Galois Unitaries, Mutually Unbiased Bases, and MUB-balanced states

TL;DR

This paper introduces Galois unitaries to better understand Mutually Unbiased Bases (MUBs), revealing new transformations that cycle through all bases in certain dimensions and identifying special symmetric states.

Contribution

It extends the concept of anti-unitary operators with Galois unitaries, enabling new basis transformations and characterizing MUB-balanced states in odd prime power dimensions.

Findings

Existence of basis-cycling transformations in odd prime power dimensions.

Identification of MUB-balanced states as eigenvectors of these transformations.

Conjecture that all such states are obtained through this construction.

Abstract

A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in every dimension which is an odd power of an odd prime. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.