Covariant mutually unbiased bases

TL;DR

This paper classifies maximal sets of mutually unbiased bases (MUBs) based on their covariance properties under symmetries of finite phase-space, revealing unique covariance in odd prime-power dimensions and subgroup covariance in even dimensions.

Contribution

It provides a classification of MUBs by their covariance with respect to finite phase-space symmetries and constructs explicit unitary operators for subgroup covariance in even dimensions.

Findings

Maximal MUBs are covariant with the full symmetry group only in odd prime-power dimensions.

In even prime-power dimensions, covariance is achieved via subgroups analogous to the oscillator group.

Explicit unitary operators are constructed for subgroup covariance in even dimensions.

Abstract

The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case their equivalence class is actually unique. Despite this limitation, we show that in even-prime power dimension covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.