Sufficient separability criteria and linear maps

TL;DR

This paper develops new criteria for determining when quantum states are separable by analyzing specific positive and completely positive maps that transform states into less entangled classes, with explicit examples especially for two-dimensional systems.

Contribution

It introduces families of sufficient separability criteria based on positive maps that transform states into less entangled classes, extending known results near the depolarized state.

Findings

Explicit criteria for arbitrary dimensions, especially M=2

Generalization of separability near the depolarized state

Entanglement classification for mixtures of polarized and pure states

Abstract

We study families of positive and completely positive maps acting on a bipartite system $\mathbb{C}^M\otimes \mathbb{C}^N$ (with $M\leq N$). 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 $\leq k$). 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.