A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

TL;DR

This paper introduces a criterion for achieving the Welch bounds with equality, providing a new necessary and sufficient condition for constructing mutually unbiased bases (MUBs) and connecting them to combinatorial structures.

Contribution

It presents a novel criterion for systems to meet Welch bounds with equality, leading to a new characterization of MUBs and linking them to combinatorial objects.

Findings

A criterion for systems to satisfy Welch bounds with equality.

A necessary and sufficient condition for MUBs formation.

Connections between MUBs and combinatorial structures like difference sets.

Abstract

The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs. This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion. Some directions for proving non-existence results are sketched as well.