Strassen's asymptotic spectrum offers a framework for analyzing the complexity of tensors. It has found applications in diverse areas, from computer science to additive combinatorics and quantum information. A long-standing open problem, dating back to 1991, asks whether Strassen's support functionals are universal spectral points, that is, points in the asymptotic spectrum of tensors. In this paper, we answer this question in the affirmative by proving that the support functionals coincide with the quantum functionals - universal spectral points that are defined via entropy optimization on entanglement polytopes. We obtain this result as a special case of a general minimax formula for convex optimization on entanglement polytopes (and other moment polytopes) that has further applications to other tensor parameters, including the asymptotic slice rank. Our proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai.

翻译：斯特拉森的渐近谱为张量复杂性分析提供了一个理论框架。该框架已在从计算机科学到加法组合学及量子信息学等多个领域得到应用。一个可追溯至1991年的长期悬而未决的开放性问题在于：斯特拉森的支撑泛函是否为普适谱点，即张量渐近谱中的点。本文通过证明支撑泛函与量子泛函——即通过纠缠多胞体上的熵优化定义的普适谱点——完全一致，对该问题给出了肯定性回答。我们将此结果作为纠缠多胞体（及其他矩多胞体）上凸优化一般极小极大公式的特例，该公式还可进一步应用于包括渐近切片秩在内的其他张量参数。我们的证明基于平井近期提出的关于哈达玛流形上的芬切尔型对偶定理。