Mutually unbiased bases for the rotor degree of freedom

TL;DR

This paper explores the construction of mutually unbiased bases for the rotor degree of freedom, introducing a new Heisenberg pair and two sets of bases, one of which is fully satisfactory and related to linear motion.

Contribution

It presents a novel approach to defining mutually unbiased bases for the rotor, including a new Heisenberg pair and two different sets of bases, one fully satisfactory.

Findings

First continuous set does not relate to Heisenberg pair

Second set is fully satisfactory and linked to linear motion

Constructs a Heisenberg pair for the rotor

Abstract

We consider the existence of a continuous set of mutually unbiased bases for the continuous and periodic degree of freedom that describes motion on a circle (rotor degree of freedom). By a singular mapping of the circle to the line, we find a first, but somewhat unsatisfactory, continuous set which does not relate to an underlying Heisenberg pair of complementary observables. Then, by a nonsingular mapping of the discrete angular momentum basis of the rotor onto the Fock basis for linear motion, we construct such a Heisenberg pair for the rotor and use it to obtain a second, fully satisfactory, set of mutually unbiased bases.