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

TL;DR

This work develops and implements practical algorithms to compute moment polytopes for tensors, extending to general reductive group actions. It leverages Franz’s characterization via supports and a Borel-polytope formulation to produce deterministic and probabilistic methods with verification, enabling exact computation for all 3×3×3 tensors and high-probability results for 4×4×4 tensors, including the matrix multiplication tensor. The approach yields new obstructions for asymptotic tensor restrictions through quantum functionals and enables structural insights into tensor marginal problems, scaling algorithms, and Kronecker polytopes. It thus provides a computational toolkit that bridges quantum information, algebraic complexity, and optimization, with potential to guide future theoretical and practical advances in moment polytope theory and its applications.

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 $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ and $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ 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 $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$, and with high probability those in $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$ (which includes the $2\times 2$ matrix multiplication tensor). We discuss how these explicit moment polytopes have led to several new theoretical directions and results.

Figures (2)

• Figure 1.0: Overview of inclusions among the moment polytopes of tensors in $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$, up to cyclic permutations of the factors. An arrow is drawn from polytope $P$ to polytope $Q$ if $P \supseteq Q'$ for a $Q'$ that can be obtained from $Q$ by a permutation of the three factors. Nodes are labeled by representative tensors with this moment polytope. The tensor $\mathsf{T}_i$ is tensor $i$ from \ref{['table:c333 unstable tensors info']}. The others are defined by the following tensors: $\mathsf{U}_{{3}} \coloneqq e_{111} + e_{222} + e_{333}$, $\mathsf{D} \coloneqq e_1 \wedge e_2 \wedge e_3$, and $\mathsf{W} \coloneqq e_{112}+e_{121}+e_{211}$ (where $e_{ijk} \coloneqq e_i \otimes e_j \otimes e_k$).
• Figure 5.26: Overview of inclusions among the moment polytopes of tensors in $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$, up to cyclic permutations of the factors. Nodes are labeled by representative tensors with this moment polytope. The tensor $\mathsf{T}_i$ is tensor $i$ from \ref{['table:c333 unstable tensors info']}. The moment polytope of $\mathsf{U}_{{3}}$ is the Kronecker polytope. The same is true for any other tensor not equivalent to the ones in the diagram. Square and circular nodes contain $\textnormal{SL}$-semistable and $\textnormal{SL}$-unstable tensors respectively. An arrow is drawn from polytope $P$ to polytope $Q$ if $P \supseteq Q'$ for a $Q'$ that can be obtained from $Q$ by a permutation of the three factors. It is dashed if there is an inclusion of moment polytopes (as above) but no degeneration between the corresponding tensors (for all permutations of the factors). It is dotted if we do not know whether such a degeneration exists. See \ref{['remark:inclusion relations']} for more details.

Theorems & Definitions (106)

• Remark 2.1
• Theorem 2.2: nessStratificationNullCone1984brion1987momentMapImage, special case
• Theorem 2.3: franz2002, special case
• Corollary 2.4
• Proposition 2.5: Borel polytopes franz2002guillemin2005convexityPropertiesguillemin2006convexityBorelburgisser2018tensorScaling
• Lemma 2.6: franz2002burgisser2018tensorScaling
• Lemma 2.7: Padding with zeros
• Proposition 2.8: Equivalence
• Remark 2.9
• Proposition 2.10: Degeneration