Witnessing PPT entanglement via rank analysis of (sub)matrices

Abstract: We develop a new method for entanglement detection in bipartite quantum states, using the violation of the rank-1 generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of only rank-1. But convex conic subsets resulting from presence of linear constraints allow extremal elements of rank > 1. The problem of deciding when a matrix is rank-1 generated, i.e a sum of rank-1 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 entangled states. We do this by extracting a matrix from the density matrix, and show that for all separable states, this matrix satisfies the rank-1 generated property. In general, the same is not true for the corresponding matrices of PPT entangled states. We also extend this approach to construct PPT entangled edge states. Finally, we provide different methods that detect the violation of rank-1 generated property.