A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6

TL;DR

This paper constructs an infinite family of triplets of mutually unbiased bases in dimension 6 using Fourier family matrices, and proves that these triplets cannot be extended to larger sets, advancing understanding of MUBs in this dimension.

Contribution

It introduces an infinite family of MUB triplets in dimension 6 and presents a novel proof method showing these cannot be extended to quartets, suggesting the maximum MUBs in dimension 6 is three.

Findings

Constructed an infinite family of MUB triplets in dimension 6.

Proved that these triplets cannot be extended to quartets.

Provided a new proof method applicable to MUB problems.

Abstract

We exhibit an infinite family of {\it triplets} of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, $F(a,b)$. However, in the main result of the paper we also prove that for any values of the parameters $(a,b)$, the standard basis and $F(a,b)$ {\it cannot be extended to a MUB-quartet}. The main novelty lies in the {\it method} of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.