Mutually Unbiased Bases in Composite Dimensions -- A Review

TL;DR

This review discusses the existence and mathematical formulations of mutually unbiased bases in composite dimensions, highlighting unresolved questions and summarizing current results and approaches.

Contribution

It provides a comprehensive overview of the problem, including formulations, existing results, and potential strategies for composite dimensions.

Findings

Multiple equivalent formulations of the existence problem are presented.

Summarizes analytic, computational, and numerical results for composite dimensions.

Reviews modifications and outlines potential solution strategies.

Abstract

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.