Mutually Unbiased Quantum Observables

TL;DR

This paper explores the concept of mutually unbiased quantum observables in finite-dimensional Hilbert spaces, examining their relationships, special cases like finite position and momentum observables, and open problems for future research.

Contribution

It introduces a formal framework for mutually unbiased observables, analyzes their properties, and discusses extensions to unsharp observables with specific examples and open questions.

Findings

Finite position and momentum observables are related by a finite Fourier transform.

Examples of parts of MU observables that are and are not value-complementary are provided.

Open problems involve extending the theory to unsharp observables.

Abstract

We begin by defining mutually unbiased (MU) observables on a finite dimensional Hilbert space. We also consider the more general concept of parts of MU observables. The relationships between MU observables, value-complementary observables and two other conditions involving sequential products of observables are discussed. We next present a special motivating case of MU observables called finite position and momentum observables. These are atomic observables related by a finite Fourier transform. Finite position and momentum observables are employed to give examples of parts of MU observables that are value-complementary and those that are not value-complementary. Various open problems involving these concepts are presented. These problems mainly involve extending this work from sharp observables to unsharp observables.