arxiv: 2603.18193 · v2 · submitted 2026-03-18 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

On Non-Existence of Stabilizer Absolutely Maximally Entangled States in Even Local Dimensions

Jakub W\'ojcik , Owidiusz Makuta , Wojciech Bruzda , Remigiusz Augusiak

Authors on Pith no claims yet

Pith reviewed 2026-05-15 08:30 UTC · model grok-4.3

classification 🪐 quant-ph

keywords absolutely maximally entangled statesstabilizer statesgraph statesquditseven local dimensionmultipartite entanglementk-uniform statesmixed quantum states

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The pith

Stabilizer absolutely maximally entangled states cannot exist for N=4n qudits of even local dimension.

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

The paper establishes that no graph state can qualify as an absolutely maximally entangled state when the total number of qudits is a multiple of four and each qudit has even dimension. This follows from an obstruction that appears only under those parity conditions and prevents the necessary uniform entanglement across all partitions. A reader would care because it rules out an entire family of stabilizer constructions for highly entangled multipartite states in dimensions such as d=2,4,6,8, thereby narrowing the search for AME states in composite-dimensional systems. The work also supplies an explicit construction of mixed k-uniform states whose purity is fixed by the best available stabilizer representation, yielding a mixed AME state of purity 1/2 for four quhexes.

Core claim

We demonstrate that absolutely maximally entangled states consisting of N=4n qudits with n in {1,2,3,...}, each of even local dimension, cannot be realized as graph states. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. In particular, this study provides an independent solution of the recently discussed case of the AME state of four quhexes and clarifies its characterization within the stabilizer formalism.

What carries the argument

Local Clifford equivalence of stabilizer states to graph states, together with an invariant that vanishes precisely when both N is a multiple of four and the local dimension is even.

If this is right

Graph-state constructions are ruled out for all stabilizer AME states when N=4n and d is even.
The four-quhex AME state cannot be a stabilizer state under the graph-state representation.
Mixed k-uniform states exist whose purity equals the optimum achievable by any stabilizer representation.
For the concrete case N=4 and d=6 the mixed AME state reaches purity exactly 1/2.
Composite local dimensions impose stricter limits on pure AME states than prime-power dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Non-stabilizer AME states may still exist for these parameters and would require genuinely non-Clifford resources.
The same parity obstruction might appear in other entanglement monotones that count stabilizer rank.
Search algorithms for AME states should now prioritize non-graph stabilizer codes or non-stabilizer pure states when d is even.
The mixed-state construction may generalize to higher uniformity orders k and yield optimal purity bounds for any N and even d.

Load-bearing premise

Every stabilizer absolutely maximally entangled state must be locally Clifford equivalent to some graph state.

What would settle it

Explicit construction of a stabilizer AME state on N=4 qudits of even dimension that cannot be transformed into a graph state by any local Clifford operation.

read the original abstract

We demonstrate that absolutely maximally entangled (AME) states consisting of $N=4n$ qudits with $n\in\{1,2,3,...\}$, each of even local dimension, cannot be realized as graph states. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. In particular, this study provides an independent solution of the recently discussed case of the AME state of four quhexes and clarifies its characterization within the stabilizer formalism, complementing the results found recently in [H. Cha, arXiv:2603.13442]. At the same time, we provide a general construction for mixed $k$-uniform states whose purity is determined by the optimal stabilizer representations. For the specific case of $(N=4,d=6)$, this yields a mixed AME state of optimal purity $1/2$, not subject to canonical graph-state constraints.

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.

Desk Editor's Note private letter to a colleague

The paper rules out graph-state AME for N=4n even d and gives a mixed construction, but extending to all stabilizer states requires confirming LC-equivalence for composite dimensions.

read the letter

The key point is that AME states for N=4n qudits with even local dimension cannot be graph states, with a separate proof for the four quhex case and a general argument. They also construct a mixed AME state for N=4 d=6 with purity 1/2. This is useful work because it limits the graph-state approach for these parameters and provides an explicit mixed alternative that achieves optimal purity under stabilizer representations. The independent take on the quhex case adds to the recent literature. The soft spot is the scope of the claim. The title talks about stabilizer AME states, yet the demonstration is for graph states. The reduction assumes that any stabilizer state is locally Clifford equivalent to a graph state, which holds for prime d but is not obviously true for composite even d. The paper does not seem to provide new justification for that step, so if a non-graph stabilizer AME exists, the non-existence result would not cover it. The mixed state shows awareness of the limits but leaves the pure case open if the equivalence fails. This paper is for people studying multipartite entanglement and stabilizer codes in quantum information. A reader looking for constraints on AME constructions will find it relevant. It deserves serious peer review because the graph-state non-existence is a concrete result and the mixed construction is checkable, though the referee should focus on whether the stabilizer claim is fully supported. I would recommend sending it out for review with attention to that equivalence assumption.

Referee Report

1 major / 1 minor

Summary. The paper claims that absolutely maximally entangled (AME) states on N=4n qudits of even local dimension d cannot be realized as graph states. It solves the open case of four quhexes within the stabilizer formalism, complements recent work on the same example, and supplies a general construction of mixed k-uniform states whose purity is fixed by optimal stabilizer representations; the (N=4,d=6) instance yields a mixed AME state of purity 1/2.

Significance. If the non-existence result for graph states is correct, the work places strong constraints on AME states in composite dimensions and clarifies the reach of graph-state constructions for multipartite entanglement. The explicit mixed-state construction with provably optimal purity is a concrete, reproducible contribution that stands independently of the graph-state claim.

major comments (1)

[Abstract and main theorem] The title asserts non-existence of stabilizer AME states, yet the abstract and main argument establish only that such states cannot be graph states. The reduction from arbitrary stabilizer states to graph states under local Clifford equivalence is invoked without proof for composite even d (e.g., d=6). If this equivalence fails, the stronger title claim does not follow from the graph-state non-existence result.

minor comments (1)

[Section on mixed k-uniform states] The mixed-state construction is presented as optimal, but the manuscript does not explicitly compare its purity bound against all possible stabilizer representations beyond the graph-state subclass.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the precise scope of our non-existence result. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and main theorem] The title asserts non-existence of stabilizer AME states, yet the abstract and main argument establish only that such states cannot be realized as graph states. The reduction from arbitrary stabilizer states to graph states under local Clifford equivalence is invoked without proof for composite even d (e.g., d=6). If this equivalence fails, the stronger title claim does not follow from the graph-state non-existence result.

Authors: We acknowledge the distinction and agree that the core technical result proves non-existence within the graph-state formalism. The reduction of general stabilizer states to graph states via local Clifford operations is standard for prime dimensions but, as the referee notes, requires explicit justification for composite even dimensions such as d=6. In the revised manuscript we will add a short subsection (or appendix) that either supplies a self-contained argument for the equivalence in this setting or, if the equivalence is not universal, explicitly restricts the title and abstract to graph states. This change will ensure the claims match the proofs without weakening the contribution on mixed-state constructions or the (N=4,d=6) example. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external stabilizer formalism

full rationale

The paper shows that AME states for N=4n qudits of even local dimension d cannot be realized as graph states, using standard properties of the stabilizer formalism and local Clifford equivalence. No step reduces a claimed prediction to a fitted parameter or self-defined quantity by construction. The mixed-state construction for (N=4,d=6) is presented separately as an explicit example rather than derived from the main non-existence argument. The cited prior work [H. Cha, arXiv:2603.13442] is external and does not form a self-citation chain that bears the central load. The argument is self-contained against the graph-state restriction without importing uniqueness theorems from the authors' own prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of graph states and the stabilizer formalism for qudits; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Any stabilizer state is locally Clifford-equivalent to a graph state
Invoked to reduce the non-existence claim to graph states

pith-pipeline@v0.9.0 · 5482 in / 1150 out tokens · 56206 ms · 2026-05-15T08:30:58.748129+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1... even local dimension d... N=4n... sum (-1)^j Δ_j =0 leading to parity contradiction (n sum even vs odd determinants)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Fact 1... det(A) not unit in Z_d implies ker(A)≠∅; used for nontrivial stabilizer solution

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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