Abstract
We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a 2-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order n exists, if there exists a skew Hadamard matrix or a symmetric conference matrix of this size. This is the case for any even \(n\le 20\), and for these dimensions we demonstrate that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix U, such that \(B_{ij}=|U_{ij}|^2\), is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order \(n \times n\). Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. In the case \(n=4\) we study geometry of the set \({\mathcal U}_4\) of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope \({\mathcal B}_4\).
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Acknowledgements
One of the authors (KŻ) had a chance to discuss the unistochasticity problem with the late Uffe Haagerup during the conference in Będlewo in July 2014, where he learned about the way to treat the \(n=4\) case presented in the “Appendix A”. It is a pleasure to thank Ingemar Bengtsson and Irina Dimitru for numerous discussions on equi-entangled bases and for sharing with us their unpublished notes. We are thankful to Dardo Goyeneche and Wojciech Tadej for numerous interactions and also to Robert Craigen and William Orrick for fruitful discussions during the workshop in Budapest in July 2017. Special thanks are due to Mate Matolcsi and Ferenc Szöllősi, for organizing such a successful conference which made these interactions possible. We are deeply obliged to the referee for a long list of constructive comments, valuable hints to the literature and for suggesting us results presented in “Appendix D”. Financial support by Narodowe Centrum Nauki under the Grant No. DEC-2015/18/A/ST2/00274 is gratefully acknowledged.
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Dedicated to the memory of Uffe Haagerup (1949–2015).
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Rajchel, G., Gąsiorowski, A. & Życzkowski, K. Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces. Math.Comput.Sci. 12, 473–490 (2018). https://doi.org/10.1007/s11786-018-0384-y
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DOI: https://doi.org/10.1007/s11786-018-0384-y
Keywords
- Hadamard matrices
- Birkhoff polytope
- Unistochasticity
Mathematics Subject Classification
- 05B20
- 05A05