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Quantum Information and Computation

Optimal deterministic entanglement concentration of polarized photons through direct sum extension

Wendong Li, Wen-Zhao Zhang, Ma Lizhen, Yongjian Gu

Published Jul 1, 2011• 1 citation
Publisher

Abstract

A scheme for optimal deterministic entanglement concentration is proposed, and its corresponding optical realization based on a cavity-assisted linear optical system is presented. In this scheme, the quantum circuit devised is simpler than that built in [Y. J. Gu, et al. (2006), Phys. Rev. A. 73, 022321], as it requires the minimum ancillary dimensions and the number of unitary operations. Moreover, we show that, by introducing a path-polarization entanglement state based on the direct sum extension method, three elementary controlled phase-flip gates between two photons are sufficient in the design of its optical realization scheme, making it easy to be implemented from the experimental point of view. Meanwhile, the scheme is verified effective to recover highly entangled pairs from mixed states.

DOI
10.26421/qic11.7-8-4

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A scheme for optimal deterministic entanglement concentration is proposed, and its corresponding optical realization based on a cavity-assisted linear optical system is presented. In this scheme, the quantum circuit devised is simpler than that built in [Y. J. Gu, et al. (2006), Phys. Rev. A. 73, 022321], as it requires the minimum ancillary dimensions and the number of unitary operations. Moreover, we show that, by introducing a path-polarization entanglement state based on the direct sum extension method, three elementary controlled phase-flip gates between two photons are sufficient in the design of its optical realization scheme, making it easy to be implemented from the experimental point of view. Meanwhile, the scheme is verified effective to recover highly entangled pairs from mixed states.
We propose a scheme for optical realization of deterministic entanglement concentration of polarized photons. To overcome the difficulty due to the lack of sufficiently strong interactions between photons, teleportation is employed to transfer the polarization states of two photons onto the path and polarization states of a third photon, which is made possible by the recent experimental realization of the deterministic and complete Bell state measurement. Then the required positive operator-valued measurement and further operations can be implemented deterministically by using a linear optical setup. All these are within the reach of current technology.
We present two realistic entanglement concentration protocols (ECPs) for pure partially entangled photons. A partially entangled photon pair can be concentrated to a maximally entangled pair with only an ancillary single photon with a certain probability, while the conventional ECPs require two copies of partially entangled pairs at least. Our first protocol is implemented with linear optics and the second protocol is implemented with cross-Kerr-nonlinearities. Compared with other ECPs, they do not need to know the accurate coefficients of the initial state. With linear optics, it is feasible with current experiments. With cross-Kerr-nonlinearities, it does not require sophisticated single-photon detectors and can be repeated to get a higher success probability. Moreover, the second protocol can get the higher entanglement transformation efficiency and it may be the most economical protocol by far. Meanwhile, both protocols are more suitable for multiphoton system concentration because they need less operations and classical communications. All these advantages make the two protocols useful in current long-distance quantum communications.
The schematic optical realization of deterministic multicopy (three-copy) entanglement concentration under local operations and classical communication (LOCC) is presented, which can resort to the linear optical elements, controlled-NOT gate that uses weak cross-Kerr nonlinearity effect and entangled qudits with different degrees of freedom. This concentration is based on Nielsen's theorem, which in principle quantifies the unit success probability. In addition, the required positive operator-valued measurement (POVM) is implemented through direct sum extension, because the least number of ancillary dimensions is demanded. The construction of POVM is systematically formulated, which results in the feasibility to generate the protocol to n-copy (n > 3) cases.
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.
We propose a single-photon-assisted entanglement concentration protocol (ECP) for nonlocal N-photon systems in a partially entangled pure W-class state with linear optical elements. Only one of the N parties in quantum communication prepares an ancillary photon and operates the entanglement concentration process for picking up the standard N-photon W state from each partially entangled pure W-class state by choosing the two-mode instances from a polarization beam splitter. Compared with other ECPs for W-class states, our protocol has some advantages. First, it requires only linear optical elements. Second, it requires an N-photon system and an ancillary photon for each round of concentration, not two systems. Third, only one party asks other parties to retain or discard their photons. All these advantages make our ECP more feasible and more convenient than others.