Maximally symmetric stabilizer MUBs in even prime-power dimensions

TL;DR

This paper explores the construction of maximally symmetric mutually unbiased bases (MUBs) in even prime-power dimensions, revealing multiple inequivalent covariant families and contrasting with the odd-dimensional case.

Contribution

It demonstrates the existence of two distinct maximal subgroups admitting covariant MUBs in even prime-power dimensions and explicitly constructs families of such bases.

Findings

Existence of two essentially different covariant MUB families in 2^n dimensions

Explicit construction of covariant MUBs for both subgroup types

Maximally covariant MUBs are not unique in even prime-power dimensions

Abstract

One way to construct a maximal set of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space is by means of finite phase-space methods. MUBs obtained in this way are covariant with respect to some subgroup of the group of all affine symplectic phase-space transformations. However, this construction is not canonical: as a consequence, many different choices of covariance sugroups are possible. In particular, when the Hilbert space is $2^n$ dimensional, it is known that covariance with respect to the full group of affine symplectic phase-space transformations can never be achieved. Here we show that in this case there exist two essentially different choices of maximal subgroups admitting covariant MUBs. For both of them, we explicitly construct a family of $2^n$ covariant MUBs. We thus prove that, contrary to the odd dimensional case, maximally covariant MUBs are very far from being unique.