect to the

of doubly stochastic matrices is

8\pi^2/105 \approx 75.2 \%

* for

n =4

explicit conditions for unistochasticity are not known yet, but there exists a numerical method to verify unistochasticity based on the algorithm by Haagerup * The Schur-Horn theorem is equivalent to the following "weak convexity" property of the set

\mathcal_n

of unistochastic

n \times n

matrices: for any vector

v \in \mathbb^n

the set

\mathcal_nv

is the convex hull of the set of vectors obtained by all permutations of the entries of the vector

v

(the permutation polytope generated by the vector

v

). * The set of

n \times n

unistochastic matrices

\mathcal_n \subset \mathbb^

has a nonempty interior. The unistochastic matrix corresponding to the unitary

n \times n

matrix with the entries

U_ = \delta_ + \frac

, where

, \theta, =1

and

\theta \neq \pm 1

, is an interior point of

\mathcal_n

References

* . * * {{Cite arXiv, arxiv=0806.2357, first=Alexander, last=Karabegov, title=A mapping from the unitary to doubly stochastic matrices and symbols on a finite set, date=2008-06-14 Matrices