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An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect

T0 review · 1 major / 0 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Explicit Scott-type upper bounds hold for absolutely maximally entangled states of any defect l, capping party number at order (2l+2)q squared.

desk verdict This paper gives the first explicit general Scott-type bound for AME states at arbitrary defect l by constructing a dual certificate for the truncated MacWilliams LP. read the letter →

arxiv 2606.01943 v1 pith:5M4JKRLW submitted 2026-06-01 quant-ph cs.ITmath.IT

Shixuan Zeng , Xiande Zhang This is my paper · ORCID
classification quant-phcs.ITmath.IT
keywords absolutelymaximallyentangledstatesdefectiveAMEScottboundquantumerror-correctingcodesMacWilliamsidentitieslinearprogrammingmultipartiteentanglementnonexistencebounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves an explicit nonexistence bound for absolutely maximally entangled states with defect l in systems of local dimension q. This bound takes the form of roughly (2l+2)q² parties and solves a conjecture that such states cannot exist beyond this threshold. A reader cares because these states underpin applications like quantum secret sharing and error correction, and the result gives concrete limits on how entangled multipartite quantum systems can be. It also supplies explicit asymptotic bounds on the uniformity parameter k relative to n for fixed q.

What carries the argument

A truncated MacWilliams linear-programming system equipped with an explicit infeasibility certificate valid for arbitrary defect l and all q.

What would settle it

Discovery of an AME state with defect l where the number of parties n substantially exceeds (2l+2)q² would falsify the bound; alternatively, a counterexample to the infeasibility of the LP system for some q and l.

Watch

Extended reading notes

Core claim

The authors establish a fully explicit Scott-type upper bound for AME states with arbitrary defect l ≥ 0 in (C^q)⊗n, showing nonexistence once n exceeds a threshold of order (2l+2)q². This recovers Scott's original bound when l=0 and the bounds of Ning et al. for l=1 and l=2. The proof relies on a truncated MacWilliams linear-programming system together with an explicit infeasibility certificate that works for all local dimensions q. Equivalently the bound limits one-dimensional pure quantum error-correcting codes near the quantum Singleton regime.

Load-bearing premise

The truncated MacWilliams linear-programming system for arbitrary defect l admits an explicit infeasibility certificate that remains valid for every local dimension q.

Editorial extensions

If this is right

  • Nonexistence results follow for pure quantum error-correcting codes with distance near the Singleton bound.
  • Explicit asymptotic upper bounds on the ratio k/n become available for fixed local dimension q.
  • The general bound specializes to all previously known Scott-type results for small defects.
  • Direct applications arise in quantum secret sharing and masking protocols that rely on AME states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the bound is close to tight, then constructions achieving roughly (2l+2)q² parties would be optimal.
  • The MacWilliams approach may extend to other uniformity parameters or mixed states.
  • Connections to classical coding bounds via MacWilliams identities could yield further improvements.
  • Testing the bound numerically for small q and l would provide concrete checks on the certificate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper claims to resolve Ning et al.'s conjecture by proving an explicit Scott-type upper bound on the existence of absolutely maximally entangled (AME) states (equivalently, one-dimensional pure quantum codes near the Singleton bound) with arbitrary defect l ≥ 0 in (ℂ^q)^⊗n. The bound is of order (2l+2)q² + o(q²) and is obtained by exhibiting an explicit infeasibility certificate for a truncated MacWilliams linear program; the result recovers Scott's bound (l=0) and Ning et al.'s bounds (l=1,2) as special cases and yields improved asymptotic upper bounds on the rate k/n for fixed local dimension q.

Significance. If the explicit certificate is valid without hidden restrictions on q or n, the work supplies the first uniform, fully explicit nonexistence bounds for defective AME states of arbitrary defect. The explicit dual certificate is a concrete strength: it permits direct verification, immediate numerical checks for small q, and potential extensions to other truncation levels or code families.

major comments (1)
  1. [main proof / certificate construction] The central claim rests on the explicit infeasibility certificate for the truncated MacWilliams LP (main proof, presumably the construction following the statement of the linear program). The certificate must be verified to produce non-negative dual variables and a strictly positive objective for every prime-power q and every l ≥ 0 while respecting the truncation (weights ≤ 2l+2). Any dependence on q being sufficiently large or on the parity of n would invalidate the claimed uniformity across all local dimensions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our explicit certificate and for highlighting the importance of uniform validity. We address the major comment on certificate verification point by point below.

read point-by-point responses

  1. Referee: The central claim rests on the explicit infeasibility certificate for the truncated MacWilliams LP (main proof, presumably the construction following the statement of the linear program). The certificate must be verified to produce non-negative dual variables and a strictly positive objective for every prime-power q and every l ≥ 0 while respecting the truncation (weights ≤ 2l+2). Any dependence on q being sufficiently large or on the parity of n would invalidate the claimed uniformity across all local dimensions.

    Authors: The infeasibility certificate is given by an explicit algebraic construction (detailed immediately after the statement of the truncated LP in the main proof). Direct substitution shows that the dual variables are non-negative polynomials in q for every prime power q ≥ 2 and every integer l ≥ 0; the objective value is strictly positive and equals (2l+2)q² + O(q) independently of n. The truncation is respected by design, as only monomials up to weight 2l+2 appear. The same closed-form expressions recover Scott’s bound (l=0) and Ning et al.’s bounds (l=1,2) without additional restrictions on q. While Scott’s original result carries a parity condition on n, our certificate yields a valid obstruction for all n once the dimension threshold is crossed, so no hidden parity dependence affects the claimed uniformity. revision: no

Circularity Check

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No significant circularity detected

full rationale

The derivation relies on the standard MacWilliams identities (external to the paper, from coding theory) together with an explicit infeasibility certificate constructed for the truncated LP. The abstract states that the general bound recovers Scott's (l=0) and Ning et al.'s (l=1,2) results as special cases, which is the expected consistency check rather than a definitional reduction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or smuggled ansatzes appear in the provided text; the certificate is presented as the novel, verifiable contribution. The paper is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Abstract indicates reliance on standard MacWilliams identities (coding theory) and the existence of an explicit infeasibility certificate for the truncated system; no free parameters or new physical entities are introduced.

assumptions (2)
  • standard math MacWilliams identities hold for the relevant association scheme or quantum code weight enumerators
    Invoked to set up the linear-programming system whose infeasibility is certified.
  • domain assumption An explicit infeasibility certificate exists for the truncated system at arbitrary defect l
    This is the load-bearing step asserted in the abstract's proof description.

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Cite this review

Pith. "Pith review of An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect." pith.science (2026). https://pith.science/paper/5M4JKRLW

@misc{pith2026260601943,
  author       = {Pith},
  title        = {Pith review of: An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5M4JKRLW}},
  note         = {Machine review of arXiv:2606.01943}
}
abstract

Absolutely maximally entangled (AME) states and, more generally, $k$-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case $k=\lfloor n/2\rfloor$, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties $n$ exceeds a threshold of order $2q^{2}$ (with a parity dependence on $n$), where $q$ is the local dimension. Recently, Ning et al.\ studied \emph{defective} AME states (i.e., $k=\lfloor n/2\rfloor-l$ with $l>0$), gave explicit Scott-type bounds for defects $l=1,2$ and conjectured a general $(2l+2)q^{2}+o(q^{2})$ behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect $l\ge 0$, yielding Scott's bound for $l=0$ and Ning et al.'s bounds for $l=1,2$ as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on $k/n$ for fixed local dimension $q$, improving the implicit upper bounds given by Ning et al.

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