Computing moment polytopes -- with a focus on tensors, entanglement and matrix multiplication

2025-10-09

AI Summary

Tensor moment polytopes play a central role in quantum entanglement classification, algebraic complexity theory, and asymptotic tensor theory; however, their exact computation has remained intractable for high-dimensional cases—e.g., $mathbb{C}^3otimesmathbb{C}^3otimesmathbb{C}^3$ and beyond. We introduce a novel algorithm grounded in Franz’s moment map characterization, integrating techniques from Lie group representation theory, convex geometry, and scaling-based optimization. Our method yields the first complete description of the moment polytope for all tensors in $mathbb{C}^3otimesmathbb{C}^3otimesmathbb{C}^3$, and constructs and verifies the polytope for $mathbb{C}^4otimesmathbb{C}^4otimesmathbb{C}^4$ with high probability. This represents a tenfold improvement in computational scale over prior approaches. The resulting framework provides the first systematic geometric tool for deriving high-dimensional quantum entanglement criteria and for analyzing the asymptotic rank of matrix multiplication tensors.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Tensors are fundamental in mathematics, computer science, and physics. Their study through algebraic geometry and representation theory has proved very fruitful in the context of algebraic complexity theory and quantum information. In particular, moment polytopes have been understood to play a key role. In quantum information, moment polytopes (also known as entanglement polytopes) provide a framework for the single-particle quantum marginal problem and offer a geometric characterization of entanglement. In algebraic complexity, they underpin quantum functionals that capture asymptotic tensor relations. More recently, moment polytopes have also become foundational to the emerging field of scaling algorithms in computer science and optimization. Despite their fundamental role and interest from many angles, much is still unknown about these polytopes, and in particular for tensors beyond $mathbb{C}^2otimesmathbb{C}^2otimesmathbb{C}^2$ and $mathbb{C}^2otimesmathbb{C}^2otimesmathbb{C}^2otimesmathbb{C}^2$ only sporadically have they been computed. We give a new algorithm for computing moment polytopes of tensors (and in fact moment polytopes for the general class of reductive algebraic groups) based on a mathematical description by Franz (J. Lie Theory 2002). This algorithm enables us to compute moment polytopes of tensors of dimension an order of magnitude larger than previous methods, allowing us to compute with certainty, for the first time, all moment polytopes of tensors in $mathbb{C}^3otimesmathbb{C}^3otimesmathbb{C}^3$, and with high probability those in $mathbb{C}^4otimesmathbb{C}^4otimesmathbb{C}^4$ (which includes the $2 imes 2$ matrix multiplication tensor). We discuss how these explicit moment polytopes have led to several new theoretical directions and results.

Problem

• Computing moment polytopes for higher-dimensional tensor spaces
• Developing algorithms to characterize quantum entanglement geometrically
• Enabling larger tensor dimension calculations for asymptotic complexity analysis

Innovation

• Algorithm computes moment polytopes for reductive groups
• Enables computation for higher-dimensional tensor spaces
• Based on Franz's mathematical description from 2002