Rank-based entanglement detection for bound entangled states

TL;DR

This paper introduces a novel entanglement detection method based on the violation of rank-1 generated matrices, enabling identification of PPT entangled states and edge states through convex cone analysis.

Contribution

It presents a new rank-based criterion for entanglement detection, extending the analysis to PPT entangled and edge states using convex cone mappings.

Findings

Separable states map to rank-1 generated matrices.

PPT entangled states often do not map to rank-1 generated matrices.

Constructed witnesses can detect violations of the rank-1 generated property.

Abstract

We develop a new method for entanglement detection in bipartite quantum states by using the violation of the rank-1-generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of rank 1. But, convex conic subsets resulting from the presence of linear constraints allow extremal elements of rank >= 2. The problem of deciding when a matrix is rank-1 generated, i.e, a sum of rank-1 positive-semidefinite (PSD) matrices, has been studied extensively in optimization theory. This rank-1 generated property acts as an entanglement criterion, and we use this property to find novel classes of PPT (Positive under partial transposition) entangled states. We do this by mapping some faces of PPT density matrices to convex cones that are not rank-1 generated. We show that all separable states map to a rank-1 generated matrix. In general, the same is not true for the corresponding mapping of PPT entangled states. We also extend this approach to construct PPT entangled edge states. Finally, we provide witnesses that detect violations of the rank-1 generated property.