Conditions for entanglement purification with general two-qubit states

TL;DR

This paper analyzes the convergence conditions of a recurrence entanglement purification protocol for arbitrary two-qubit states, revealing new criteria and families of states that can be purified efficiently.

Contribution

It introduces a generalized analysis of entanglement purification conditions using an invariant subspace approach and identifies states that can be purified without high fidelity requirements.

Findings

Purifiable states do not need fidelity > 0.5 with maximally entangled states.

A family of states can be converted to Bell states in one step with probability proportional to concurrence.

The protocol maps all two-qubit states onto a seven-parameter invariant subspace.

Abstract

We present the convergence study of a recurrence entanglement purification protocol using arbitrary two-qubit initial states. The protocol is based on a rank two projector in the Bell basis which serves as a two-qubit operation replacing the usual controlled-NOT gate. We show that the whole space of two-qubit density matrices is mapped onto an invariant subspace characterized by seven real parameters. By analyzing this type of density matrices we are able to find general conditions for entanglement purification in the form of two inequalities between pairs of diagonal elements and pairs of coherences. We show that purifiable initial states do not necessary require a fidelity larger than one half with respect to any maximally entangled pure state. Furthermore, we find a family of states parametrized by their concurrence that can be perfectly converted into a Bell state in just one step of the protocol with probability proportional to the square of the concurrence.