Mutually Unbiased Bases, Generalized Spin Matrices and Separability

TL;DR

This paper provides a constructive proof for the existence of maximal sets of mutually unbiased bases in prime power dimensions using generalized spin matrices, and analyzes their separability properties.

Contribution

It offers explicit constructions of MUBs in prime power dimensions and introduces techniques to analyze their separability using algebraic field extensions.

Findings

Explicit representations of commuting unitary matrices for MUBs

Formulas for computing orthogonal bases

Analysis of basis separability using algebraic methods

Abstract

A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: |<v,w>| ^{2}=1/d. The MUB problem is to prove or disprove the the existence of a maximal set of d+1 bases. It has been shown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381, (1989)] that such a collection exists if d is a power of a prime number p. We revisit this problem and use dX d generalizations of the Pauli spin matrices to give a constructive proof of this result. Specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the MUB problem. Additionally we give formulas from which the orthogonal bases can be readily computed. We show how the techniques developed here provide a natural way to analyze the separability of the bases. The techniques used require properties of algebraic field extensions, and the relevant part of that theory is included in an Appendix.