On Obtaining New MUBs by Finding Points on Complete Intersection Varieties over $\mathbb{R}$

TL;DR

This paper explores the mathematical structure of Mutually Unbiased Bases (MUBs) by linking their extendability to real points on certain algebraic varieties and establishing a correspondence with maximal commuting classes of matrices.

Contribution

It introduces algebraic criteria for extending MUBs using affine varieties and connects MUBs with maximal commuting classes of orthogonal matrices.

Findings

Equivalent criteria for MUB extension via algebraic varieties

Identification of parts of the variety as complete intersection domains

One-to-one correspondence between MUBs and maximal commuting classes

Abstract

Mutually Unbiased Bases (MUBs) are closely connected with quantum physics, and the structure has a rich mathematical background. We provide equivalent criteria for extending a set of MUBs for $C^n$ by studying real points of a certain affine algebraic variety. This variety comes from the relations that determine the extendability of a system of MUBs. Finally, we show that some part of this variety gives rise to complete intersection domains. Further, we show that there is a one-to-one correspondence between MUBs and the maximal commuting classes (bases) of orthogonal normal matrices in $\mathcal M_n({\mathbb{C}})$. It means that for $m$ MUBs in $C^n$, there are $m$ commuting classes, each consisting of $n$ commuting orthogonal normal matrices and the existence of maximal commuting basis for $\mathcal M_n({\mathbb{C}})$ ensures the complete set of MUBs in $\mathcal M_n({\mathbb{C}})$.