Upper and lower quantum functionals, introduced by Christandl, Vrana and Zuiddam (STOC 2018, J. Amer. Math. Soc. 2023), are families of monotone functions of tensors indexed by a weighting on the set of subsets of the tensor legs. Inspired by quantum information theory, they were crafted as obstructions to asymptotic tensor transformations, relevant in algebraic complexity theory. For tensors of order three, and more generally for weightings on singletons for higher-order tensors, the upper and lower quantum functionals coincide and are spectral points in Strassen's asymptotic spectrum. Moreover, the singleton quantum functionals characterize the asymptotic slice rank, whereas general weightings provide upper bounds on asymptotic partition rank. It has been an open question whether the upper and lower quantum functionals also coincide for other cases, or more generally, how to construct further spectral points, especially for higher-order tensors. In this work, we show that upper and lower quantum functionals generally do not coincide, but that they anchor new spectral points. With this we mean that there exist new spectral points, which equal the quantum functionals on the set of tensors on which upper and lower coincide. The set is shown to include embedded three-tensors and W-like states and concerns all laminar weightings, significantly extending the singleton case.

翻译：上、下量子泛函由Christandl、Vrana和Zuiddam引入（STOC 2018, J. Amer. Math. Soc. 2023），它们是张量单调函数族，由张量腿子集集上的权重索引。受量子信息理论启发，这些函数被设计为渐近张量变换的障碍，与代数复杂性理论相关。对于三阶张量，以及更一般地对于高阶张量的单元素权重，上、下量子泛函一致，并且是Strassen渐近谱中的谱点。此外，单元素量子泛函刻画了渐近切片秩，而一般权重则提供了渐近划分秩的上界。一个悬而未决的问题是：在其他情况下，上、下量子泛函是否也一致，或者更一般地，如何构造进一步的谱点，特别是对于高阶张量。在这项工作中，我们表明上、下量子泛函通常不一致，但它们锚定了新的谱点。这意味着存在新的谱点，这些谱点在上下泛函一致的张量集上等于量子泛函。该集合被证明包含嵌入三张量和W类态，并涉及所有层状权重，显著扩展了单元素情形。