Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six

TL;DR

This paper investigates the construction of mutually unbiased bases in dimension six using a numerical method, revealing new symmetries and providing evidence that only up to three such bases can be formed from certain complex Hadamard matrices.

Contribution

It introduces an efficient numerical approach to find mutually unbiased bases and uncovers new symmetries in non-affine complex Hadamard matrices in dimension six.

Findings

Discovery of new symmetries in Karlsson's non-affine family

Evidence limiting the number of mutually unbiased bases to three

Analytical confirmation of the symmetries found

Abstract

We study the problem of constructing mutually unbiased bases in dimension six. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family $K_6^{(2)}$ which we confirm analytically. Also, we obtain strong evidence that no more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.