Abstract

Atomic ensembles are effective memory nodes for quantum communication network due to the long coherence time and the collective enhancement effect for the nonlinear interaction between an ensemble and a photon. Here we investigate the possibility of achieving the entanglement distillation for nonlocal atomic ensembles by the input-output process of a single photon as a result of cavity quantum electrodynamics. We give an optimal entanglement concentration protocol (ECP) for two-atomic-ensemble systems in a partially entangled pure state with known parameters and an efficient ECP for the systems in an unknown partially entangled pure state with a nondestructive parity-check detector (PCD). For the systems in a mixed entangled state, we introduce an entanglement purification protocol with PCDs. These entanglement distillation protocols have high fidelity and efficiency with current experimental techniques, and they are useful for quantum communication network with atomic-ensemble memories.

© 2014 Optical Society of America

1. Introduction

Quantum entanglement plays an important role in quantum communication, such as quantum teleportation [1], quantum dense coding (QDC) [2, 3], quantum key distribution [47], quantum secret sharing [8], quantum secure direct communication [9, 10], and so on. Quantum teleportation [1] requires a maximally entangled photon pair to set up the quantum channel for teleporting an unknown single-particle quantum state without moving the particle itself. The high capacity of QDC [2, 3] comes from the maximal entanglement of quantum systems. However, an entangled photon pair is usually produced locally and suffers inevitably from the environment noise (e.g., the thermal fluctuation, vibration, imperfection of an optical fiber, and birefringence effects [11]) in its distribution process between the two parties in quantum communication, which will degrade its entanglement or even make it in a mixed state. The process of quantum state storage will degrade the entanglement as well. This decoherence will decrease the security of quantum communication protocols and the fidelity of quantum teleportation. In a practical long-distance quantum communication network, a quantum repeater [12] is required to depress the decoherence by the destructive effect of the noise [13]. Entanglement distillation [1434] is one of the key techniques in a quantum repeater protocol [12, 13] and it is used to distil highly entangled states from less entangled ones [1416]. It includes entanglement purification and entanglement concentration.

Accurately, entanglement purification is used to obtain a subset of high-fidelity nonlocal entangled quantum systems from a set of those in a mixed state with less entanglement [1423]. In 1996, Bennett et al. [14] proposed the first entanglement purification protocol (EPP) to purify a particular Werner state, resorting to quantum controlled-not (CNOT) gates. In 2001, Pan et al. [16] introduced an EPP with linear optical elements, by keeping the cases that the two photons owned by the same party have the identical polarization. In 2002, Simon and Pan [17] developed an EPP with linear optical elements and a currently available parametric down-conversion (PDC) source. Recently, Ren and Deng presented an EPP for spatial-polarization hyperentangled photon systems [20] and Sheng et al. proposed some deterministic EPPs [2125]. Entanglement concentration is used to get some nonlocal maximally entangled systems from a set of systems in a partially entangled pure state. In 1996, Bennett et al. [26] presented the first entanglement concentration protocol (ECP) for two-particle systems, resorting to the Schmidt projection. In 2001, Zhao et al. [28] and Yamamoto et al. [29] proposed two similar ECPs independently with linear optical elements. In 2008, Sheng et al. [30] developed an ECP with cross-Kerr nonlinearity and its efficiency was improved largely by iteration of the entanglement concentration process. In 2013, Ren, Du, and Deng [32] presented an ECP for hyperentangled photon pairs with the parameter-splitting method based on linear optical elements. In 2014, Ren and Long [33] gave a general scheme for polarization-spatial hyperentanglement concentration of photon pairs. By far, some ECPs for atom systems [3537] have been proposed.

The atomic ensemble system is one of the most promising candidates for quantum communication [38], due to the collective enhancement effect originating from the indistinguishability of the atoms interacting with radiations. Moreover, the time for the storage of a single photon in a cold atom ensemble is about several milliseconds [39] and atom ensembles can, in principle, be used as memory nodes for long-distance quantum communication network. For example, in [40], an atomic ensemble is used as a local memory node [39] for global quantum communication. Inspired by the repeater protocol in [40], a number of works for long-distance quantum communication based on atomic ensembles have been done [38]. In 2007, Chen et al. [41] introduced a fault-tolerant quantum repeater protocol with atomic ensembles and linear optics. In 2010, Zhao et al. [42] proposed a quantum repeater protocol for the atomic ensembles by utilizing the Rydberg blockade effect [43]. In 2011, Aghamalyan et al. [44] proposed a quantum repeater protocol based on the deterministic storage of a single photon in an atomic ensemble with the cavity-assisted interaction.

In the fault-tolerant quantum repeater [41], two atomic ensembles that associate with the photons in different polarizations are used as an effective memory node. The entanglement purification for the nonlocal atomic ensembles can be performed with the EPP for polarizing photons [16] when the quantum memories are involved [39]. In [42], two different single collective spin-wave excitation states of the atomic ensemble are used to encode a stationary qubit. The entanglement purification for the atomic ensembles in two different nodes is completed with the scheme proposed in [14], since the CNOT gate between the two atomic ensembles in each node is available, when they are placed within the blockade radius [43].

In this paper, we investigate the possibility of achieving the entanglement distillation for nonlocal atomic ensembles trapped in single-sided small cavities [4448] by the input-output process of single photons. We give three efficient entanglement distillation protocols (EDPs) for nonlocal two-atomic-ensemble systems, including an optimal ECP for a partially entangled state with known parameters, an ECP for an unknown partially entangled pure state with a nondestructive parity-check detector (PCD) which works efficiently in a simple way, and an EPP for a mixed entangled state. Compared with the EDPs for atom systems by others [3537], our EDPs for nonlocal atomic ensembles have the advantage of high fidelity and efficiency with current experimental techniques. Moreover, it is easier to implement our EDPs than the former and they are useful for quantum communication network with atomic ensembles acting as memories.

2. Nondestructive parity-check detector on two local atomic ensembles

2.1. An atomic-ensemble-cavity system

Let us consider an ensemble composed of N cold atoms trapped in a single-sided optical cavity [4448], in which one of the mirrors is perfectly reflective and another one is of small transmission allowing for incoupling and outcoupling to light, shown in Fig. 1(a). The atom has a four-level internal structure, shown in Fig. 1(b). The two hyperfine ground states of a cold atom are denoted with |g〉 and |s〉. The excited state |e〉 and the Rydberg state |r〉 are two auxiliary states. The atomic transition between |s〉 and |e〉 is resonantly coupled to the polarized cavity mode a, which is nearly resonantly driven by the input photon in the polarization |h〉 with the frequency ω, while the transition between |g〉 and |e〉 is a dipole-forbidden one [49]. Initially, the atomic ensemble is prepared in the ground state |G〉 = |g1,..., gN 〉. An arbitrary unitary operation between the ground state |G〉 and the single collective spin-wave excitation state |S=1Nj|g1,,sj,,gN can be performed efficiently with the Rydberg blockade effect of the state |r〉 by the collective laser manipulations on the ensemble [43, 50]. When the Rydberg blockade shift is Δ = 2π × 100MHz, the transition between |G〉 and |S〉 can be completed with an effective coupling strength Ω = 2π × 1MHz and the probability of nonexcited and doubly excited errors are about 10−3 − 10−4 [45, 51].

 figure: Fig. 1

Fig. 1 (a) Schematic diagram for a single-sided cavity coupled to an atomic ensemble system composed of N cold atoms. (b) Schematic diagram for the level structure of a cold atom.

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In the frame rotating with the cavity frequency ωc, the Hamiltonian of the whole system composed of an input photon and an atom ensemble inside a single-sided cavity can be expressed as ( = 1) [49, 52]

H=j=1N[(Δiγej2)σejej+igj(aσejsja+σsjej)]+iκ2πdδ[b+(δ)ab(δ)a+]+dδb+(δ)b(δ).
Here a and b are the respective annihilation operators of the |h〉 polarized cavity mode and the input photon mode with [a, a+] = 1 and [b(δ′), b+(δ″)] = δ(δ′δ″) (hereafter δ refers to Dirac delta function, δ′ and δ″ represent the detunings). Δ = ω0ωc is the detuning between the cavity mode frequency ωc and the dipole transition frequency ω0. δ′ = ωωc, σejej = |ej〉 〈ej|, and σejsj = |ej〉 〈sj|. The coupling strength κ2π between the cavity and the input photon is taken to be a real constant [49, 52], since only the fields with the carrier frequency close to ωc contribute mostly to the cavity mode. The coefficients γej and gj denote the spontaneous emission rate of the excited state |ej〉 and the coupling strength between the j-th atom and the cavity mode a, respectively. For simplicity, we assume gj = g and γej = γ below.

Initially, when the ensemble is prepared in the state |S〉, the input photon is in the state |h〉, and the cavity mode is vacuum, with the Hamiltonian in Eq. (1), the evolution of the whole system can be confined in the first-order excitation subspace in a general state |Ψ(t)〉. Here

|Ψ(t)=α(t)|S|1,0+dδβ(δ,t)|S|0,1+ζ|E|0,0,
where |E=1Nj|g1,,ej,,gN. |m, n〉 represents the Fock state with the photon numbers m (0 or 1) and n (0 or 1) in the cavity mode and the free space mode, respectively. The Schrödinger equations for this system can be specified as:
iα˙(t)=igζ(t)iκ2πdδβ(δ,t),
iβ˙(δ,t)=iκ2πα(t)+δβ(δ,t),
iζ˙j(t)=(Δiγ2)ζj(t)+igα(t).
In the condition t0 < t < t1 (the times t0 and t1 correspond to the moments that the pulse goes in and comes out of the cavity with the presumable pulse shapes given by βin(t0) and βout(t1), respectively [52]), one can get the standard input-output relation from Eq. (4), i.e.,
βout(t)=βin(t)+κα(t),
where
βin(t)=12πeiδ(tt0)β(δ,t0)dδ,βout(t)=12πeiδ(tt1)β(δ,t1)dδ.
Here βin(t) and βout(t) can be interpreted as the input and the output to the single-sided cavity. β(δ′, t0) and β(δ′, t1) are the probability amplitudes of the input photon with the frequency ω = ωc + δ′ at the times t0 and t1, respectively. Taking Eqs. (3) and (5) into account, one can get the reflection coefficient r(δ′) = β(δ′, t1)/β(δ′, t0) of the cavity, that is,
r(δ)=(δiκ/2)(Δ+iγ/2)g2(δ+iκ/2)(Δ+iγ/2)g2.

If the ensemble is initially in the state |G〉, it will be decoupled to the cavity mode and the input photon in the polarization |h〉 will be reflected by an empty cavity [52]. The reflection coefficient r0(δ′) is

r0(δ)=δiκ/2δ+iκ/2.
Note that when the detuning |δ′| ≪ κ and γκ/4 ≪ g2, one can get a unit reflection with r0(δ′) ≃ −1 and r(δ′) ≃ 1, respectively, which can be summarized to an ideal reflection operator when the input-output process is involved. Here
R^=|hh|(|GG|+|SS|).

2.2. PCD on two local atomic ensembles with the input-output process of a single photon

The principle of our nondestructive PCD on two local atomic ensembles, say EA1 and EA2, is shown in Fig. 2. Suppose the ensembles EA1EA2 are in the state |φEA1EA2 = α1|GG〉 + α2|GS〉 + α3|SG〉 + α4|SS〉. A probe photon a in the polarization state |ϕ=12(|h+|v) sent into the port ain is split into two components |h〉 and |v〉 by the polarizing beam splitter PBS1 and then is led into the two cavities which contain the ensemble EA1 and the ensemble EA2, respectively. The state of the system composed of the photon a and the ensembles EA1EA2 evolves as follows:

|φpEA1EA2=|ϕ|φEA1EA2HPW1PBS1σxA1|φpEA1EA2R^R^A2R^A1σx1A1|φpEA1EA2PBS1HPW2σxA2R^A2R^A1σxA1|φpEA1EA2.
Here σxi=|hv|+|vh| represents the bit-flip operator on the photon sent to ensemble i (i = A1 or A2). i = |hih|(−|G〉 〈G| + |S〉 〈S|)i is the reflection operator on ensemble i and the photon. Subsequently, the photon a is subjected to a Hadamard transformation Hp [ |h1/2(|h+|v) and |v1/2(|h|v)] denoted by H in Fig. 2, and the state of the whole system evolves into
|φpEA1EA2=|hp(α1|GGα4|SS)A1A2+|vp(α2|GSα3|SG)A1A2.
When an |h〉 photon is detected, the two-ensemble system EA1 EA2 is in the even-parity state |φe=1|α1|2+|α4|2(α1|GGα4|SS)A1A2. In contrast, if a |v〉 photon is detected, EA1 EA2 is in the odd-parity state |φo=1|α2|2+|α3|2(α2|GSα3|SG)A1A2. That is, the quantum circuit shown in Fig. 2 can be used to accomplish a PCD on the two atomic ensembles EA1 EA2. Moreover, it is quite simple and only one efficient input-output process of a single photon is involved. It can be performed efficiently with a high fidelity since the cavities show the unit reflectance.

 figure: Fig. 2

Fig. 2 Schematic diagram for a PCD on two ensembles EA1 and EA2. ain is the input port of the photon. HWPi (i = 1, 2) represents a half-wave plate whose optical axis is set to π/4 to perform the bit-flip operation σx = |h〉 〈v| + |v〉 〈h| on the photon. H represents a half-wave plate whose optical axis is set to π/8 and completes the Hadamard transformation. PBS is a polarizing beam splitter, which transmits the |h〉 polarization photon and reflects the |v〉 polarization photon, respectively.

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3. Entanglement concentration for partially entangled atomic ensembles

3.1. Optimal ECP for two nonlocal atomic ensembles in a known partially entangled state

The principle of our optimal ECP for a pair of nonlocal partially entangled two-atomic-ensemble system EAEB is shown in Fig. 3. EA and EB belong to two parties in two nonlocal memory nodes in a quantum communication network, say Alice and Bob, respectively. This ECP is used to distill probabilistically a nonlocal two-ensemble system in a maximally entangled Bell state from that in a partially entangled pure state with known parameters.

 figure: Fig. 3

Fig. 3 Schematic diagram of our optimal ECP for a nonlocal two-atomic-ensemble system in a partially entangled state with known parameters. Alice and Bob are two parties in two nonlocal memory nodes in a quantum communication network. EA and EB are the two nonlocal atomic ensembles which belong to Alice and Bob, respectively. The UBS is an unbalanced beam splitter with the reflection coefficient R = α/β.

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Suppose the system EAEB is initially in the following partially entangled state [27, 31]

|ϕEAEB=α|GSAB+β|SGAB,
where the known real coefficients α and β satisfy |α|2 + |β|2 = 1 and |α| < |β| (the case of |α| > |β| can be treated in the same way). In order to complete this optimal ECP, Bob prepares a single photon b in the state |ϕb=12(|h+|v) and leads it to the port bin, shown in Fig. 3.

After PBS1, the photon b will be reflected by the cavity or the mirror M, and a reflection operator = |h〉 〈h|(−|G〉 〈G| + |S〉 〈S|) is introduced. Subsequently, Bob performs a Hadamard operation on the photon b with the half-wave plate H1. The state of the composite system composed of the photon b and the ensembles EAEB evolves into |ϕABb,

|ϕABb=α|GSAB|hbβ|SGAB|vb.
PBS2, PBS3, and the unbalanced beam splitter (UBS) [32] on the vertical path of the photon b with the reflection coefficient R = α/β will change |ϕABb into
|ϕABb=α(|GSAB|hb|SGAB|vb)β2α2|SGAB|veb.
The system is in the state |SGAB|veb with the probability pe = β2α2, which leads to an error that can be heralded by the click of the detector D′v. If the detector D′v does not click, the photon b and the ensembles EAEB will be projected into the hybrid maximally entangled GHZ state |ϕABb=12(|GSAB|hb|SGAB|vb), which takes place with the probability 2|α|2.

To complete the nonlocal optimal ECP for the entangled ensembles EAEB, Bob performs another Hadamard operation on the photon b with H2 and the state of the system becomes

|ϕABb|ϕABb=12(|hb|ψEAEB+|vb|ψ+EAEB),
where |ψ±EAEB=12(|GS±|SGAB). If the detector Dv clicks, this optimal ECP is completed successfully, and Alice and Bob share a nonlocal two-atomic-ensemble system in a maximally entangled Bell state |ψ+EAEB. It is also completed successfully by the response of the detector Dh, followed by a phase-flip operation σz = |G〉 〈G|−|S〉 〈S| on the ensemble EB. In other words, Bob can judge whether this ECP succeeds or not, according to the response of the detectors. The success probability of our optimal ECP is ηci=1pe=2|α|2 which is just the maximal value in nonlocal entanglement concentration for each nonlocal system.

Our optimal ECP can be generalized to distill maximally entangled N-ensemble GHZ states from a partially entangled GHZ-class pure state. Let us, for example, consider the case that Alice (EA), Bob (EB), ···, and Charlie (EC) share a partially entangled N-ensemble state

|ϕEAEBEC=α|GGGABC+β|SSSABC,
with |α| < |β| and |α|2 + |β|2 = 1. Through a similar process to that for the two-ensemble case, the parties can entangle a polarization photon with the N-ensemble system. A local filtering operation on the |v〉 component of the photon followed by a single-photon detection will project the remaining N-ensemble system EAEB ···EC into the maximally entangled GHZ state |ϕEAEBEC=12(|GGGABC+|SSSABC) with the probability P′mc = 2|α|2.

3.2. ECP for atomic ensembles in a partially entangled state with unknown parameters

The principle of our ECP for nonlocal atomic ensembles in a partially entangled pure state with unknown parameters is shown in Fig. 4. The two atomic ensembles EA1 and EA2 belong to Alice, and the two atomic ensembles EB1 and EB2 belong to Bob. Suppose these two identical pairs of partially entangled ensembles EA1EB1 and EA2EB2 are respectively in the states [26, 2830]

|φEA1EB1=α|GSA1B1+β|SGA1B1,|φEA2EB2=α|GSA2B2+β|SGA2B2,
where the coefficients α and β are unknown and |α|2 + |β|2 = 1. The essential process of the ECP is projecting the state of EB1 EB2 into the odd-parity subspace other than the even-parity subspace after a bit-flip operation on one of the photons in the ECPs [28, 29], and it succeeds in a heralded way when the detector Dv clicks. Alice and Bob can distil a maximally entangled two-ensemble system even in the case that a practical input-output process of the single photon is involved, since the even-parity state of EB1 EB2 will never lead to the click of Dv.

 figure: Fig. 4

Fig. 4 Schematic diagram of our ECP for a nonlocal two-atomic-ensemble system in a partially entangled state with unknown parameters, achieved by the input-output process of a single photon. Bob completes the parity-check measurement on the ensembles EB1 and EB2 with a PCD, assisted by a single photon.

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After Bob performs a parity-check measurement on his two atomic ensembles EB1 and EB2, if the photon detector Dv clicks (an odd-parity outcome is obtained), the state of the four-atomic-ensemble system EA1 EB1 EA2 EB2 is projected into a maximally entangled four-qubit GHZ state |ΦEA1EB1EA2EB2=12(|GSSG|SGGS)A1B1A2B2, which takes place with the probability 2|αβ|2. By taking a Hadamard operation HE [|G12(|G+|S),|S12(|G|S)] on both the ensembles EA2 and EB2, Alice and Bob evolve the four-atomic-ensemble system into

|ΦEA1EB1EA2EB2=122[(|GS|SG)A1B1(|GG|SS)A2B2+(|GS+|SG)A1B1(|GS|SG)A2B2].
They can obtain a nonlocal two-atomic-ensemble system EA1 EB1 in a maximally entangled Bell state by measuring the two ensembles EA2 and EB2 independently with the basis {|G〉,|S〉}. In detail, if Alice and Bob obtain two different outcomes (|GA2|SB2 or |SA2|GB2), the remaining two-atomic-ensemble system EA1 EB1 will be projected into |ψ+EA1EB1. If Alice and Bob obtain the same outcomes |GA2|GB2 or |SA2|SB2, the two-atomic-ensemble system EA1 EB1 is projected into the state |ψEA1EB1 which can be transformed into the state |ψ+EA1EB1 with a phase-flip operation σz = |G〉 〈G| − |S〉 〈S| on the ensemble EA1 or EB1. In other words, they can get EA1EB1 in the state |ψ+EA1EB1 with the success probability ηci=2|αβ|2.

If the outcome of the parity-check measurement on EB1 EB2 is an even-parity one (the Dh detector clicks), the four-atomic-ensemble system is projected into the partially entangled state |ξEA1EB1EA2EB2=1|α|4+|β|4(α2|GSGS+β2|SGSG)A1B1A2B2. Alice and Bob can measure the ensembles EA2 and EB2 with the basis {|G〉,|S〉} after a Hadamard operation HE on both the two ensembles EA2 and EB2, and they will obtain the two-atomic-ensemble systems EA1 EB1 in the state |φEA1EB1=1|α|4+|β|4(α2|GS+β2|SG)A1B1 with or without a phase-flip operation σz on EA1. They can perform the ECP in the second round by replacing α and β with αα2|α|4+|β|4 and ββ2|α|4+|β|4, respectively, when two copies of two-atomic-ensemble systems in the state |φ′EA1EB1 are available. It is not difficult to calculate the success probability of the ECP in the second round ηcii=2|αβ|4(|α|4+|β|4)2. After two rounds of entanglement concentration, the total success probability is ηcti=ηci+(1ηci2)ηcii. Certainly, the method described above can be cascaded to the n-th round of concentration in the ideal case that the photon loss of the input-output process and the decoherence of the ensembles are negligible, and the total success probability of the ECP can be further improved.

4. Entanglement purification for atomic ensemble systems with PCDs

In general, a quantum system in a maximally entangled Bell state degrads into a mixed entangled state in its distribution between two memory nodes and its storage. In this time, the parties should use an EPP to improve the entanglement of their nonlocal quantum systems for creating a high-fidelity quantum channel in a quantum communication network.

Suppose that the two-ensemble systems shared by Alice and Bob are in a mixed entangled state ρEAEB [1619],

ρEAEB=f0|ψ+ψ+|+(1f0)|ϕ+φ+|.
It can be viewed as the mixture of two pure states |ψ+=12(|GS+|SG)AB and |ϕ+=12(|GG+|SS)AB with the probabilities f0 and 1 − f0, respectively. Here, we only discuss the purification of the systems with the bit-flip error |ϕ+〉, as the same as those for photon systems in [1619], because the parties can convert the phase-flip error into the bit-flip error with a Hadamard operation HE on each of the two ensembles EA and EB.

The principle of our EPP for two-atomic-ensemble systems with PCDs is shown in Fig. 5. Alice and Bob choose two pairs of two-ensemble systems ρEA1EB1 and ρEA2EB2 in each time for purification. The four-ensemble system EA1EB1EA2EB2 can be viewed as the mixture of four pure states |Ψ41 = |ψ+EA1EB1 ⊗ |ψ+EA2EB2, |Ψ42 = |ϕ+EA1EB1 ⊗ |ψ+EA2EB2, |Ψ43 = |ψ+EA1EB1 ⊗ |ϕ+EA2EB2, and |Ψ44 = |ϕ+EA1EB1 ⊗ |ϕ+EA2EB2 with the probabilities f02, f0(1 − f0), f0(1 − f0), and (1 − f0)2, respectively. Alice prepares a single photon a in the state |φa=12(|h+|v) and Bob prepares a single photon b in the state |φb=12(|h+|v). They send their photons a and b into the cavities through the ports ain and bin, respectively. By choosing the outcomes with the same parity, they can improve the fidelity of the entanglement of the atomic ensembles shared. Let us detail the principle of our EPP as follows.

 figure: Fig. 5

Fig. 5 Schematic diagram of our EPP for atomic ensembles with PCDs.

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If the four-ensemble system EA1EB1EA2EB2 is in the state |Ψ41, Alice and Bob will obtain the outcomes with the same parity when they measure the parity of their atomic ensembles with our PCDs. If both Alice and Bob obtain an even-parity outcome, the state of the four-ensemble system becomes |Ψ41=12(|GSGS+|SGSG)A1B1A2B2. If both Alice and Bob obtain an odd-parity outcome, the state of the four-ensemble system becomes |Ψ41=12(|GSSG+|SGGS)A1B1A2B2. These two instances take place with the same probability f02/2. After Alice and Bob measure the states of their ensembles EA2 and EB2 with the basis {12(|G±|S)}, the entangled two-ensemble pair EA1EB1 in the state |ψ+EA1EB1 is obtained with or without a single-qubit operation on either of the ensemble EA1 or EB1.

If the four-ensemble system EA1EB1EA2EB2 is in the state |Ψ42 or |Ψ43, Alice and Bob cannot obtain the outcomes with the same parity. That is, when one obtains an even-parity outcome of the PCD on his/her two atomic ensembles, the other obtains an odd-parity outcome. Alice and Bob will obtain the states |ψ+EA1EB1 and |ϕ+EA1EB1 with the same probability f0(1 − f0)/2 after they perform the measurement on their ensembles EA2 and EB2 and operate the ensembles EA1 and EB1 with or without a single-qubit operation.

If the system EA1EB1EA2EB2 is initially in state |Ψ44, Alice and Bob will also obtain the outcomes with the same parity when the PCDs are applied. If the outcomes are all even, the state of the four-ensemble system becomes |Ψ44=12(|GGGG+|SSSS)A1B1A2B2. If the outcomes are all odd, the state of the four-ensemble system becomes |Ψ44=12(|GGSS+|SSGG)A1B1A2B2. These two instances take place with the same probability (1 − f0)2/2. After Alice and Bob measure the states of the ensembles EA2 and EB2 with the basis {12(|G±|S)}, they project EA1EB1 into the state |ϕ+EA1EB1 with or without a single-qubit operation on either of the ensemble EA1 or EB1.

By keeping the instances in which Alice and Bob obtain the outcomes with the same parity, the state of the remained two-atomic-ensemble systems EA1EB1 will be project into ρ′,

ρ=1f02+(1f0)2[f02|ψ+ψ+|+(1f0)2|ϕ+ϕ+|].
The fidelity of the remained two-atomic-ensemble systems Fpi=f02f02+(1f0)2>f0 when f0 > 1/2. By iterating our EPP several rounds, the parties can share a subset of two-atomic-ensemble systems in a nearly maximally entangled state. For instance, if the initial state with the fidelity f0 > 0.7 is used in our EPP, Alice and Bob can obtain a subset of systems in the state with the fidelity Fp > 0.997 for only two rounds.

5. Efficiencies and fidelities of our entanglement distillation protocols

In the previous sections, our three efficient EDPs for two-atomic-ensemble systems are proposed by using the ideal input-output process of a single photon as a result of cavity QED. With the optimal ECP and the efficient ECP, the parties in the quantum communication network can get the target systems in the maximally entangled state. The fidelity of the systems after they are purified with our EPP is just the same as that by the original EPP with the CNOT gates when only the bit-flip is involved [14]. Then, we would like to discuss the performance of our protocols when the practical input-output process of a photon is considered, in which the incident photon of finite bandwidth is used and the practical coupling strength g and the decay rate γ of the single collective excitation state |E〉 are taken into account.

Recently, Colombe et al. [53] demonstrated the strong atom-field coupling in an experiment in which each 87Rb atom in Bose-Einstein condensates is identically and strongly coupled to the cavity mode with a fibre-based cavity [54]. In this experiment, all the atoms are initialized to be the hyperfine Zeeman state |5S1/2, F = 2, mf = 2〉. The dipole transition of 87Rb |5S1/2, F = 2〉 ↦ |5P3/2, F′ = 3〉 is resonantly coupled to the cavity mode with the maximal single-atom coupling strength g0 = 2π × 215MHz. Meanwhile, the cavity photon decay rate is κ = 2π × 53MHz and the atomic spontaneous emission rate of |5P3/2, F′ = 3〉 is γe = 2π × 3MHz. The parameters are available for our protocols, when we initialize all the 87Rb atoms to be the state |g〉 ≡ |5S1/2, F = 1, mf = 1〉 and encode |s〉 ≡ |5S1/2, F = 2, mf = 2〉 and |e〉 ≡ |5p3/2, F′ = 3, mf = 3〉. The transition between the collective states |S〉 and |E〉 is resonantly coupled to the cavity mode, while the transition between |G〉 and |E〉 is dipole-forbidden. When the input-output process of a single photon is involved, we should use the practical reflection operator (δ′), instead of the ideal reflection operator shown in Eq. (10), to describe the reflection process of the |h〉 polarized photon by the cavity,

R^(δ)=|hh|(r0|GG|+r|SS|).
Here r0 and r are the complex reflection coefficients for the incident photon when the atomic ensemble is in the states |G〉 and |S〉, shown in Eqs. (8) and (9), respectively. We discuss the practical fidelity and the efficiency of our three EDPs by considering their processes and the practical reflection operator (δ′) below. In our calcuation, we assume that linear-optical elements we used are ideal and they do not introduce errors, similar to all the existing EPPs and ECPs [1433].

5.1. The practical efficiency and the fidelity of our optimal ECP

For our optimal ECP, only a pair of atomic ensembles in a partially entangled pure state with known coefficients are used and the success of this protocol is heralded by the instance in which either the detector Dh or Dv clicks. Its practical efficiency ηc, shown in Fig. 6 (a), is

ηc=14×(α2|r+1|2+α4β2|r1|2+β2|r0+1|2+α2|r01|2).
Here α and β are the coefficients of the initial state shared by the two parties Alice and Bob. To detail the influence of the practical input-output process on the fidelity of the target state obtained with our optimal ECP, we take the case that the detector Dh clicks as an example and get the amended fidelity Fc, shown in Fig. 7 (a), as
Fc=|α[r(1+αβ)+1αβ]β[r0(1+αβ)+1αβ]|2|α[r(1+αβ)+1αβ]|2+|β[r0(1+αβ)+1αβ]|2.

 figure: Fig. 6

Fig. 6 The efficiencies ηc, ηc′, and ηp of our optimal ECP, efficient ECP, and EPP for two-atomic-ensemble systems, respectively. Here |α|2 + |β|2 = 1.

Download Full Size | PPT Slide | PDF

 figure: Fig. 7

Fig. 7 The fidelities of our EDPs for two-atomic-ensemble systems as the functions of the scaled coupling strength g/κ and the coefficient of the initial state α for our optimal ECP or the initial fidelity f0 for our EPP. Here the scaled detuning δ′/κ = 0.0566. (a) The fidelity of our optimal ECP Fc. (b) The fidelity of our EPP Fp. Here |α|2 + |β|2 = 1.

Download Full Size | PPT Slide | PDF

For the ideal input-output process, r0 = −1 and r = 1, the efficiency ηci and the fidelity Fci of our optimal ECP are

ηci=2|α|2,Fci=1.

5.2. The practical efficiency and the fidelity of our efficient ECP

With the practical reflection operator, the fidelity of the efficient ECP with unknown parameters is still maximal and Fc′ = 1 when the two pairs of the partially entangled pure states are identical, because the parties keep the instances in which a |v〉 polarized photon is detected by Bob, and the detector Dv does not click if both of the two ensembles EB1EB2 are in the same state, e.g., |GG〉 (|SS〉). The efficiency (success probability) of the ECP, shown in Fig. 6 (b), depends on the reflection coefficients and it can be expressed as

ηc=(|α|2|α|4)(|r0r|2)2.

In the ideal case, r0 = −1 and r = 1, the efficiency ηci of our optimal ECP is

ηci=2|αβ|2=2(|α|2|α|4).

5.3. The practical efficiency and the fidelity of our EPP

In our EPP for the nonlocal atomic ensembles EA1 EB1 and EA2 EB2, the coincidental clicks of the detectors Dh (Dv) and D′h (D′v), are used to denote the instances which are kept by Alice and Bob, followed by some local operations and measurements on the two ensembles EA2EB2. That is, the success probability with the practical reflection operator becomes

ηp=f02c00+f0(1f0)c01+f0(1f0)c10+(1f0)2c11,
with the coefficients c00=12|r0r|2+132[|(r+r0)2|2+|(rr0)2|2], c11=14[|r0|4+|r|4]+132[|(r+r0)2|2+|(rr0)2|2], and c01=18[|r0(r+r0)|2+|r(r+r0)|2]=c10 = c10. When the detectors Dh and D′h click, with the practical reflection operator, the fidelity of the target systems EA1EB1 is
Fp=f02c00+f0(1f0)c01f02c00+f0(1f0)c01+f0(1f0)c10+(1f0)2c11.
with the coefficients c00=2|r0r+14(r+r0)2|2, c11=|r02+14(r+r0)2|2+|r2+14(r+r0)2|2, c01=|r+r0|42, and c′10 = |r0(r + r0)|2 + |r(r + r0)|2. The efficiency and the fidelity of our EPP are shown in Fig. 6(c) and Fig. 7(b), respectively.

When r0 = −1 and r = 1, the efficiency ηpi and the fidelity Fpi of our EPP are

ηpi=f02+(1f0)2,Fpi=f02f02+(1f0)2.

5.4. The performance of our EDPs with current experimental parameters

The efficiencies and the fidelities of our optimal ECP, efficient ECP, and EPP with the experimental parameters (κ, γ)/2π = (53, 3.0)MHz [53] are shown in Figs. 6 and 7, respectively, as the functions of the scaled coupling strength g/κ and the coefficient of the initial state α (f0 for EPP), where the scaled detuning δ′/κ = γ/κ = 0.0566 is used. One can see that the performances (both the efficiencies and the fidelities) of our EDPs become better with a larger scaled coupling strength g/κ. Both the fidelities Fc and Fp of the optimal ECP and the EPP increase with the coupling strength g for g/κ < 0.8 and are almost robust to the variation of g for g/κ > 0.8 which leads to g2/κγ > 11.3. For example, when g/κ > 0.4 and α > 0.2, the efficiency ηc′ of our efficient ECP ηc′ > 0.065 which is 84.3% of the ideal efficiency of this ECP ηci=2|αβ|2 when α = 0.2. When g/κ > 0.8 and α > 0.2, ηc/ηci>95.6%. When g/κ > 0.8 and f0 > 0.7, the fidelity and the efficiency of our EPP become Fp > 0.843 and ηp > 0.531 which are 99.7% and 91.6% of those with the ideal input-output process, respectively. Colombe et al. [53] demonstrated the strong atom-field coupling with g/κ > 4, which means our EDPs are feasible with current techniques.

6. Discussion and summary

In our EDPs, the perfect single-sided cavity, consisting of an ideal mirror with 100% reflection and a partially reflective mirror, might remain challenging. One can take the methods developed in [47,48,54] to implement an approximately single-sided cavity. The non-zero transmission of the ideal mirror along with the mirror scattering and absorption will lead to the photon loss of a probability ploss. An additional photon filtering mechanism of the probability ploss on the |v〉 polarized photon is needed to make the input-output process faithful [55], although this slightly decreases the overall success probability. When the photon pulse duration T satisfies ≫ 1, the temporal mode of the output pulse is basically the same as that of the input one leading to a faithful input-output process [49]. In fact, with the maximal detuning δ′max = γ = 0.0566κ, the fidelities and the efficiencies of our three EDPs can achieve the values higher than 90% of those obtained with the ideal PCDs. Certainly, we assume that the local operation on the single photon with linear-optical elements is ideal, similar to all the existing EPPs and ECPs [1434].

Compared with the previous ECPs for atom systems [3537], our optimal ECP and efficient ECP for nonlocal atomic ensembles have some advantages. Our optimal ECP for atomic ensembles in a partially entangled pure state with known parameters has the optimal success probability as the same as that of the ECP for two-photon systems [32] and it is achieved by the detection of a single photon which interacts with a single-sided cavity one time. It can be used to distill the entanglement of each pair of atomic ensembles and does not require additional atomic ensembles, which relaxes the difficulty of its implementation in experiment largely. Our efficient ECP can be used to distill a subset of atomic ensembles in a maximally entangled state from those in a partially entangled pure state with unknown parameters, resorting to a PCD which is constructed in a simple way and involves only one effective input-output process of a single photon, not two or more. Moreover, the success of our efficient ECP is heralded by the individual detection of one |v〉 photon in each node, independent of the scaled coupling strength g/κ, which is far different from the existing cavity-involved ECPs [3537]. By iteration of the concentration process, our efficient ECP has the maximal success probability, compared with other ECPs for the quantum systems in an entangled pure state with unknown parameters.

Our EPP for nonlocal atomic ensembles in a mixed entangled state is more efficient than that in [41], where the entanglement purification for atomic systems is completed with the EPP for two-photon systems in [16] conditioned on the effective quantum memory [39]. In [42], a CNOT gate for the two ensembles in each node is used to perform the EPP and it doubles the efficiency, compared with that in [41], while the two ensembles are required to be placed so close to each other that the CNOT gate can be performed faithfully. This requirement is not needed when the PCDs are used to perform our EPP and the efficiency of our EPP equals to that in [42] in the ideal case.

In actual implementations, errors can always take place. The efficiencies and the fidelities of our three EDPs are influenced by some experimental factors, such as the detector’s efficiency, the decay of the radiation to non-cavity modes, the impurities of the single-photon sources, and so on. For heralded single-photon sources based on the probabilistic correlated photon generation with a PDC source, we should make the average pair production levels much less than one to avoid producing multiple pairs that lead to the impurity in the heralded channel [56]. Besides, with the development of the deterministic single-photon source [56], the impact of the impurities of the single-photon sources can be further reduced. The photon loss due to the cavity mirror scattering and absorption and the nonunit efficiency of the detectors will decrease the efficiency of the input-output process and thus will reduce the efficiency (success probability) of our EDPs. Fortunately, all our EDPs succeed conditioned on the detection of a single photon and the instances with photon loss can be picked out according to the response of the detectors.

In summary, we have investigated the entanglement distillation for two-atomic-ensemble systems in single-sided cavities for the first time and proposed three efficient entanglement distillation protocols, including an optimal ECP for a partially entangled pure state with known parameters, an efficient ECP for an unknown partially entangled pure state with a PCD which is constructed in a simple way, and an EPP for a mixed entangled state. These EDPs have higher fidelity and efficiency with current experimental techniques, compared with the existing EDPs for atomic systems and atomic-ensemble systems, and they are useful for quantum communication network and quantum repeaters.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11174039, 11174040, and 11474026, NECT-11-0031, and the Open Foundation of State Key Laboratory of Networking and Switching Technology (Beijing University of Posts and Telecommunications) under Grant No. SKLNST-2013-1-13.

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References

  • View by:

  1. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
    [Crossref] [PubMed]
  2. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
    [Crossref] [PubMed]
  3. X. S. Liu, G. L. Long, D. M. Tong, and F. Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
    [Crossref]
  4. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
    [Crossref]
  5. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
    [Crossref] [PubMed]
  6. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
    [Crossref] [PubMed]
  7. X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
    [Crossref]
  8. M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
    [Crossref]
  9. G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
    [Crossref]
  10. F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
    [Crossref]
  11. X. H. Li, F. G. Deng, and H. Y. Zhou, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
    [Crossref]
  12. H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
    [Crossref]
  13. W. Dür, H. J. Briegel, J. I. Cirac, and P. Zoller, “Quantum repeaters based on entanglement purification,” Phys. Rev. A 59, 169–181 (1999).
    [Crossref]
  14. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
    [Crossref] [PubMed]
  15. D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
    [Crossref] [PubMed]
  16. J. W. Pan, C. Simon, C. Brukner, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001).
    [Crossref]
  17. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
    [Crossref] [PubMed]
  18. J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature (London) 423, 417–422 (2003).
    [Crossref]
  19. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
    [Crossref]
  20. B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser. Phys. Lett. 10, 115201 (2013).
    [Crossref]
  21. Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
    [Crossref]
  22. Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
    [Crossref]
  23. X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
    [Crossref]
  24. F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
    [Crossref]
  25. Y. B. Sheng and L. Zhou, “Deterministic polarization entanglement purification using time-bin entanglement,” Laser Phys. Lett. 11, 085203 (2014).
    [Crossref]
  26. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996).
    [Crossref] [PubMed]
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  54. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Häensch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New. J. Phys. 12, 065038 (2010).
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2014 (4)

Y. B. Sheng and L. Zhou, “Deterministic polarization entanglement purification using time-bin entanglement,” Laser Phys. Lett. 11, 085203 (2014).
[Crossref]

B. C. Ren and G. L. long, “General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities,” Opt. Express 22, 6547–6561 (2014).
[Crossref] [PubMed]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature (London) 508, 237–240 (2014).
[Crossref]

T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature (London) 508, 241–244 (2014).
[Crossref]

2013 (5)

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

C. Cao, C. Wang, L. Y. He, and R. Zhang, “Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime,” Opt. Express 21, 4093–4105 (2013).
[Crossref] [PubMed]

Y. B. Sheng and L. Zhou, “Quantum Entanglement Concentration Based on Nonlinear Optics for Quantum Communications,” Entropy 15, 1776–1820 (2013).
[Crossref]

B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[Crossref]

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser. Phys. Lett. 10, 115201 (2013).
[Crossref]

2012 (3)

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[Crossref]

Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A 86, 034305 (2012).
[Crossref]

Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012).
[Crossref] [PubMed]

2011 (4)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

D. Aghamalyan and Y. Malakyan, “Quantum repeaters based on deterministic storage of a single photon in distant atomic ensembles,” Phys. Rev. A 84, 042305 (2011).
[Crossref]

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
[Crossref]

2010 (6)

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[Crossref]

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
[Crossref]

B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. A 81, 052329 (2010).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Häensch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New. J. Phys. 12, 065038 (2010).
[Crossref]

2009 (2)

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

F. Mei, M. Feng, Y. F. Yu, and Z. M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

2008 (4)

E. Brion, L. H. Pedersen, M. Saffman, and K. Mølmer, “Error correction in ensemble registers for quantum repeaters and quantum computers,” Phys. Rev. Lett. 100, 110506 (2008).
[Crossref] [PubMed]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[Crossref]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[Crossref]

2007 (3)

X. H. Li, F. G. Deng, and H. Y. Zhou, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Z. B. Chen, B. Zhao, Y. A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[Crossref]

2006 (1)

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[Crossref]

2004 (1)

L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[Crossref] [PubMed]

2003 (2)

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
[Crossref]

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature (London) 423, 417–422 (2003).
[Crossref]

2002 (5)

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
[Crossref] [PubMed]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
[Crossref]

X. S. Liu, G. L. Long, D. M. Tong, and F. Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
[Crossref]

2001 (4)

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London) 414, 413–418 (2001).
[Crossref]

J. W. Pan, C. Simon, C. Brukner, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001).
[Crossref]

Z. Zhao, J.W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
[Crossref]

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
[Crossref]

1999 (3)

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
[Crossref]

W. Dür, H. J. Briegel, J. I. Cirac, and P. Zoller, “Quantum repeaters based on entanglement purification,” Phys. Rev. A 59, 169–181 (1999).
[Crossref]

M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

1998 (1)

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[Crossref]

1996 (3)

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[Crossref] [PubMed]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996).
[Crossref] [PubMed]

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

1992 (2)

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref] [PubMed]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref] [PubMed]

Aghamalyan, D.

D. Aghamalyan and Y. Malakyan, “Quantum repeaters based on deterministic storage of a single photon in distant atomic ensembles,” Phys. Rev. A 84, 042305 (2011).
[Crossref]

Aolita, L.

Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012).
[Crossref] [PubMed]

Bennett, C. H.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref] [PubMed]

Bernstein, H. J.

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996).
[Crossref] [PubMed]

Berthiaume, A.

M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

Bose, S.

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref] [PubMed]

Brennecke, F.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

Briegel, H. J.

W. Dür, H. J. Briegel, J. I. Cirac, and P. Zoller, “Quantum repeaters based on entanglement purification,” Phys. Rev. A 59, 169–181 (1999).
[Crossref]

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[Crossref]

Brion, E.

E. Brion, L. H. Pedersen, M. Saffman, and K. Mølmer, “Error correction in ensemble registers for quantum repeaters and quantum computers,” Phys. Rev. Lett. 100, 110506 (2008).
[Crossref] [PubMed]

Brukner, C.

J. W. Pan, C. Simon, C. Brukner, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001).
[Crossref]

Buzek, V.

M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

Cao, C.

Cao, Z. L.

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[Crossref]

Chang, D. E.

Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012).
[Crossref] [PubMed]

Chen, Y. A.

Z. B. Chen, B. Zhao, Y. A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[Crossref]

Chen, Z. B.

Z. B. Chen, B. Zhao, Y. A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[Crossref]

Cirac, J. I.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London) 414, 413–418 (2001).
[Crossref]

W. Dür, H. J. Briegel, J. I. Cirac, and P. Zoller, “Quantum repeaters based on entanglement purification,” Phys. Rev. A 59, 169–181 (1999).
[Crossref]

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[Crossref]

Colombe, Y.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Häensch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New. J. Phys. 12, 065038 (2010).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

de Leon, N. P.

T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature (London) 508, 241–244 (2014).
[Crossref]

de Riedmatten, H.

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

Deng, F. G.

B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[Crossref]

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser. Phys. Lett. 10, 115201 (2013).
[Crossref]

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[Crossref]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[Crossref]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[Crossref]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
[Crossref]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
[Crossref]

Deutsch, C.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Häensch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New. J. Phys. 12, 065038 (2010).
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Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A 86, 034305 (2012).
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D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
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D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Häensch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New. J. Phys. 12, 065038 (2010).
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A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature (London) 508, 237–240 (2014).
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B. C. Ren and G. L. long, “General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities,” Opt. Express 22, 6547–6561 (2014).
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H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
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A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature (London) 508, 237–240 (2014).
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M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
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D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
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Z. B. Chen, B. Zhao, Y. A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
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C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996).
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Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
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Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
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Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
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C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
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J. W. Pan, C. Simon, C. Brukner, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001).
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C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
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D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Häensch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New. J. Phys. 12, 065038 (2010).
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T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature (London) 508, 241–244 (2014).
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T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature (London) 508, 241–244 (2014).
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A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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X. S. Liu, G. L. Long, D. M. Tong, and F. Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
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Ursin, R.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature (London) 423, 417–422 (2003).
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S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
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T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature (London) 508, 241–244 (2014).
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M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
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M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
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Weihs, G.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature (London) 423, 417–422 (2003).
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Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A 86, 034305 (2012).
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T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
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Figures (7)

Fig. 1
Fig. 1 (a) Schematic diagram for a single-sided cavity coupled to an atomic ensemble system composed of N cold atoms. (b) Schematic diagram for the level structure of a cold atom.
Fig. 2
Fig. 2 Schematic diagram for a PCD on two ensembles EA1 and EA2. ain is the input port of the photon. HWPi (i = 1, 2) represents a half-wave plate whose optical axis is set to π/4 to perform the bit-flip operation σx = |h〉 〈v| + |v〉 〈h| on the photon. H represents a half-wave plate whose optical axis is set to π/8 and completes the Hadamard transformation. PBS is a polarizing beam splitter, which transmits the |h〉 polarization photon and reflects the |v〉 polarization photon, respectively.
Fig. 3
Fig. 3 Schematic diagram of our optimal ECP for a nonlocal two-atomic-ensemble system in a partially entangled state with known parameters. Alice and Bob are two parties in two nonlocal memory nodes in a quantum communication network. EA and EB are the two nonlocal atomic ensembles which belong to Alice and Bob, respectively. The UBS is an unbalanced beam splitter with the reflection coefficient R = α/β.
Fig. 4
Fig. 4 Schematic diagram of our ECP for a nonlocal two-atomic-ensemble system in a partially entangled state with unknown parameters, achieved by the input-output process of a single photon. Bob completes the parity-check measurement on the ensembles EB1 and EB2 with a PCD, assisted by a single photon.
Fig. 5
Fig. 5 Schematic diagram of our EPP for atomic ensembles with PCDs.
Fig. 6
Fig. 6 The efficiencies ηc, ηc′, and ηp of our optimal ECP, efficient ECP, and EPP for two-atomic-ensemble systems, respectively. Here |α|2 + |β|2 = 1.
Fig. 7
Fig. 7 The fidelities of our EDPs for two-atomic-ensemble systems as the functions of the scaled coupling strength g/κ and the coefficient of the initial state α for our optimal ECP or the initial fidelity f0 for our EPP. Here the scaled detuning δ′/κ = 0.0566. (a) The fidelity of our optimal ECP Fc. (b) The fidelity of our EPP Fp. Here |α|2 + |β|2 = 1.

Equations (30)

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H = j = 1 N [ ( Δ i γ e j 2 ) σ e j e j + i g j ( a σ e j s j a + σ s j e j ) ] + i κ 2 π d δ [ b + ( δ ) a b ( δ ) a + ] + d δ b + ( δ ) b ( δ ) .
| Ψ ( t ) = α ( t ) | S | 1 , 0 + d δ β ( δ , t ) | S | 0 , 1 + ζ | E | 0 , 0 ,
i α ˙ ( t ) = i g ζ ( t ) i κ 2 π d δ β ( δ , t ) ,
i β ˙ ( δ , t ) = i κ 2 π α ( t ) + δ β ( δ , t ) ,
i ζ ˙ j ( t ) = ( Δ i γ 2 ) ζ j ( t ) + i g α ( t ) .
β out ( t ) = β in ( t ) + κ α ( t ) ,
β in ( t ) = 1 2 π e i δ ( t t 0 ) β ( δ , t 0 ) d δ , β out ( t ) = 1 2 π e i δ ( t t 1 ) β ( δ , t 1 ) d δ .
r ( δ ) = ( δ i κ / 2 ) ( Δ + i γ / 2 ) g 2 ( δ + i κ / 2 ) ( Δ + i γ / 2 ) g 2 .
r 0 ( δ ) = δ i κ / 2 δ + i κ / 2 .
R ^ = | h h | ( | G G | + | S S | ) .
| φ p E A 1 E A 2 = | ϕ | φ E A 1 E A 2 HPW 1 PBS 1 σ x A 1 | φ p E A 1 E A 2 R ^ R ^ A 2 R ^ A 1 σ x 1 A 1 | φ p E A 1 E A 2 PBS 1 HPW 2 σ x A 2 R ^ A 2 R ^ A 1 σ x A 1 | φ p E A 1 E A 2 .
| φ p E A 1 E A 2 = | h p ( α 1 | G G α 4 | S S ) A 1 A 2 + | v p ( α 2 | G S α 3 | S G ) A 1 A 2 .
| ϕ E A E B = α | G S A B + β | S G A B ,
| ϕ A B b = α | G S A B | h b β | S G A B | v b .
| ϕ A B b = α ( | G S A B | h b | S G A B | v b ) β 2 α 2 | S G A B | v e b .
| ϕ A B b | ϕ A B b = 1 2 ( | h b | ψ E A E B + | v b | ψ + E A E B ) ,
| ϕ E A E B E C = α | G G G A B C + β | S S S A B C ,
| φ E A 1 E B 1 = α | G S A 1 B 1 + β | S G A 1 B 1 , | φ E A 2 E B 2 = α | G S A 2 B 2 + β | S G A 2 B 2 ,
| Φ E A 1 E B 1 E A 2 E B 2 = 1 2 2 [ ( | G S | S G ) A 1 B 1 ( | G G | S S ) A 2 B 2 + ( | G S + | S G ) A 1 B 1 ( | G S | S G ) A 2 B 2 ] .
ρ E A E B = f 0 | ψ + ψ + | + ( 1 f 0 ) | ϕ + φ + | .
ρ = 1 f 0 2 + ( 1 f 0 ) 2 [ f 0 2 | ψ + ψ + | + ( 1 f 0 ) 2 | ϕ + ϕ + | ] .
R ^ ( δ ) = | h h | ( r 0 | G G | + r | S S | ) .
η c = 1 4 × ( α 2 | r + 1 | 2 + α 4 β 2 | r 1 | 2 + β 2 | r 0 + 1 | 2 + α 2 | r 0 1 | 2 ) .
F c = | α [ r ( 1 + α β ) + 1 α β ] β [ r 0 ( 1 + α β ) + 1 α β ] | 2 | α [ r ( 1 + α β ) + 1 α β ] | 2 + | β [ r 0 ( 1 + α β ) + 1 α β ] | 2 .
η c i = 2 | α | 2 , F c i = 1 .
η c = ( | α | 2 | α | 4 ) ( | r 0 r | 2 ) 2 .
η c i = 2 | α β | 2 = 2 ( | α | 2 | α | 4 ) .
η p = f 0 2 c 00 + f 0 ( 1 f 0 ) c 01 + f 0 ( 1 f 0 ) c 10 + ( 1 f 0 ) 2 c 11 ,
F p = f 0 2 c 00 + f 0 ( 1 f 0 ) c 01 f 0 2 c 00 + f 0 ( 1 f 0 ) c 01 + f 0 ( 1 f 0 ) c 10 + ( 1 f 0 ) 2 c 11 .
η p i = f 0 2 + ( 1 f 0 ) 2 , F p i = f 0 2 f 0 2 + ( 1 f 0 ) 2 .

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