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

2025-07-03

AI Summary

This study addresses the scalability problem of mutually unbiased bases (MUBs) in complex Hilbert spaces. We recast MUB construction as a real algebraic geometry problem: characterizing MUB extendibility over the reals is equivalent to determining the existence of real points on a specific complete intersection variety. Our method integrates real affine variety theory, spectral analysis of orthogonal normal matrices, and the structure of maximal commutative classes. We establish a geometric criterion for MUB extendibility, prove for the first time that the associated variety is a complete intersection, and rigorously demonstrate a one-to-one correspondence between MUB extensions and maximal commutative classes of orthogonal normal matrices. This framework transcends conventional constructive paradigms and provides the first systematic, algebraic-geometric approach to discovering novel MUBs.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $mathcal M_n({mathbb{C}})$.

Problem

• Extending MUBs in C^n via real points on algebraic varieties
• Linking MUBs to maximal commuting classes of orthogonal matrices
• Establishing complete intersection domains for MUB variety relations

Innovation

• Studying real points on affine algebraic varieties
• Linking MUBs to maximal commuting matrix classes
• Using complete intersection domains for MUB extension