On quantum functionals for higher-order tensors

2026-04-20

AI Summary

This work investigates whether the upper and lower quantum functionals—originating from quantum information theory—coincide in the setting of higher-order tensors, aiming to construct new asymptotic spectrum points. For layered weight structures, it establishes for the first time that these functionals generally differ, yet they do coincide on specific regions where novel asymptotic spectrum points can be anchored. By integrating tools from quantum information theory, tensor rank theory, and asymptotic spectral analysis—and employing a third-order tensor embedding technique—the study proves the existence of new spectrum points encompassing W-class states and embedded third-order tensors. This significantly extends the class of tractable tensors beyond those with singleton-weight structures and provides a crucial theoretical framework for understanding barriers in asymptotic tensor transformations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Game Theory and Economic Paradigms: Imperfect Information
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• quantum functionals
• higher-order tensors
• asymptotic spectrum
• tensor rank
• spectral points

Innovation

• quantum functionals
• spectral points
• higher-order tensors
• asymptotic tensor transformations
• laminar weightings