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arxiv: 2605.30426 · v1 · pith:CGTB24LNnew · submitted 2026-05-28 · 🪐 quant-ph

Detecting bipartite entanglement with PnCP maps and non-negative polynomials

Ga\"el Mass\'e , Mounir Rezig , Paul Catala , Santiago Scheiner , Laia Serradesanferm C\'ordoba , Enky Oudot , Damian Markham This is my paper

Pith reviewed 2026-06-29 06:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords bipartite entanglementPnCP mapspositive mapsPPT entangled statesnon-sum-of-squares polynomialsentanglement witnessesindecomposable maps
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The pith

Maps from positive non-sum-of-squares polynomials detect bipartite entangled states missed by most other tests.

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

The paper implements an algorithm that turns certain positive polynomials into positive but not completely positive maps on quantum states. It proves that the resulting maps are indecomposable, lie on the boundary of the cone of positive maps, and differ from most previously known examples. Numerical checks show that these maps flag some positive-partial-transpose entangled states that standard criteria overlook. A sympathetic reader would care because PPT states are the hardest form of bipartite entanglement to certify, and better witnesses directly improve the ability to confirm quantum correlations in experiments.

Core claim

The maps generated from the polynomial construction are indecomposable PnCP maps that sit on the boundary of the positive cone and are inequivalent to most known PnCP maps. These maps have sufficient entanglement power to detect PPT entangled states that most other criteria fail to detect.

What carries the argument

PnCP maps obtained by converting positive non-sum-of-squares polynomials into linear maps on density operators, which act as entanglement witnesses when the output fails to be positive.

If this is right

  • The maps are indecomposable.
  • They lie on the boundary of the positive cone.
  • They are inequivalent to most other known PnCP maps.
  • They detect PPT entangled states that most criteria fail to detect.

Where Pith is reading between the lines

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

  • Combining several such maps could raise the fraction of detectable PPT states beyond what single-map criteria achieve.
  • The same polynomial route might produce witnesses for multipartite or continuous-variable systems if the construction generalizes.
  • The numerical robustness of the implementation makes it feasible to scan many random states and measure the fraction of newly detected entanglement.

Load-bearing premise

The algorithm produces actual PnCP maps whose indecomposability, boundary location, and numerical detection performance on PPT states hold as claimed.

What would settle it

An explicit decomposition of any generated map into a sum of a completely positive map and a positive map, or a concrete PPT state on which the map returns a non-negative value despite the numerical tests.

Figures

Figures reproduced from arXiv: 2605.30426 by Damian Markham, Enky Oudot, Ga\"el Mass\'e, Laia Serradesanferm C\'ordoba, Mounir Rezig, Paul Catala, Santiago Scheiner.

Figure 1
Figure 1. Figure 1: Pictorial view of the topology of separable and entangled states. The separable states form a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Choi-Jamiołkowski isomorphism between cones of maps and cones of operators. Positive [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: KSMZ isomorphism between nonnegative biquadratic forms and positive maps. The cone [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic representation of the DPS correspondence between biquadratic forms and Her [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Triple correspondence between Hermitian biquadratic forms, positive maps, and block-positive [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cone of Positive maps (in red) that contain the cone of Decomposable map (in green), [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hierarchy of geometric properties of entanglement witnesses. Here [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Compatibility diagram for the polynomial interpretation of the witness family [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Geometric interpretation of decomposability and PPT detection. Left: in operator space, [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results of the IsSeparable function for the entanglement detection of the 2,000 PPT entan [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The DPS hierarchy converges to the set of separable states. At infinity, it can thus detect [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Affine chart of the Segre variety S1,1 ⊂ P 3 , the image of the Segre embedding P 1×P 1 ,→ P 3 . In the chart X00 ̸= 0, it is described by the equation z = xy. The red and blue lines show the two families of rulings, obtained by fixing one of the two projective coordinates. Definition B.2 (Segre variety). The Segre variety (or Segre embedding) is the projective subvariety Sn−1,m−1 := σn,m P n−1 R × P m−1 … view at source ↗
Figure 13
Figure 13. Figure 13: Geometric interpretation of decomposability and PPT detection. Left: in operator space, [PITH_FULL_IMAGE:figures/full_fig_p067_13.png] view at source ↗
read the original abstract

Positive non-Completely Positive (PnCP) maps are an essential tool to detect entanglement since their characterization is a dual aspect of the separability problem. A recent algorithm proposed by Kelp et al. explains how to generate PnCP maps based on the construction of certain positive non-Sum-of-Squares polynomials. We implement this algorithm in a numerically robust way and propose a working version on GitHub. We theoretically demonstrate that the maps produced by the algorithm are indecomposable, localized on the boundary of the positive cone and show that they are inequivalent with most other known PnCP maps. We numerically investigate their entanglement power, demonstrating notably that they are capable of detecting PPT entangled states that most criteria fail to detect.

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

0 major / 2 minor

Summary. The paper implements the Kelp et al. algorithm for constructing PnCP maps from positive non-sum-of-squares polynomials in a numerically robust manner, releasing the code on GitHub. It theoretically establishes that the generated maps are indecomposable and lie on the boundary of the positive cone, demonstrates their inequivalence to most known PnCP maps, and provides numerical evidence that these maps detect certain PPT entangled states missed by other criteria.

Significance. If the theoretical demonstrations hold, the work supplies new explicit examples of indecomposable PnCP maps with demonstrated utility for PPT entanglement detection, complementing existing criteria. The open-source code release is a clear strength that supports reproducibility and further investigation in quantum information.

minor comments (2)
  1. [Abstract] Abstract: the claim of inequivalence 'with most other known PnCP maps' should be made precise by stating the number of maps compared and the criteria used for inequivalence.
  2. The manuscript should include the exact GitHub repository URL in the main text (not only the abstract) to ensure accessibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of its significance in providing new explicit indecomposable PnCP maps and the open-source code, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core workflow implements the external Kelp et al. algorithm to construct PnCP maps from positive non-Sum-of-Squares polynomials, then supplies independent theoretical proofs of indecomposability, boundary localization, and inequivalence, plus numerical entanglement-power tests on PPT states. No step reduces a claimed prediction or property to a fitted parameter, self-definition, or load-bearing self-citation; the cited algorithm is treated as an external black box whose outputs are analyzed separately. Open GitHub code further decouples the numerical results from any internal fitting loop. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no indication of new free parameters or invented entities; work rests on standard quantum information duality between separability and PnCP maps plus the cited algorithm.

axioms (1)
  • domain assumption Duality between separability problem and characterization of PnCP maps
    Stated in first sentence of abstract as foundational to the approach.

pith-pipeline@v0.9.1-grok · 5672 in / 1240 out tokens · 36830 ms · 2026-06-29T06:38:58.541428+00:00 · methodology

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Reference graph

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