An improvement on the Delsarte-type LP-bound with application to MUBs

TL;DR

This paper introduces a Fourier analytic method to refine the Delsarte LP-bound and applies it to demonstrate that a specific family of MUBs in dimension 6 cannot be extended to a complete set.

Contribution

A novel Fourier analytic approach that improves the Delsarte LP-bound and provides new results on the limitations of MUBs in dimension 6.

Findings

Improved LP-bound via Fourier analysis

Proved the Fourier family in dimension 6 cannot form a complete MUB system

Enhanced understanding of MUB limitations in quantum information

Abstract

The linear programming (LP) bound of Delsarte can be applied to several problems in various branches of mathematics. We describe a general Fourier analytic method to get a slight improvement on this bound. We then apply our method to the problem of mutually unbiased bases (MUBs) to prove that the Fourier family $F(a,b)$ in dimension 6 cannot be extended to a full system of MUBs.