Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.14644

Generating Non-Decomposable Maps with Differentiable Semidefinite Programming

Angela Rosy Morgillo, Davide Poderini, Fabio Anselmi, Fabio Benatti, Massimiliano F. Sacchi, Chiara Macchiavello

quant-phSubmitted May 14, 2026• Announced May 15, 2026• 0 citations
arXivHTMLPDF

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.

Comments
12 pages, 7 figures

Related papers

Semidefinite Programming for Optimal Quantum Cloning: A Computational Framework

Jörg Hettel

While algebraic derivations establish theoretical limits for quantum cloning, practical implementations require explicit operator representations that are often unavailable analytically. We present a computational framework that reformulates cloning optimization as a search over completely positive trace-preserving maps using the Choi-Jamiolkowski isomorphism and Semidefinite Programming. The framework (i) numerically certifies global optimality through primal-dual strong duality and (ii) automatically extracts operational Kraus operators from the optimal Choi matrix via spectral decomposition. We systematically treat universal, phase-covariant, asymmetric, and entanglement cloning scenarios, providing -for the first time - a unified computational catalogue of explicit, implementable Kraus representations across all major cloning families, including higher-order processes and arbitrary input state distributions. As an application, we analyse optimal cloning attacks on BB84 under depolarizing noise, demonstrating how the extracted operators enable quantitative security analysis in realistic noisy quantum channels. An open-source implementation enables community validation and extension.

quant-pharXiv:2605.21274•May 21, 2026•0 citations

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

Davide Poderini, Angela Rosy Morgillo, Fabio Benatti +3 more

We study a family of sparse positive maps on qutrits for which positivity, decomposability, and PPT entanglement can all be analysed explicitly. The block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form and leads to exact positivity boundaries for three representative parametric families. For the same families we determine the exact transition between decomposable and non-decomposable maps and construct associated PPT states of two classes. The first consists of witness-adapted deformations naturally tied to the non-decomposability analysis. The second consists of analytically tractable families whose full PPT-entangled branch is detected by fixed positive maps, yielding exact thresholds between separability and bound entanglement. For the trace-preserving subclass, we further compare positivity with a recent eigenvalue bound for 2-positive maps, thereby making the gap between positivity and higher-order positivity fully explicit within this family.

quant-pharXiv:2606.19765•Quantum Science and Technology•Jun 19, 2026•0 citations

Optimizing completely positive maps using semidefinite programming

Koenraad M. R. Audenaert, Bart De Moor

Recently, a lot of attention has been devoted to finding physically realizable operations that realize as closely as possible certain desired transformations between quantum states, e.g., quantum cloning, teleportation, quantum gates, etc. Mathematically, this problem boils down to finding a completely positive trace-preserving (CPTP) linear map that maximizes the (mean) fidelity between the map itself and the desired transformation. In this communication, we want to draw attention to the fact that this problem belongs to the class of so-called semidefinite programming (SDP) problems. As SDP problems are convex, it immediately follows that they do not suffer from local optima. Furthermore, this implies that the numerical optimization of the CPTP map can, and should, be done using methods from the well-established SDP field, as these methods exploit convexity and are guaranteed to converge to the real solution. Finally, we show how the duality inherent to convex and SDP problems can be exploited to prove analytically the optimality of a proposed solution. We give an example of how to apply this proof method by proving the optimality of Hardy and Song's optimal qubit $\ensuremath{\theta}$ shifter (e-print quant-ph/0102100).

arXiv:quant-ph/0109155•Physical Review A•Feb 20, 2002•67 citations

Some applications of Choi polynomials of linear maps

Minh Toan Ho, Thanh Hieu Le, Cong Trinh Le +1 more

This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in $M_m(\mathbb{C}) \otimes M_n(\mathbb{C})$. Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.

quant-pharXiv:2604.27034•Linear Algebra and its Applications•Apr 30, 2026•0 citations

k-Positive Maps: New Characterizations and a Generation Method

Frederik vom Ende, Sumeet Khatri, Sergey Denisov

We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.

arXiv:2508.21348•Open Systems & Information Dynamics•Dec 1, 2025•1 citation

Constructing positive maps from block matrices

Yu Jing Guo, Heng Fan

Positive maps are useful for detecting entanglement in quantum information theory. Any entangled state can be detected by some positive map. In this paper, the relation between positive block matrices and completely positive trace-preserving maps is characterized. Consequently, a new method for constructing decomposable maps from positive block matrices is derived. In addition, a method for constructing positive but not completely positive maps from Hermitian block matrices is also obtained.

arXiv:1212.5851•Journal of Physics A Mathematical and Theoretical•Jun 12, 2013•1 citation
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.
While algebraic derivations establish theoretical limits for quantum cloning, practical implementations require explicit operator representations that are often unavailable analytically. We present a computational framework that reformulates cloning optimization as a search over completely positive trace-preserving maps using the Choi-Jamiolkowski isomorphism and Semidefinite Programming. The framework (i) numerically certifies global optimality through primal-dual strong duality and (ii) automatically extracts operational Kraus operators from the optimal Choi matrix via spectral decomposition. We systematically treat universal, phase-covariant, asymmetric, and entanglement cloning scenarios, providing -for the first time - a unified computational catalogue of explicit, implementable Kraus representations across all major cloning families, including higher-order processes and arbitrary input state distributions. As an application, we analyse optimal cloning attacks on BB84 under depolarizing noise, demonstrating how the extracted operators enable quantitative security analysis in realistic noisy quantum channels. An open-source implementation enables community validation and extension.
We study a family of sparse positive maps on qutrits for which positivity, decomposability, and PPT entanglement can all be analysed explicitly. The block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form and leads to exact positivity boundaries for three representative parametric families. For the same families we determine the exact transition between decomposable and non-decomposable maps and construct associated PPT states of two classes. The first consists of witness-adapted deformations naturally tied to the non-decomposability analysis. The second consists of analytically tractable families whose full PPT-entangled branch is detected by fixed positive maps, yielding exact thresholds between separability and bound entanglement. For the trace-preserving subclass, we further compare positivity with a recent eigenvalue bound for 2-positive maps, thereby making the gap between positivity and higher-order positivity fully explicit within this family.
Recently, a lot of attention has been devoted to finding physically realizable operations that realize as closely as possible certain desired transformations between quantum states, e.g., quantum cloning, teleportation, quantum gates, etc. Mathematically, this problem boils down to finding a completely positive trace-preserving (CPTP) linear map that maximizes the (mean) fidelity between the map itself and the desired transformation. In this communication, we want to draw attention to the fact that this problem belongs to the class of so-called semidefinite programming (SDP) problems. As SDP problems are convex, it immediately follows that they do not suffer from local optima. Furthermore, this implies that the numerical optimization of the CPTP map can, and should, be done using methods from the well-established SDP field, as these methods exploit convexity and are guaranteed to converge to the real solution. Finally, we show how the duality inherent to convex and SDP problems can be exploited to prove analytically the optimality of a proposed solution. We give an example of how to apply this proof method by proving the optimality of Hardy and Song's optimal qubit $\ensuremath{\theta}$ shifter (e-print quant-ph/0102100).
This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in Mm(C)⊗Mn(C)M_m(\mathbb{C}) \otimes M_n(\mathbb{C})Mm​(C)⊗Mn​(C). Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.
We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.