Rank-based entanglement detection for bound entangled states

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.

15 pages. Published in PRA. For a better formatted article with less errors, look at the arxiv version

References in corpus (24)

• Entanglement detection
• Separability criterion and inseparable mixed states with positive partial transposition
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Optimization of entanglement witnesses
• Entanglement Measures under Symmetry
• Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
• Entanglement detection via mutually unbiased bases
• Optimal entanglement criterion for mixed quantum states
• Entanglement and permutational symmetry
• Irreversibility in asymptotic manipulations of entanglement
• Quantum states with a positive partial transpose are useful for metrology
• Class of PPT bound entangled states associated to almost any set of pure entangled states
• Entanglement thresholds for random induced states
• Separability of diagonal symmetric states: a quadratic conic optimization problem
• High-Dimensional Entanglement in States with Positive Partial Transposition
• Diagonal unitary and orthogonal symmetries in quantum theory
• Classification of bi-qutrit positive partial transpose entangled edge states by their ranks
• Bound entangled singlet-like states for quantum metrology
• The PPT${}^2$ conjecture holds for all Choi-type maps
• A class of genuinely high-dimensionally entangled states with a positive partial transpose
• Can entanglement hide behind triangle-free graphs?
• Random covariant quantum channels
• Entangled edge states of corank one with positive partial transposes
• Quantum Grid States and Hybrid Graphs