Three numerical approaches to find mutually unbiased bases using Bell inequalities

TL;DR

This paper explores the existence of mutually unbiased bases in quantum information by numerically solving an optimization problem related to Bell inequalities, providing evidence supporting Zauner's conjecture in specific dimensions.

Contribution

It introduces three numerical methods to test Zauner's conjecture by formulating it as an optimization problem involving Bell inequalities in various dimensions.

Findings

No evidence of four MUBs in dimension six from three methods.

Numerical optimizers match known optimal bases in low dimensions.

Results suggest at most three MUBs in dimension ten.

Abstract

Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers $n,d \ge 2$ that can be maximally violated in dimension $d$ if and only if $n$ MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the "four most distant bases" in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. L\"u, B.-G. Englert, Phys. Rev. A, 83 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.