Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a doubly stochastic matrix $B'$ with entries $\beta_{ii}$, does there exist a unitary matrix with entries $\alpha_{ii}$ such that $|\alpha|^2_{ii} = \beta_{ii}$?

share|improve this question
    
add comment

4 Answers

up vote 1 down vote accepted

A brief googling yielded another interesting paper:

http://www.sciencedirect.com/science/article/pii/0024379578900228

Topological properties of orthostochastic matrices ☆

Tony F. Heinz

"In this paper, it is shown that for n⩾3 the orthostochastic matrices are not everywhere dense in the set of doubly stochastic matrices, thus answering a question of L. Mirsky in his survey article on doubly stochastic matrices [2]"

share|improve this answer
    
That answers the question directly - thank you! –  Ben Lerner Mar 29 '12 at 1:25
add comment

Counterexample: $$\begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0 \end{pmatrix}.$$ The question is of interest in quantum probability. Your map from unitary matrices to doubly stochastic matrices defines an interesting region which has non-zero volume, and which cannot be convex because it visits all of the vertices.

share|improve this answer
2  
Stochasm is the lowest form of wit. –  Will Jagy Nov 7 '11 at 2:53
add comment

Interesting topic. Today we do not have a clear picture about the relationship between being uni-stochastic (or ortho-stochastic, if you restrict your attention to orthogonal matrices) and doubly-stochastic.

A couple of references:

Defect of a unitary matrix, Wojciech Tadej, Karol Zyczkowski http://arxiv.org/abs/math/0702510

Recent work, it contains a number of references on the discussion.

On the digraph of a unitary matrix, Simone Severini http://arxiv.org/abs/math/0205187

A more combinatorial perspective (but superficial)

share|improve this answer
add comment

Yet another reference

I. Bengtsson, A. Ericsson, M. Kus, W. Tadej, and K. Zyczkowski, Birkhoff's polytope and unistochastic matrices, N=3 and N=4, Comm. Math. Phys. 259, 307-324 (2005).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.