Geometrical approach to mutually unbiased bases

TL;DR

This paper introduces a geometric phase-space method for constructing mutually unbiased bases in two-qubit systems, classifying compatible structures and transformations, with potential generalizations to higher prime-power dimensions.

Contribution

It presents a unifying geometric framework for mutually unbiased bases, including classification of structures and transformations, and discusses generalization to higher-dimensional systems.

Findings

Classified geometrical structures compatible with unbiasedness.

Linked transformations to local rotations on the Bloch sphere.

Suggested extension to systems with prime-power dimensions.

Abstract

We propose a unifying phase-space approach to the construction of mutually unbiased bases for a two-qubit system. It is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the Bloch-sphere principal axes. We suggest how to generalize the method to systems in dimensions that are powers of a prime.