Mutually Unbiased Bases and The Complementarity Polytope

Details Mutually Unbiased Bases and The Complementarity Polytope Mutually Unbiased Bases and The Complementarity Polytope

2004-10-15

arxiv.org/abs/quant-ph/0410120v1

Open Sys. & Information Dyn. (2005) 12: 107-120

Physics

Quantum Physics

18 pages, 3 figures

Scientific paper

A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N=p^k, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N^2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N=p^k; perhaps they do not exist otherwise. However, the link to the existence of MUBs--if any--remains to be found.

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