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Journal of Optics B Quantum and Semiclassical Optics

The entanglement purification for entangled multi-particle states

Ye Liu, Chunmei Yao, Guang‐Can Guo

Published Apr 18, 2002• 3 citations
Publisher

Abstract

We present two purification schemes for nonmaximally entangled states. We first show that two parties, Alice and Bob, start with shared less-entangled three-particle states to probabilistically produce a three-particle Greenberger–Horne–Zeilinger state by Bell state measurements and positve operator valued measure (POVM) or a unitary transformation. Then, by a straightforward generalization of the schemes, the purification of a multi-particle entangled state can be realized.

DOI
10.1088/1464-4266/4/3/308

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We present two purification schemes for nonmaximally entangled states. We first show that two parties, Alice and Bob, start with shared less-entangled three-particle states to probabilistically produce a three-particle Greenberger–Horne–Zeilinger state by Bell state measurements and positve operator valued measure (POVM) or a unitary transformation. Then, by a straightforward generalization of the schemes, the purification of a multi-particle entangled state can be realized.
It is known that entanglement swapping can be used to realize entanglement purification. In this way, two particles belonging to different nonmaximally entangled pairs can be projected probabilistically to a maximally entangled state or to a less entangled state. In this paper, we show, when the less entangled state is obtained, then a maximally entangled state can be obtained probabilistically from this less entangled state if a unitary transformation is introduced locally. The probability of success of our scheme is equal to the entanglement of a single pair purification (if two original pairs are in the same nonmaximally entangled states) or to the smaller entanglement of a single pair purification of these two pairs (if two original pairs are not in the same nonmaximally entangled states). The advantage of our scheme is that no continuous indefinite iterative procedure is needed to achieve optimal purification.
We present a new scheme for investigating the usefulness of non-maximally entangled states for multi-party quantum state sharing in a simple and elegant manner. In our scheme, the sender, Alice shares n various probabilistic channels composed of non-maximally entangled states with n agents in a network. Our protocol involves only Bell-basis measurements, single qubit measurements, and a two-qubit unitary transformation operated by free optional agents. Our scheme is a more convenient realization because no other multipartite joint measurements are needed. Furthermore, in our scheme various probabilistic channels lessen the requirement for quantum channels, which makes it more practical for physical implementation.
We introduce a class of multiparticle entanglement purification protocols that allow us to distill a large class of entangled states. These include cluster states, Greenberger-Horne-Zeilinger states, and various error correction codes all of which belong to the class of two-colorable graph states. We analyze these schemes under realistic conditions and observe that they are scalable; i.e., the threshold value for imperfect local operations does not depend on the number of parties for many of these states. When compared to schemes based on bipartite entanglement purification, the protocol is more efficient and the achievable quality of the purified states is larger. As an application we discuss an experimental realization of the protocol in optical lattices which allows one to purify cluster states.
We propose a measure of entanglement between two subsystems of arbitrary (but finite) number of levels. This measure is invariant under unitary transformations of the subsystems and varies between 0 (product states) and 1 (maximum entangled states). The case of two two-level systems is worked out explicitly, where the known complementarity between one- and two-particle interference results in a very natural way from a sum rule. Generalizations to more than two subsystems are discussed. Here we show that for three particles there cannot exist a pure state that is completely characterized by three-particle entanglement alone. For the example of the Greenberger-Horne-Zeilinger (GHZ) state, an experimental setup is proposed, in which its corresponding two-particle entanglement would show up.