Mubs and Hadamards of Order Six

TL;DR

This paper investigates the existence of mutually unbiased bases (MUBs) in 6-dimensional complex space, finding only triplets and exploring the landscape of complex Hadamard matrices to understand their distribution.

Contribution

It introduces a new notion of distance between bases, maps the landscape of MUB triplets, and provides evidence for a larger family of complex Hadamard matrices in order six.

Findings

Only triplets of MUBs found in 6 dimensions

Developed a detailed map of MUBs within the Hadamard landscape

Evidence suggesting a four-dimensional family of Hadamard matrices exists

Abstract

We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organise our results. Finally we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.