Some attempts at proving the non-existence of a full set of mutually unbiased bases in dimension 6

TL;DR

This paper explores various mathematical approaches to prove the non-existence of a complete set of four mutually unbiased bases in six-dimensional space, a problem unresolved in quantum information theory.

Contribution

It applies multiple advanced mathematical techniques, including algebraic geometry and semidefinite programming, to address the open problem of mutually unbiased bases in dimension 6.

Findings

Proposes new approaches to the non-existence problem

Analyzes the problem using Grassmannian distance and quadratic matrix programming

Utilizes tools from algebraic geometry for the proof effort

Abstract

Complete sets of mutually unbiased bases are only known to exist in prime-power dimensions. We will describe a few approaches to the problem proving the (non)-existence of four mutually unbiased bases in dimension 6. These will include the notions of Grassmannian distance, quadratic matrix programming, semidefinite relaxations to polynomial programming, as well as various tools from algebraic geometry.