Reduction criterion of separability and limits for a class of protocols of entanglement distillation

TL;DR

This paper introduces a new separability criterion for mixed states in higher-dimensional systems, linking it to entanglement distillation protocols and providing a method to identify distillable states.

Contribution

It develops a separability criterion based on positive maps and relates it to the ability to distill entanglement in higher dimensions.

Findings

States violating the criterion can be distilled using generalized protocols.

All states distillable by the protocol violate the criterion.

Constructs a family of invariant states related to the depolarizing channel.

Abstract

We analyse the problem of distillation of entanglement of mixed states in higher dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223 1 (1996)] we introduce and analyse a criterion of separability which relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol which distills any inseparable two-qubit state. Conversely, all the states which can be distilled by such a protocol must violate the criterion. The proof involves construction of the family of states which are invariant under transformation $\varrho\to U\otimes U^*\varrho U^\dagger\otimes U^{*\dagger}$ where $U$ is a unitary transformation and star denotes complex conjugation. The states are related to the depolarizing channel generalized to non-binary case.