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Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4

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The set of bistochastic or doubly stochastic N×N matrices is a convex set called Birkhoff’s polytope, which we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff’s polytope. For N=3 we present fairly complete results. For N=4 partial results are obtained. An interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der Waerden matrix for N=3, while this is not the case for N=4.

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Authors and Affiliations

  1. Stockholm University, AlbaNova, Fysikum, 106 91, Stockholm, Sweden

    Ingemar Bengtsson & Åsa Ericsson

  2. Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/44, 02–668, Warszawa, Poland

    Marek Kuś & Karol Życzkowski

  3. Cardinal Stefan Wyszynski University, Warszawa, Poland

    Wojciech Tadej

  4. Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, ul. Reymonta 4, 30–059, Kraków, Poland

    Karol Życzkowski

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  1. Ingemar Bengtsson
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  2. Åsa Ericsson
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  3. Marek Kuś
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  4. Wojciech Tadej
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  5. Karol Życzkowski
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Correspondence to Ingemar Bengtsson.

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Communicated by M.B. Ruskai

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Bengtsson, I., Ericsson, Å., Kuś, M. et al. Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4. Commun. Math. Phys. 259, 307–324 (2005). https://doi.org/10.1007/s00220-005-1392-8

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  • DOI: https://doi.org/10.1007/s00220-005-1392-8

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