Nonexistence of sharply covariant mutually unbiased bases in odd prime dimensions

TL;DR

This paper proves that in odd prime dimensions, mutually unbiased bases cannot be generated by a group of the specified order, revealing limitations on their symmetry and contributing to the understanding of quantum state space geometry.

Contribution

It establishes the nonexistence of sharply covariant MUB in odd prime dimensions, advancing the theoretical understanding of MUB symmetry properties.

Findings

No MUB in odd prime dimensions is sharply covariant.

Supports the conjecture that only dimensions 2 and 4 have sharply covariant MUB.

Provides insights into the symmetry and geometry of quantum state space.

Abstract

Mutually unbiased bases (MUB) are useful in a number of research areas. The symmetry of MUB is an elusive and interesting subject. A (complete set of) MUB in dimension $d$ is sharply covariant if it can be generated by a group of order $d(d+1)$ from a basis state. Such MUB, if they exist, would be most appealing to theoretical studies and practical applications. Unfortunately, they seem to be quite rare. Here we prove that no MUB in odd prime dimensions is sharply covariant, by virtue of clever applications of Mersenne primes, Galois fields, and Frobenius groups. This conclusion provides valuable insight about the symmetry of MUB and the geometry of quantum state space. It complements and strengthens the earlier result of the author that only two stabilizer MUB are sharply covariant. Our study leads to the conjecture that no MUB other than those in dimensions 2 and 4 is sharply covariant.