Generating Non-Decomposable Maps with Differentiable Semidefinite Programming

TL;DR

This paper presents a differentiable semidefinite programming framework to systematically generate and explore non-decomposable positive maps, advancing entanglement detection and quantum information theory.

Contribution

It introduces a novel optimization approach combining SDP certificates and gradient methods to construct non-decomposable maps with flexible structures.

Findings

Generated previously unknown non-decomposable maps.

Identified a parametrized family of maps from masked Choi matrices.

Adapted the approach to address open questions in quantum information theory.

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.