Sufficient separability criteria and linear maps

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), 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 states, partial transposes, and/or local transformations thereof.

Figure 1

The range of $\alpha ,\beta$ such that for any state $\rho$ acting on ${\mathbb{C}}^2\otimes {\mathbb{C}}^N, {\mathrm{\Lambda }}_{\alpha ,\beta }\left(\rho \right)$ is separable.