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arxiv: 2605.14644 · v1 · pith:SJGRWSBNnew · submitted 2026-05-14 · 🪐 quant-ph

Generating Non-Decomposable Maps with Differentiable Semidefinite Programming

Angela Rosy Morgillo , Davide Poderini , Fabio Anselmi , Fabio Benatti , Massimiliano F. Sacchi , Chiara Macchiavello This is my paper

Pith reviewed 2026-06-30 20:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords positive mapsnon-decomposable mapssemidefinite programmingChoi matricesbound entanglementdifferentiable optimizationquantum information
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The pith

Differentiable semidefinite programming generates positive non-decomposable maps under chosen constraints on their Choi matrices.

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

The paper introduces an optimization method that uses differentiable semidefinite programming to search for positive maps that cannot be decomposed into simpler forms. These maps matter in entanglement theory because they can witness bound entangled states that standard tests miss. The approach combines SDP certificates for positivity and non-decomposability with gradient descent, allowing searches over maps of varying input and output dimensions while enforcing structural constraints on the Choi matrix. It produces new numerical examples, a parametrized family from masked Choi matrices, and explicit real maps. The same machinery adapts to probe open problems such as the PPT square conjecture and eigenvalue bounds on 2-positive trace-preserving maps.

Core claim

We introduce an optimization framework based on differentiable semidefinite programming (SDP) for generating positive non-decomposable maps under flexible structural constraints on their Choi matrices. The method combines SDP-based certificates of non-decomposability and positivity with gradient-based optimization, enabling a systematic search over maps with different input and output dimensions. Within this framework, we generate previously unknown numerical examples, identify a parametrized family of maps arising from masked Choi matrices, and construct real non-decomposable maps. We further show that the same approach can be adapted to explore open questions in quantum information theory,

What carries the argument

Differentiable semidefinite programming that optimizes Choi matrices subject to SDP certificates of positivity and non-decomposability.

If this is right

  • New numerical examples of positive non-decomposable maps become available in chosen dimensions.
  • A parametrized family of maps is obtained directly from masked Choi matrices.
  • Real (non-numerical) non-decomposable maps can be constructed within the same procedure.
  • The framework extends to test the PPT square conjecture and proposed eigenvalue bounds on 2-positive maps.

Where Pith is reading between the lines

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

  • The method may scale to higher-dimensional maps once the SDP solvers are replaced by faster approximations.
  • The generated families could supply concrete counter-examples or supporting evidence for conjectures on map decomposability.
  • Similar differentiable-SDP pipelines might apply to other positivity constraints, such as k-positivity for k>1.

Load-bearing premise

The SDP certificates correctly certify that the optimized maps remain positive and non-decomposable, and the gradient optimization stays inside the feasible set.

What would settle it

An explicit example in which the output of the optimization procedure yields a map that the SDP certificate labels non-decomposable yet an independent analytic check shows it is decomposable.

Figures

Figures reproduced from arXiv: 2605.14644 by Angela Rosy Morgillo, Chiara Macchiavello, Davide Poderini, Fabio Anselmi, Fabio Benatti, Massimiliano F. Sacchi.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic overview of the optimization framework. A parametrized Choi matrix [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Example of a Choi matrix associated with a positive [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Examples of real Choi matrices corresponding to non-decomposable maps on [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Number of epochs required to reach convergence, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of the loss in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Evolution of the loss in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Evolution of the SDP objectives [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

Positive maps that are not decomposable are a key resource in entanglement theory because they can detect bound entangled states, yet systematic methods for constructing them remain limited. We introduce an optimization framework based on differentiable semidefinite programming (SDP) for generating positive non-decomposable maps under flexible structural constraints on their Choi matrices. The method combines SDP-based certificates of non-decomposability and positivity with gradient-based optimization, enabling a systematic search over maps with different input and output dimensions. Within this framework, we generate previously unknown numerical examples, identify a parametrized family of maps arising from masked Choi matrices, and construct real non-decomposable maps. We further show that the same approach can be adapted to explore open questions in quantum information theory, including the PPT square conjecture and recently proposed eigenvalue bounds for 2-positive trace-preserving maps.

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

2 major / 2 minor

Summary. The paper claims to introduce an optimization framework based on differentiable semidefinite programming (SDP) for generating positive non-decomposable maps under flexible structural constraints on their Choi matrices. It combines SDP-based certificates of non-decomposability and positivity with gradient-based optimization to enable systematic search over maps with varying dimensions, reports previously unknown numerical examples, identifies a parametrized family arising from masked Choi matrices, constructs real non-decomposable maps, and adapts the method to explore the PPT square conjecture and eigenvalue bounds for 2-positive trace-preserving maps.

Significance. If the central claims hold, the framework offers a systematic numerical tool for constructing positive maps that are not decomposable, which are important resources for detecting bound entanglement. The combination of SDP certificates with gradient optimization under flexible constraints on Choi matrices is a methodological strength that could facilitate discovery of new families and exploration of open problems such as the PPT square conjecture.

major comments (2)
  1. [§3] §3 (Optimization framework): The method backpropagates through SDP solution maps to optimize Choi-matrix parameters for positivity and non-decomposability. No verification is provided that the converged optima satisfy conditions required for reliable differentiability (e.g., uniqueness of the optimal solution or strict complementarity). Standard SDP theory shows that the solution map fails to be differentiable when these fail; without analysis or numerical checks confirming the reported examples lie in the differentiable regime, the validity of the generated maps and the systematic search rests on an unverified assumption.
  2. [§4] §4 (Numerical results and examples): The reported examples and parametrized families are obtained via the differentiable SDP procedure, yet the manuscript does not name the SDP solver, report any regularization or uniqueness safeguards, or include post-optimization checks (e.g., eigenvalue gaps or multiplicity) that would confirm gradients were reliable at the reported points. This directly affects the load-bearing claim that new non-decomposable maps have been systematically generated.
minor comments (2)
  1. [Introduction] The distinction between 'numerical' and 'real' non-decomposable maps is mentioned in the abstract but receives insufficient clarification in the introduction or methods regarding how the real maps are obtained from the optimization output.
  2. [§2] Notation for the Choi matrix constraints and the precise form of the non-decomposability certificate could be made more explicit with an additional equation or table summarizing the SDP formulations used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments on the differentiability and numerical verification of our differentiable SDP framework. We address each point below and will revise the manuscript accordingly to strengthen the reliability claims.

read point-by-point responses

  1. Referee: [§3] §3 (Optimization framework): The method backpropagates through SDP solution maps to optimize Choi-matrix parameters for positivity and non-decomposability. No verification is provided that the converged optima satisfy conditions required for reliable differentiability (e.g., uniqueness of the optimal solution or strict complementarity). Standard SDP theory shows that the solution map fails to be differentiable when these fail; without analysis or numerical checks confirming the reported examples lie in the differentiable regime, the validity of the generated maps and the systematic search rests on an unverified assumption.

    Authors: We agree that the manuscript does not explicitly verify the differentiability conditions (uniqueness and strict complementarity) required by SDP theory for the solution map to be differentiable. While the reported examples were obtained via standard backpropagation through an SDP solver and produced consistent results, this constitutes a genuine gap in the presentation. In the revised manuscript we will add a discussion of these theoretical requirements together with post-optimization numerical checks (eigenvalue gaps and multiplicity) confirming that the converged points satisfy the conditions for reliable gradients. These additions will directly support the validity of the generated maps. revision: yes

  2. Referee: [§4] §4 (Numerical results and examples): The reported examples and parametrized families are obtained via the differentiable SDP procedure, yet the manuscript does not name the SDP solver, report any regularization or uniqueness safeguards, or include post-optimization checks (e.g., eigenvalue gaps or multiplicity) that would confirm gradients were reliable at the reported points. This directly affects the load-bearing claim that new non-decomposable maps have been systematically generated.

    Authors: The referee is correct that the manuscript omits the SDP solver name, regularization details, and post-optimization diagnostics. We will revise §4 to name the solver, describe any regularization used to promote uniqueness, and incorporate the eigenvalue-gap and multiplicity checks referenced in our response to the §3 comment. These changes will provide the requested safeguards and reinforce the systematic-generation claim without altering the reported examples or families. revision: yes

Circularity Check

0 steps flagged

No circularity; computational optimization framework uses external SDP solvers

full rationale

The paper presents a method that combines SDP certificates of positivity and non-decomposability with gradient-based optimization over Choi-matrix parameters to generate examples of positive non-decomposable maps. No derivation reduces a claimed result to a fitted parameter or self-citation by construction; the outputs are numerical maps and parametrized families obtained via the external optimization procedure rather than internal redefinitions. The approach is self-contained against external benchmarks (SDP solvers and gradient methods) with no load-bearing self-referential steps in the abstract or described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework depends on the domain assumption that SDP provides reliable certificates for positivity and non-decomposability, with no free parameters or invented entities explicitly introduced in the abstract.

axioms (1)
  • domain assumption SDP-based certificates reliably detect positivity and non-decomposability of maps
    The optimization framework combines these certificates with gradient-based search as stated in the abstract.

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Forward citations

Cited by 1 Pith paper

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  1. Sparse positive maps on qutrits with exact nondecomposability thresholds and PPT-entanglement transitions

    quant-ph 2026-06 unverdicted novelty 7.0

    Exact positivity boundaries, nondecomposability transitions, and PPT-entanglement thresholds are derived for three parametric families of sparse positive maps on qutrits.

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