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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]].
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| A square matrix ''B'' of size ''n'' is doubly stochastic (or ''bistochastic'') if all its entries are non-negative [[real number]]s and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix ''U'' such that
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| :<math> B_{ij}=|U_{ij}|^2 \text{ for } i,j=1,\dots,n. \, </math>
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| All 2-by-2 doubly stochastic matrices are unistochastic and [[orthostochastic matrix|orthostochastic]], but for larger ''n'' it is not the case.
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| Already for <math> n=3 </math> there exist a [[Doubly stochastic matrix|bistochastic matrix]] B which is not unistochastic:
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| :<math>
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| B= \frac{1}{2}
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| \begin{bmatrix}
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| 1 & 1 & 0 \\
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| 0 & 1 & 1 \\
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| 1 & 0 & 1 \end{bmatrix}
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| </math>
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| since any two vectors with moduli equal to the square root of the entries of two columns (rows)
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| of B cannot be made orthogonal by a suitable choice of phases.
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| == Properties ==
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| * the set of unistochastic matrices contains all [[permutation matrix|permutation matrices]]
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| * for <math> n \ge 3 </math> this set is not convex
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| * for <math> n =3 </math> the set of unistochastic matrices is [[star shaped]].
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| * for <math> n =3 </math> the relative volume of the set of unistochastic matrices with respect to the [[Birkhoff polytope]] of [[doubly stochastic matrix|bistochastic matrices]] is <math> 8\pi^2/105 \approx 75.2 \% </math>
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| == References ==
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| * {{Citation | last1=Bengtsson | first1=Ingemar | last2=Ericsson | first2=Åsa | last3=Kuś | first3=Marek | last4=Tadej | first4=Wojciech | last5=Życzkowski | first5=Karol | title=Birkhoff's Polytope and Unistochastic Matrices, N = 3 and N = 4 | doi=10.1007/s00220-005-1392-8 | year=2005 | journal=Communications in Mathematical Physics | volume=259 | issue=2 | pages=307–324 | arxiv=math/0402325}}.
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| [[Category:Matrices]]
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Latest revision as of 12:22, 5 May 2014
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