Sufficient separability criteria and linear maps

Abstract

We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^M\otimes {\mathbb{C}}^N$ (with $M\le N$). The maps have a property that when applied to any state (of a given entanglement class) they result in a separable state, or more generally a state of another certain entanglement class (e.g., Schmidt number $\le k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary $M,N$, with a special emphasis on $M=2$. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to "weakly" entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure state projectors, partially transposed and locally transformed pure state projectors.

12 pages, 1 figure; V2: some minor typos corrected; comments welcome

References in corpus (4)

• Optimal entanglement criterion for mixed quantum states
• Structural approximations to positive maps and entanglement breaking channels
• The volume of separable states is super-doubly-exponentially small
• General construction of noiseless networks detecting entanglement with help of linear maps

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• Optimal entanglement witnesses from limited local measurements
• Generating and detecting bound entanglement in two-qutrits using a family of indecomposable positive maps
• Separability criteria based on Heisenberg-Weyl representation of density matrices
• The hierarchies of "witnesses" and the properties and characterization of entangled states as super entanglement witnesses
• Entangled states in the role of witnesses
• Introduction to quantum entanglement in many-body systems
• Internal Boundary between Entanglement and Separability Within a Quantum State
• Sufficient criteria for absolute separability in arbitrary dimensions via linear map inverses
• Linear maps as sufficient criteria for entanglement depth and compatibility in many-body systems
• New Partial Trace Inequalities and Distillability of Werner States
• On Sufficient and Necessary Criteria of Multipartite Quantum Entanglement