Constructions of Mutually Unbiased Bases

TL;DR

This paper discusses the construction and limitations of mutually unbiased bases in complex vector spaces, providing a simplified proof for their maximal number in prime power dimensions and exploring open problems in the field.

Contribution

It offers a simplified proof for the existence of maximal sets of mutually unbiased bases in prime power dimensions using exponential sum estimates.

Findings

Maximal sets of mutually unbiased bases exist in prime power dimensions.

The paper discusses conjectures for the maximum number of such bases in arbitrary dimensions.

Provides a simplified proof technique based on exponential sums.

Abstract

Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if |<b|b'>|^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.