Maximal Sets of Mutually Unbiased Quantum States in Dimension Six

TL;DR

This paper investigates the existence of maximal sets of mutually unbiased quantum states in six-dimensional space, providing numerical evidence that no seven such bases exist, thus advancing understanding of quantum state configurations.

Contribution

It introduces a numerical approach to identify MU constellations in dimension six and provides evidence against the existence of seven MU bases in this dimension.

Findings

Identified 18 MU constellations out of 35 possible in dimension six.

Numerical evidence suggests no seven MU bases exist in dimension six.

Supports the conjecture that maximal MU sets are limited in dimension six.

Abstract

We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the squares of the moduli of their scalar products are equal to zero, one, or 1/d. These sets will be called a MU constellation, and if four MU bases were to exist for d=6, they would give rise to 35 different MU constellations. Using a numerical minimisation procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension six.