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arXiv:1506.09062

Cliffordtori and unbiased vectors

Ole Andersson, Ingemar Bengtsson

Published Jun 30, 2015• 1 citation
arXivHTMLPDF

Abstract

The existence problem for mutually unbiased bases is an unsolved problem in quantum information theory. A related question is whether every pair of bases admits vectors that are unbiased to both. Mathematically this translates to the question whether two Lagrangian Clifford tori intersect, and a body of results exists concerning it. These (deep!) results are however rather weak when viewed from the point of view of the first problem. We make a detailed study of how the intersections behave in the simplest non-trivial case, that of complex projective 2-space (the qutrit), for which the set of Clifford tori can be usefully parametrized by the unistochastic subset of Birkhoff's polytope. An interesting picture emerges. A foray into higher dimensions is included.

DOI
10.48550/arxiv.1506.09062

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Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.
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