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Physical Review A

Protocol and quantum circuits for realizing deterministic entanglement concentration

Yongjian Gu, Wendong Li, Guang‐Can Guo

Published Feb 13, 2006• 37 citations
Publisher

Abstract

Entanglement concentration can be achieved in a deterministic fashion [Morikoshi, Phys. Rev. Lett. 84, 3189 (2000); Phys. Rev. A 64, 022316 (2001)]. We present a protocol for deterministic extraction of an Einstein-Podolsky-Rosen pair from two partially entangled pairs via local operations and classical communication. This protocol consists of a generalized measurement described by a positive operator-valued measure (POVM), one-way classical communication, and a corresponding unitary operation or a choice between the two pairs. Based on the theory of majorization, we explicitly construct the required POVM, which can be realized by performing a unitary operation in the extended space and a conventional Von Neuman orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates, and thus provide a physical means to realize the deterministic entanglement concentration. Our method for constructing quantum circuits differs from the usual method based on decomposition of unitary matrices, and is convenient for a large class of quantum processes involving generalized measurements.

DOI
10.1103/physreva.73.022321

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Entanglement concentration can be achieved in a deterministic fashion [Morikoshi, Phys. Rev. Lett. 84, 3189 (2000); Phys. Rev. A 64, 022316 (2001)]. We present a protocol for deterministic extraction of an Einstein-Podolsky-Rosen pair from two partially entangled pairs via local operations and classical communication. This protocol consists of a generalized measurement described by a positive operator-valued measure (POVM), one-way classical communication, and a corresponding unitary operation or a choice between the two pairs. Based on the theory of majorization, we explicitly construct the required POVM, which can be realized by performing a unitary operation in the extended space and a conventional Von Neuman orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates, and thus provide a physical means to realize the deterministic entanglement concentration. Our method for constructing quantum circuits differs from the usual method based on decomposition of unitary matrices, and is convenient for a large class of quantum processes involving generalized measurements.
Deterministic and exact teleportation can be achieved via two partially entangled pairs of particles [Gu Y J 2005 Opt. Comm. ]. The key point of the protocol is a generalized measurement described by a positive operator-valued measure, which can be realized by performing a unitary operation in the extended space and a conventional Von Neumann orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates and thus provide a physical means to realize the teleportation. Our method for constructing quantum circuits differs from the usual methods based on decomposition of unitary matrices and is convenient for a large class of quantum processes involving generalized measurements.
We present an explicit protocol for extraction of an EPR pair from two partially entangled pairs in a deterministic fashion via local operations and classical communication. This protocol is constituted by a local measurement described by a positive operator-valued measure (POVM), one-way classical communication, and a corresponding local unitary operation or a choice between the two pairs. We explicitly construct the required POVM by the analysis of the doubly stochastic matrix connecting the initial and the final states. Our scheme might be useful in future quantum communication.
According to the Nielsens theorem and the basic idea of entanglement concentration, we improve and extend the deterministic entanglement concentration protocol that extracts one pair of maximally entangled particles out of two pairs of non-maximally entangled particles, thus establishing a deterministic entanglement concentration protocol that extracts two pairs of maximally entangled particles out of three pairs of non-maximally entangled particles in the way of the classical communication and local operation and the specific positive operator-valued measure. This is helpful for establishing a deterministic entanglement concentration protocol that extracts k pairs of maximally entangled particles out of n pairs of non-maximally entangled particles.
It deserves mentioning that the quantum circuit, i.e. quantum logic network, is essential to the practical realization of teleportation in experiment. Using only one partially entangled pair, we first propose two novel strategies for probabilistically teleporting any partially entangled state of a bipartite system. The feature of the present protocol is to weaken the requirement for the quantum channel, and also to cut down the number of entangled particles initially shared by the sender and receiver. On the other hand, we explicitly construct the generalized measurement described by the positive operator-valued measure (POVM). Two kinds of efficient quantum circuits for implementing the teleportation are offered. In addition, we generalize the two-particle probabilistic teleportation to the system of many particles.
The determinisitc entanglement concentration (DEC) protocol under local operations and classical communication (LOCC) for three-copy partially entangled states in bipartite systems is presented. The explicit elements of the operators used are calculated due to the general construction of the generalized measurement operator and the corresponding permutation matrices. The measurement is implemented based on the direct sum extension under the motivation to extend the initial state space with minimum number of ancillary dimensions. Morever, the concentration protocol is generalized to the DEC for n-copy multi-partite GHZ-class states. It is also pointed out that the formation of the measurement operator and the algorithms used to calculate its elements can be utilized in more general cases achieving the feasible entanglement transformation that is clarified by Nielsen's theorem, such as the transformation between arbitrary bipartite states in multicopy and high-dimensional cases and so does multi-partite GHZ-class states.