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

Abstract

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.

5 pages, 2 figures

References in corpus (2)

• Security of quantum key distribution using d-level systems
• Mubs and Hadamards of Order Six

Cited by in corpus (9)

• On mutually unbiased bases
• Constructing Mutually Unbiased Bases in Dimension Six
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Two-parameter complex Hadamard matrices for N=6
• States that "look the same" with respect to every basis in a mutually unbiased set
• Families of complex Hadamard matrices
• A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids
• The problem of mutually unbiased bases in dimension 6
• How much complementarity?