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arXiv · 1506.09062

Cliffordtori and unbiased vectors

Ole Andersson·Ingemar Bengtsson
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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.

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Ole Andersson, Ingemar Bengtsson. 2015-06-30. Cliffordtori and unbiased vectors. https://arxiv.org/abs/1506.09062

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