Rank-based entanglement detection for bound entangled states
Phys. Rev. A 113, 022415 – Published 10 February, 2026
DOI: https://doi.org/10.1103/4r1h-njb6
Abstract
We develop a 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. However, convex conic subsets resulting from the presence of linear constraints allow extremal elements of rank greater than or equal to 2. The problem of deciding when a matrix is rank-1 generated, i.e., a sum of rank-1 positive-semidefinite 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 different classes of positive under partial transposition (PPT) 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.
