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arxiv: 2604.18283 · v1 · submitted 2026-04-20 · 🧮 math.AG · cs.CC· math.RT· quant-ph

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On quantum functionals for higher-order tensors

Alonso Botero , Matthias Christandl , Thomas C. Fraser , Itai Leigh , Harold Nieuwboer
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Pith reviewed 2026-05-10 03:47 UTC · model grok-4.3

classification 🧮 math.AG cs.CCmath.RTquant-ph
keywords quantum functionalsasymptotic spectrumhigher-order tensorslaminar weightingsStrassen spectrumembedded three-tensorsW-like statespartition rank
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The pith

Upper and lower quantum functionals do not generally coincide but anchor new spectral points for laminar weightings on higher-order tensors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines families of monotone functions called upper and lower quantum functionals, indexed by weightings on subsets of tensor legs, which act as obstructions to asymptotic tensor transformations. It establishes that these upper and lower versions fail to coincide in general for higher-order tensors. They do coincide, however, on the set of embedded three-tensors and W-like states under every laminar weighting. This common value then defines new points in Strassen's asymptotic spectrum, extending the known case where weightings are restricted to singletons. A sympathetic reader cares because the new points supply additional invariants that bound asymptotic partition rank and refine understanding of slice rank in algebraic complexity.

Core claim

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.

What carries the argument

Quantum functionals indexed by laminar weightings on subsets of tensor legs, which serve as monotone obstructions to asymptotic tensor transformations and anchor spectral points precisely where the upper and lower versions agree.

Load-bearing premise

The upper and lower quantum functionals coincide on embedded three-tensors and W-like states for all laminar weightings.

What would settle it

An explicit embedded three-tensor or W-like state together with a laminar weighting for which the numerically computed upper quantum functional differs from the lower quantum functional would disprove the existence of the anchored spectral points.

Figures

Figures reproduced from arXiv: 2604.18283 by Alonso Botero, Harold Nieuwboer, Itai Leigh, Matthias Christandl, Thomas C. Fraser.

Figure 1
Figure 1. Figure 1: The left figure illustrates the laminar family of bipartitions {AB|CDE, C|ABDE, ABC|DE, E|ABCD, DE|ABC}, while the right figure illustrates a fam￾ily {AB|CDE, BCD|AE} that is not laminar. The laminar quantum functionals can provide more information than the singleton-supported quantum functionals. For instance, it is known that the minimum over the singleton-supported quantum functionals is precisely the a… view at source ↗
Figure 2
Figure 2. Figure 2: Venn diagram for the semirings T3 ⊊ Tlq ⊊ T of k-tensors (when k ≥ 4) and the set of Absolutely Maximally Entangled (AME) tensors. That Tlq ⊊ T is witnessed by an explicit tensor Sp (Proposition 3.8). Similarly, W4 ∈ T 4 lq \ T 4 3 by Theorem 4.17. For the definition of the tensor L, see Eq. (3.12). For the inclusion of detk = e1 ∧ · · · ∧ ek ∈ (C k ) ⊗k and AME in Tlq, see Eq. (3.12) and Remark 4.5. For t… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper studies upper and lower quantum functionals for higher-order tensors, indexed by weightings on subsets of the legs. Building on Christandl-Vrana-Zuiddam, it shows that for laminar weightings these functionals generally fail to coincide, yet they agree on the subclass of embedded three-tensors and W-like states; the common value on this subclass therefore defines new points in Strassen's asymptotic spectrum, extending the singleton-weighting case.

Significance. If correct, the construction supplies additional spectral points for the asymptotic spectrum of tensors of order greater than three. These points furnish new monotone, submultiplicative obstructions that are tight on embedded three-tensors and W-states, thereby refining upper bounds on asymptotic partition rank and slice rank beyond what singleton functionals provide. The argument re-uses the monotonicity and submultiplicativity framework of the 2018/2023 papers without introducing new free parameters or circular definitions.

major comments (1)
  1. §4 (main anchoring theorem): the proof that the common value on the anchor set is a spectral point requires verifying that the resulting functional remains submultiplicative on the whole space of tensors, not merely on the anchor set; the current argument sketch does not explicitly address the extension step, which is load-bearing for the claim that a new spectral point is obtained.
minor comments (3)
  1. The abstract states that the functionals 'anchor new spectral points' but does not recall the precise definition of a spectral point (monotonicity, submultiplicativity, normalization) used in the paper; a one-sentence reminder would help readers.
  2. Notation for the weighting function w and the associated quantum functionals Q^w_upper / Q^w_lower is introduced without a consolidated table of symbols; adding such a table after the preliminaries would improve readability.
  3. The bibliography entry for the JAMS 2023 paper is incomplete (missing volume and page numbers); the STOC 2018 citation should likewise be expanded to the full conference proceedings reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. The single major comment identifies a point in the proof of the anchoring theorem that merits clarification, and we will revise the manuscript accordingly to make the argument fully explicit.

read point-by-point responses

  1. Referee: §4 (main anchoring theorem): the proof that the common value on the anchor set is a spectral point requires verifying that the resulting functional remains submultiplicative on the whole space of tensors, not merely on the anchor set; the current argument sketch does not explicitly address the extension step, which is load-bearing for the claim that a new spectral point is obtained.

    Authors: We agree that the extension of submultiplicativity from the anchor set to the full tensor space is essential and that the current sketch in §4 is too brief on this step. The functional is constructed as the common value of the upper and lower quantum functionals on the anchor set (embedded three-tensors and W-like states), and submultiplicativity on the whole space follows from the monotonicity and submultiplicativity properties already established for the quantum functionals in the earlier sections, together with the fact that arbitrary tensors can be related to the anchor set via embeddings and asymptotic transformations that preserve the relevant inequalities. Nevertheless, we acknowledge that this reduction is not written out explicitly. In the revised manuscript we will insert a short dedicated paragraph (or lemma) immediately after the statement of the anchoring theorem that spells out the extension: any tensor T is bounded above and below by anchor-set tensors via the monotonicity of the quantum functionals, and the common value on the anchor set therefore yields a submultiplicative functional on all tensors. This addition will be self-contained and will not alter any definitions or introduce new parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper extends the definitions of upper and lower quantum functionals from the cited prior work (Christandl-Vrana-Zuiddam STOC 2018 / JAMS 2023) to laminar weightings on higher-order tensors. The central results—non-coincidence in general, but coincidence on embedded three-tensors and W-like states for all laminar weightings, thereby anchoring new spectral points—are established via explicit constructions, monotonicity, and submultiplicativity arguments that do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The singleton case is referenced only for context and comparison; the new proofs for the generalized laminar setting are self-contained and independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper extends an existing mathematical framework without introducing new free parameters or invented entities; it relies on standard background from algebraic complexity and quantum information.

axioms (2)
  • standard math Definitions and properties of upper and lower quantum functionals as introduced in Christandl, Vrana and Zuiddam (STOC 2018, J. Amer. Math. Soc. 2023).
    The paper builds directly on these prior definitions for the analysis of coincidence and spectral points.
  • standard math Existence and basic properties of Strassen's asymptotic spectrum.
    Used as the ambient structure in which new spectral points are anchored.

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Reference graph

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