Quantum Entanglement Concentration Based on Nonlinear Optics for Quantum Communications

Yu-Bo Sheng, Lan Zhou
2013 Entropy  
Entanglement concentration is of most importance in long distance quantum communication and quantum computation. It is to distill maximally entangled states from pure partially entangled states based on the local operation and classical communication. In this review, we will mainly describe two kinds of entanglement concentration protocols. One is to concentrate the partially entangled Bell-state, and the other is to concentrate the partially entangled W state. Some protocols are feasible in
more » ... are feasible in current experimental conditions and suitable for the optical, electric and quantum-dot and optical microcavity systems. Keywords: quantum entanglement; entanglement concentration; quantum communication 1. Introduction Quantum communication and quantum computation have attracted much attention over the last 20 years, due to the absolute safety in the information transmission for quantum communication and the super fast factoring for quantum computation [1,2]. However, in most of the communication protocols, such as quantum teleportation [3], quantum key distribution (QKD) [4-6], quantum secure Entropy 2013, 15 1777 direct communication (QSDC) [7] [8] [9] and quantum secret sharing [10] [11] [12] , the basic requirement is to set up the quantum entanglement channel via entanglement distribution. Unfortunately, the entanglement decreases exponentially with the length of the connecting channel, because of the optical absorption and inevitable channel noise. In this way, quantum repeaters are used [13] [14] [15] [16] . Moreover, in order to obtain a high quality of entanglement for long distance communication, the entanglement distillation is required. Usually, the entanglement distillation mainly includes three different classes [17, 18] . The first one is the entanglement purification that focuses on the general mixed state [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] . Entanglement purification is to distill highly entangled states from mixed states. The second one is the Procrustean method. The third one is the Schmidt decomposition method. These two methods also are called the entanglement concentration [17, 18, [30] [31] [32] [33] [34] [35] [36] [37] . In optical fiber transmission, the dielectric constant acquires a temporal and spatial dependence. Therefore, if the time delay between the photons is small, the effect of the noise is known as the unitary collective noise model. Some entanglement distribution protocols based on collective noise are proposed [38] [39] [40] [41] . Entanglement concentration is to distill maximally entangled states from pure partially entangled states. It has been used as a basis for theoretically-oriented results in quantum information and gives operational meaning to the von Neumann entropy [42] . The Schmidt decomposition method is a powerful way for realizing the entanglement concentration. The first entanglement concentration protocol (ECP) is proposed by Bennett et al. using collective measurement [30]. Using linear optical elements, Zhao et al. and Yamamoto et al. developed the Schmidt decomposition method and proposed two similar ECPs, respectively [33,34]. Two independent experiments were reported for linear optical entanglement concentration [35,36]. Most ECPs are based on linear optics, for the photons can be manipulated and controlled easily. However, in current quantum communication and computation, linear optics has an inherent defect, because it is usually based on the post-selection principle. After the photons are detected by the single photon detectors, the photons are destroyed simultaneously, and they cannot be further used. Using nonlinear optics, such as the cross-Kerr nonlinearity, or other systems, as the auxiliary, can greatly improve such protocols. In this review, we will mainly describe some ECPs based on the nonlinear optical elements. The review is organized as follows: In Section 2, we describe the ECPs with Bell states. In Section 3, we explain the ECPs for W states. In Section 4, we mainly describe some ECPs for NOON states, ECPs for electrons and quantum dot and optical microcavities systems. In Section 5, we will provide a discussion and make a conclusion. ECPs for Bell States Cross-Kerr nonlinearity provides us with a good tool to construct nondestructive quantum nondemolition detectors (QND), which have the potential for conditioning the evolution of our system without necessarily destroying the single photon. The Hamiltonian of a cross-Kerr nonlinear medium can be written by the form [43] [44] [45] : These two qubits are transmitted into the spatial modes, a 1 and a 2 , respectively, and they interact with cross-Kerr nonlinearities. The polarization beam splitter (PBS) transmits the |H⟩ polarization photon and reflects the |V ⟩ polarization photon. The action of the PBS's and cross-Kerr nonlinearities will make the whole state of the two photons evolve to: |Ψ T ⟩ ab = α 1 β 1 |HH⟩ ab |α⟩ + α 1 β 2 |HV ⟩ ab |αe iθ ⟩ + α 2 β 1 |V H⟩ ab |αe −iθ ⟩ + α 2 β 2 |V V ⟩ ab |α⟩ (5) One can observe immediately that the items, |HH⟩ and |V V ⟩, make the coherent state pick up no phase shift and remain as a coherent state with respect to each other. However, the items, |HV ⟩ and |V H⟩, pick up phase shifts, θ and −θ, respectively. The different phase shifts can be distinguished by a
doi:10.3390/e15051776 fatcat:26ojefelrjgtlonngy3jnpszlu