Mutually Unbiased Bases and Semi-definite Programming

TL;DR

This paper reviews computational methods, including semidefinite programming, to analyze the maximum number of mutually unbiased bases in a six-dimensional complex Hilbert space, confirming the non-existence of a complete set of seven such bases.

Contribution

It introduces a novel application of semidefinite programming to determine bounds on mutually unbiased bases in composite dimensions.

Findings

Confirmed that no more than three mutually unbiased bases exist in dimension six

Reviewed discretization and Grobner basis methods for tightening bounds

Demonstrated the non-existence of a complete set of seven bases in dimension six

Abstract

A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Grobner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.