Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier

Details Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier

2008-10-09

arxiv.org/abs/0810.1761v1

Physics

Quantum Physics

5 pages, 2 figures

Scientific paper

10.1063/1.3059166

In quantum mechanics some properties are maximally incompatible, such as the position and momentum of a particle or the vertical and horizontal projections of a 2-level spin. Given any definite state of one property the other property is completely random, or unbiased. For N-level systems, the 6-level ones are the smallest for which a tomographically efficient set of N+1 mutually unbiased bases (MUBs) has not been found. To facilitate the search, we numerically extend the classification of unbiased bases, or Hadamards, by incrementally adjusting relative phases in a standard basis. We consider the non-unitarity caused by small adjustments with a second order Taylor expansion, and choose incremental steps within the 4-dimensional nullspace of the curvature. In this way we prescribe a numerical integration of a 4-parameter set of Hadamards of order 6.

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