$H_2$-reducible matrices in six-dimensional mutually unbiased bases

TL;DR

This paper investigates the structure of six-dimensional mutually unbiased bases (MUBs), showing that if four such bases containing the identity exist, then the associated $H_2$-reducible matrix must have exactly nine $2 imes2$ Hadamard submatrices, and uses this to exclude certain known matrices.

Contribution

The paper establishes a necessary condition for the existence of four six-dimensional MUBs involving $H_2$-reducible matrices, advancing understanding of their structure and non-existence.

Findings

$H_2$-reducible matrix in four MUBs has exactly nine $2\times2$ Hadamard submatrices if they exist.

Certain known CHMs are excluded from four MUBs based on the derived condition.

Progress towards resolving the open problem of six-dimensional MUBs existence.

Abstract

Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the $H_2$-reducible matrix in the four MUBs has exactly nine $2\times2$ Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric $H_2$-reducible matrix, the Hermitian matrix, Dita family, Bjorck's circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs.