Doubly stochastic matrices as squares of entires of unitary matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:33:39Z http://mathoverflow.net/feeds/question/80259 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80259/doubly-stochastic-matrices-as-squares-of-entires-of-unitary-matrices Doubly stochastic matrices as squares of entires of unitary matrices Ben Lerner 2011-11-06T22:58:55Z 2012-03-28T15:08:39Z <p>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 $|\alpha|^2_{ii} = \beta_{ii}$?</p> http://mathoverflow.net/questions/80259/doubly-stochastic-matrices-as-squares-of-entires-of-unitary-matrices/80260#80260 Answer by Greg Kuperberg for Doubly stochastic matrices as squares of entires of unitary matrices Greg Kuperberg 2011-11-06T23:32:53Z 2011-11-06T23:32:53Z <p>Counterexample: <code>$$\begin{pmatrix} 0 &amp; 1/2 &amp; 1/2 \\ 1/2 &amp; 0 &amp; 1/2 \\ 1/2 &amp; 1/2 &amp; 0 \end{pmatrix}.$$</code> 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.</p> http://mathoverflow.net/questions/80259/doubly-stochastic-matrices-as-squares-of-entires-of-unitary-matrices/80261#80261 Answer by Simone Severini for Doubly stochastic matrices as squares of entires of unitary matrices Simone Severini 2011-11-06T23:52:11Z 2011-11-06T23:52:11Z <p>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. </p> <p>A couple of references: </p> <p>Defect of a unitary matrix, Wojciech Tadej, Karol Zyczkowski <a href="http://arxiv.org/abs/math/0702510" rel="nofollow">http://arxiv.org/abs/math/0702510</a></p> <p>Recent work, it contains a number of references on the discussion.</p> <p>On the digraph of a unitary matrix, Simone Severini <a href="http://arxiv.org/abs/math/0205187" rel="nofollow">http://arxiv.org/abs/math/0205187</a></p> <p>A more combinatorial perspective (but superficial)</p> http://mathoverflow.net/questions/80259/doubly-stochastic-matrices-as-squares-of-entires-of-unitary-matrices/82818#82818 Answer by Karol Zyczkowski for Doubly stochastic matrices as squares of entires of unitary matrices Karol Zyczkowski 2011-12-06T20:16:58Z 2011-12-06T20:16:58Z <p>Yet another reference</p> <p>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).</p> http://mathoverflow.net/questions/80259/doubly-stochastic-matrices-as-squares-of-entires-of-unitary-matrices/92462#92462 Answer by Felix Goldberg for Doubly stochastic matrices as squares of entires of unitary matrices Felix Goldberg 2012-03-28T15:08:39Z 2012-03-28T15:08:39Z <p>A brief googling yielded another interesting paper:</p> <p><a href="http://www.sciencedirect.com/science/article/pii/0024379578900228" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0024379578900228</a></p> <p>Topological properties of orthostochastic matrices ☆</p> <pre><code>Tony F. Heinz </code></pre> <p>"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]"</p>