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.
Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters
w and
z. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square
0≤w,z≤2/3, complete positivity on the smaller square
0≤w,z≤1/3, and decomposability is lost precisely outside a quarter circle in the corner
w,z≥1/3. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint
W∗=W(2/3,2/3) we construct a four-parameter family of PPT edge states of rank type
(5,5), derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although
W∗ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.
Abstract A
∗-linear map
Φ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations
In⊗Φ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.
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.
Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of biseparable states but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitly construct examples of multipartite non-positive maps, obtained from positive maps via a lifting procedure, that in this fashion can reveal genuine multipartite entanglement in a robust way.