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.

翻译：张量是数学、计算机科学和物理学中的基本对象。通过代数几何和表示论研究张量，在代数复杂性理论和量子信息领域已证明极具成果。其中，矩多面体已被理解发挥着关键作用。在量子信息中，矩多面体（亦称纠缠多面体）为单粒子量子边际问题提供了理论框架，并对纠缠现象给出几何刻画。在代数复杂性理论中，它们构成了捕捉渐近张量关系的量子泛函的数学基础。近年来，矩多面体更成为计算机科学与优化领域中新兴的尺度化算法研究的基础。尽管这些多面体具有根本重要性且受到多角度关注，其性质仍存在大量未知领域，特别是对于超越 $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ 和 $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ 的张量，相关计算仅零散存在。基于 Franz (J. Lie Theory 2002) 的数学描述，我们提出了一种计算张量矩多面体（实际上适用于约化代数群的一般类别）的新算法。该算法能够计算比以往方法高一个数量级维度的张量矩多面体，首次确凿计算了 $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$ 中所有张量的矩多面体，并以高概率计算了 $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$ 中的矩多面体（包含 $2\times 2$ 矩阵乘法张量）。我们进一步探讨了这些显式矩多面体如何引向若干新的理论方向与研究成果。