Overview
This research investigates quantum functionals for higher-order tensors, specifically addressing the long-standing question of whether upper and lower quantum functionals coincide beyond previously established cases. Introduced by Christandl, Vrana, and Zuiddam, these functionals represent families of monotone functions of tensors, indexed by a weighting on the set of subsets of tensor legs. They were conceived as obstructions to asymptotic tensor transformations, a concept relevant to algebraic complexity theory, and are inspired by quantum information theory.
Research Context
Quantum functionals play a role in the asymptotic spectrum of Strassen, particularly in the context of tensor transformations. For order-three tensors, and more broadly for weightings on singletons in higher-order tensors, the upper and lower quantum functionals have been shown to coincide. In these specific instances, they align with spectral points within Strassen's asymptotic spectrum. Furthermore, singleton quantum functionals are known to characterize the asymptotic slice rank. When considering general weightings, these functionals provide upper bounds on the asymptotic partition rank.
A central open question in this domain has concerned the broader coincidence of upper and lower quantum functionals. Researchers have also sought methods to construct additional spectral points, particularly for higher-order tensors, to deepen the understanding of tensor properties and transformations.
Approach
The study examines the behavior of upper and lower quantum functionals in diverse scenarios beyond the singleton weighting case. It explores their relationship and conditions under which they align or diverge, aiming to identify and characterize new spectral points.
Findings
The research demonstrates that upper and lower quantum functionals generally do not coincide. Instead, they are found to anchor new spectral points. This means that there exist novel spectral points that align with the quantum functionals on the specific set of tensors where the upper and lower quantum functionals do coincide.
The set of tensors on which these quantum functionals coincide, and thus equal the new spectral points, is shown to be significantly expanded beyond the singleton case. This extended set includes embedded three-tensors and W-like states. Crucially, this coincidence is observed for all laminar weightings, representing a substantial generalization from the previously known singleton scenario.
Moreover, these newly identified spectral points are shown to provide obstructions to asymptotic restriction. This extends the current understanding beyond obstructions offered by previously known spectral points.
Why This Matters
The findings advance the foundational understanding of quantum functionals and their role within Strassen's asymptotic spectrum. By identifying new spectral points and demonstrating their relationship to quantum functionals, the research provides new tools for analyzing asymptotic tensor transformations. The extension of coincidence conditions to laminar weightings broadens the applicability of these theoretical constructs, offering more comprehensive obstructions to asymptotic restriction.