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

TL;DR

This paper develops a numerical method to classify and construct mutually unbiased bases (Hadamards) for 6-level quantum systems, addressing a key gap in quantum information theory.

Contribution

It introduces a novel numerical approach to extend the classification of unbiased bases for 6-level systems by adjusting phases within a nullspace, aiding the search for MUBs.

Findings

Extended classification of unbiased bases for 6-level systems.

Numerical method using phase adjustments and Taylor expansion.

Prescribed a 4-parameter set of Hadamards for N=6.

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.