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

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 two-level spin. Given any definite state of one property, the other property is completely random or unbiased. For N-level systems, the six-level ones are the smallest for which a tomographically efficient set of N+1 mutually unbiased bases 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 nonunitarity caused by small adjustments with a second order Taylor expansion and choose incremental steps within the four-dimensional null space of the curvature. In this way, we prescribe a numerical integration of a four-parameter set of Hadamards of order of 6