Constructing Mutually Unbiased Bases in Dimension Six

TL;DR

This paper investigates the existence of mutually unbiased bases in dimension six, using algebraic computations to test all known Hadamard matrices, and finds strong evidence supporting the conjecture that no seven such bases exist.

Contribution

It provides computational evidence that only up to two mutually unbiased bases can be constructed in dimension six, supporting the conjecture of their non-existence beyond that.

Findings

Never found more than two mutually unbiased bases in dimension six

Supports the conjecture that seven mutually unbiased bases do not exist in dimension six

Uses algebraic computations on known Hadamard matrices

Abstract

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.