Structural and computational understanding of tensors is the driving force behind faster matrix multiplication algorithms, the unraveling of quantum entanglement, and the breakthrough on the cap set problem. Strassen's asymptotic spectra program (FOCS 1986) characterizes optimal matrix multiplication algorithms through monotone functionals. Our work advances and makes novel connections among two recent developments in the study of tensors, namely - the slice rank of tensors, a notion of rank for tensors that emerged from the resolution of the cap set problem (Ann. of Math. 2017), - and the quantum functionals of tensors (STOC 2018), monotone functionals defined as optimizations over moment polytopes. More precisely, we introduce an extension of slice rank that we call weighted slice rank and we develop a minimax correspondence between the asymptotic weighted slice rank and the quantum functionals. Weighted slice rank encapsulates different notions of bipartiteness of quantum entanglement. The correspondence allows us to give a rank-type characterization of the quantum functionals. Moreover, whereas the original definition of the quantum functionals only works over the complex numbers, this new characterization can be extended to all fields. Thereby, in addition to gaining deeper understanding of Strassen's theory for the complex numbers, we obtain a proposal for quantum functionals over other fields. The finite field case is crucial for combinatorial and algorithmic problems where the field can be optimized over.

翻译：张量的结构与计算理解是推动更快速矩阵乘法算法、量子纠缠解析以及cap集问题突破的核心驱动力。斯特拉森渐近谱程序（FOCS 1986）通过单调泛函刻画了最优矩阵乘法算法。本文在张量研究的两个近期进展之间建立联系并推进其发展，即：张量的切片秩——源于cap集问题（Ann. of Math. 2017）解法的张量秩概念，以及张量量子泛函（STOC 2018）——定义为矩多面体上优化的单调泛函。具体而言，我们引入了一种称为加权切片秩的切片秩扩展，并建立了渐近加权切片秩与量子泛函之间的最小最大对应关系。加权切片秩囊括了量子纠缠中二分性的不同概念。该对应关系使我们能够给出量子泛函的秩类型刻画。此外，量子泛函的原始定义仅适用于复数域，而这一新刻画可扩展至所有域。因此，在深化对复数域斯特拉森理论理解的同时，我们获得了其他域上量子泛函的提案。其中有限域情形对于可优化域的组合与算法问题至关重要。