On quantum functionals for higher-order tensors

TL;DR

This paper investigates quantum functionals for higher-order tensors, revealing they generally do not coincide but can define new spectral points, extending understanding of tensor asymptotics.

Contribution

It demonstrates that upper and lower quantum functionals typically differ but can anchor new spectral points for higher-order tensors.

Findings

Upper and lower quantum functionals generally do not coincide.

New spectral points are anchored by quantum functionals on specific tensor sets.

The set includes embedded three-tensors and W-like states, extending the singleton case.

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.