Abstract

Hyperentanglement is a promising resource in quantum information processing, especially for increasing the channel capacity of long-distance quantum communication. Here we present a general hyper-entanglement concentration protocol (hyper-ECP) for nonlocal partially hyperentangled Bell states that decay with the interrelationship between the polarization and the spatial-mode degrees of freedom of two-photon systems, which is not taken into account in other hyper-ECPs, resorting to the optical property of the quantum-dot spins inside one-side optical microcavities. We show that the success probability of our hyper-ECP is largely increased by iteration of the hyper-ECP process. Our hyper-ECP can be straightforwardly generalized to distill nonlocal maximally hyperentangled N-photon Greenberger-Horne-Zeilinger (GHZ) states from arbitrary partially hyperentangled GHZ-class states.

© 2014 Optical Society of America

1. Introduction

Entanglement is an essential resource in quantum information processing and it always acts as the quantum channel in some important long-distance quantum communication applications, such as quantum key distribution [14], quantum teleportation [5], quantum dense coding [6,7], quantum secret sharing [8,9], quantum secure direct communication [1012], and so on. Hyperentanglement, defined as the entanglement in multiple degrees of freedom (DOFs) of quantum systems [1315], is a promising resource in quantum information processing, especially for increasing the channel capacity of quantum communication [1621] and improving the power of quantum computation [22]. Hyperentanglement has been used for assisting Bell-state analysis [2326], quantum repeater [27], and deterministic entanglement purification [2831] on the polarizations of photon pairs. In 2008, Barreiro et al. [16] beat the channel capacity limit of the photonic superdense coding with the polarization-orbital-angular-momentum hyperentanglement in linear optics. Now, there are some interesting works for increasing the channel capacity of long-distance quantum communication with hyperentanglement, such as quantum teleporation with two DOFs of photon pairs [17], entanglement swapping with photonic spatial-polarization hyperentanglement [17,18], hyperentangled Bell-state analysis [1721], and so on.

The entangled photon systems used in quantum communication are always produced locally, and their entanglement will decrease due to the environment noise in the distribution process among the remote parties. Moreover, the photon signals can be transmitted only several hundreds of kilometers either in an optical fiber or in a free space with current technology and quantum repeaters are required to connect the neighboring nodes for long-distance quantum communication. The entanglement of photon systems will decrease in the storage process as well. Therefore, the fidelity and the security of long-distance quantum communication protocols will decrease, and entanglement purification and entanglement concentration are required to depress the noise effect on the entanglement.

Entanglement purification is used to extract nonlocally some high-fidelity entangled systems from a less-entangled ensemble in a mixed entangled state [2840]. So far, some important entanglement purification protocols (EPPs) were proposed for photon systems entangled either in one DOF [2839] or in two DOFs [40], including those assisted by the hyperentanglement [2831,36,37]. Entanglement concentration is used to distill a subset of quantum systems in a maximally entangled pure state from those in a partially entangled pure state [4049]. In 1996, Bennett et al. [41] proposed the first entanglement concentration protocol (ECP) with the Schmidt projection method and collective measurements. Since this pioneering work, many interesting ECPs were proposed for photon systems in a partially entangled pure state with its parameters either known [4146] or unknown [4649] to the remote users. In 2008, Sheng and Deng [49] presented a high-efficiency ECP for photon pairs by iteration of their ECP process. This interesting idea is very useful in the ECPs for other photon systems [43, 44]. In 2013, Ren et al. [46] proposed some hyperentanglement concentration protocols (hyper-ECPs) for two-photon systems in polarization-spatial hyperentangled states with the parameters either known or unknown to the remote users, resorting to linear optical elements. With the parameter-splitting method [46], the hyper-ECP for a partially hyperentangled pure state with known parameters is the optimal one as it has the maximal success probability and it requires linear optical elements only. With the Schmidt projection method [41], the success probability of the hyper-ECP for partially hyperentangled pure states with unknown parameters is relatively low resorting to linear optical elements, and it can be improved largely by using the parity-check quantum nondemolition detectors (QNDs) based on nonlinear optics [40]. In essence, all these hyper-ECPs are focused on the partially hyperentangled pure states with the independent de-coherence of the entanglement in the two DOFs. That is, the optical fibers for the two spatial modes of the photons should be exactly the same ones in the long-distance quantum communication proposals based on hyperentanglement.

In a practical quantum communication, there is difference between the two optical fibers for the transmission of the two photons from the two spatial modes of the photon systems in hyperentangled states and the decay rates of the polarization states in the two spatial modes are usually different, so the decoherence of the entanglement in both the spatial-mode and the polarization DOFs of photon systems are not independent. In this paper, we investigate a general hyper-ECP for two-photon systems in an arbitrary partially hyperentangled unknown Bell state that decays with the interrelationship between the polarization and the spatial-mode DOFs. Our hyper-ECP is achieved by the Schmidt projection method and some parity-check QNDs that are constructed with the optical property of the quantum dots (QDs) embedded in one-side optical microcavities (QD-cavity), which can preserve the photon pairs in the failed hyper-ECP processes. By iteration of the hyper-ECP process, the success probability of our hyper-ECP becomes much higher than that in the hyper-ECP with linear optics. At last, we show that our hyper-ECP is suitable for arbitrary partially hyperentangled multipartite GHZ-class states.

2. Hyperentanglement concentration for arbitrary partially hyperentangled Bell states

In a long-distance quantum communication, the maximally hyperentangled Bell state |ψAB may decay to the partially hyperentangled Bell-type state |ϕ0AB by the independent decoherence of the entanglement in these two DOFs. If the interrelationship between the polarization and the spatial-mode DOFs is taken into account, the maximally hyperentangled Bell state |ψAB will decay to an arbitrary partially hyperentangled Bell state |ϕAB. Here

|ψAB=12(|RR+|LL)AB(|a1b1+|a2b2),|ϕ0AB=(α0|RR+β0|LL)AB(γ0|a1b1+δ0|a2b2),|ϕAB=(α|RR|a1b1+β|LL|a1b1+γ|RR|a2b2+δ|LL|a2b2)AB.
The two subscripts A and B represent the two photons that belong to the two remote users, say Alice and Bob, respectively. R and L represent the left and the right circularly polarized photons, respectively, and i1 and i2 represent the two spatial modes of the photon i (i = a, b). α0, β0, γ0, and δ0 are four parameters unknown to the two remote users, and they satisfy the relations |α0|2 + |β0|2 = 1 and |γ0|2 + |δ0|2 = 1. α, β, γ, and δ are another four parameters unknown to the two remote users, and they satisfy the relation |α|2 + |β|2 + |γ|2 + |δ|2 = 1. In this section, we present a general hyper-ECP for two-photon systems in an arbitrary partially hyperentangled Bell state |ϕAB with the parity-check QNDs for the polarization and the spatial-mode DOFs, resorting to the optical property of QD-cavity systems.

2.1. Parity-check QNDs for the polarization and spatial-mode DOFs of two-photon systems

The parity-check QND for the polarization (spatial-mode) DOF of a photon pair is constructed with a singly charged QD (e.g., a self-assembled In(Ga)As QD or a GaAs interface QD) embedded in the center of a one-side optical resonant microcavity (the bottom distributed Bragg reflectors are 100% reflective and the top distributed Bragg reflectors are partially reflective). If an excess electron is injected into a QD, the negatively charged exciton X (consists of two antiparallel electrons bound to one hole) [50] is created by resonantly absorbing a circularly polarized light, and there are two spin-dependent transitions in the exciton X according to Pauli’s exclusion principle [51], shown in Fig. 1. That is, when the excess electron spin is in the state |↑〉 (|↓〉), the negatively charged exciton in the state |↑↓⇑〉 (|↓↑⇓〉) will be created by resonantly absorbing a left (right) circularly polarized light |L〉 (|R〉). Here |⇑〉 (|⇓〉) represents the heavy-hole spin state |+32 ( |32).

 figure: Fig. 1

Fig. 1 The optical transitions for a negatively charged exciton X with circularly polarized photons. (a) A charged QD embedded in a one-side micropillar microcavity with a circular cross section. (b) The spin-dependent transition rules of a negatively charged exciton Xaccording to Pauli’s exclusion principle. L and R represent the left and the right circularly polarized photons, respectively. ↑↓⇑ (↓↑⇓) represents the negatively charged exciton X in the spin state |+32 ( |32). ↑ and ↓ represent the excess electron spin states |+12 and |12, respectively.

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The input-output process of the circularly polarized light with a QD-cavity system can be represented by Heisenberg equations for the cavity field operator â and X dipole operator σ̂ in the interaction picture [52]. That is,

da^dt=[i(ωcω)+κ2+κs2]a^gσ^κa^in,dσ^dt=[i(ωXω)+γ2]σ^gσ^za^,a^out=a^in+κa^,
where ωc, ω, and ωX are the frequencies of the cavity mode, the input probe photon, and the X transition, respectively. κ/2 and κs/2 are the decay rate and the side leakage rate of the cavity mode, respectively. g is the coupling strength between X and the cavity mode, and γ/2 is the decay rate of X. In the weak excitation condition (X is dominantly in the ground sate and 〈σ̂z〉 = −1), the reflection coefficient for this QD-cavity system can be obtained [53],
r(ω)=a^ina^=1κ[i(ωXω)+γ2][i(ωXω)+γ2][i(ωcω)+κ2+κs2]+g2.
If the cavity side leakage rate is negligible, the reflection coefficient is |r0(ω)| ≃ 1 for the case g = 0. In the strong coupling regime (gκ, γ), the reflection coefficient is |r(ω)| ≃ 1.

The |L〉 and |R〉 photons will get different phase shifts when they are reflected by a QD-cavity system. With the excess electron spin in the state |↑〉, the |L〉 photon gets a phase shift φ and the |R〉 photon gets a phase shift φ0. Otherwise, the |L〉 photon gets a phase shift φ0 and the |R〉 photon gets a phase shift φ. Here φ0 = arg[r0(ω)] and φ = arg[r(ω)], and θF=(φ0φ)/2=θF is the Faraday rotation angle. If the frequencies are adjusted to be ωωcκ/2 (ωc = ωX), the phase shifts of the |L〉 and |R〉 photons are φ0=π2 and φ = 0. If the photon interacts with the QD-cavity system twice (shown in Fig. 2), the output states of the |L〉 and |R〉 photons can be obtained as

|L,|L,,|L,|L,,|R,|R,,|R,|R,.

Our parity-check QNDs for the polarization and the spatial-mode DOFs of a two-photon system are constructed with the optical property of QD-cavity systems, shown in Fig. 2. The polarization parity-check QND is composed of QD1, CPBSs, X1, X2, Z1, X3, X4, and Z2. The spatial-mode parity-check QND is composed of QD2, Z3, and Z4.

 figure: Fig. 2

Fig. 2 Schematic diagram of the general hyper-ECP for two-photon systems in an arbitrary partially hyperentangled Bell state. The optical elements in the blue dashed box are used to perform the polarization parity-check QND on a two-photon system, and the optical elements in the pink dashed box are used to perform the spatial-mode parity-check QND on a two-photon system. The small mirror is used to reflect the photon, which makes the photon interact with the cavity twice. Zi (i = 1, 2, 3, 4) represents a half-wave plate which is used to perform a polarization phase-flip operation σzp=|RR||LL|. Xj (j = 1, 2, 3, 4) represents a half-wave plate which is used to perform a polarization bit-flip operation σxp=|RL|+|LR|. R45 represents a half-wave plate which is used to perform the polarization Hadamard operation. CPBS represents a polarizing beam splitter in the circular basis, which transmits the photon in the right-circular polarization |R〉 and reflects the photon in the left-circular polarization |L〉, respectively. BS represents a 50:50 beam splitter which is used to perform the spatial-mode Hadamard operation. DL represents a time-delay device which makes the two wave packets reach the last CPBS in each Mach-Zehnder interferometer simultaneously. Dk (k = L1, R1, R2, L2) represents a single-photon detector. D represents the same operations as the ones performed by Alice in the green dotted box.

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  • (1) Polarization parity-check QND (P-QND). Suppose that the initial state of QD1 is |+1(|±1=(|1±|1)/2) and the photon A is in the state |χA = (α0|R〉 + β0|L〉)A(γ0|a1〉 +δ0|a2〉). Alice makes the photon in the |R〉 polarization from the spatial modes |a1〉 and |a2〉 interact with QD1 by performing X1 and CPBSs on these two spatial modes. After the photon A coming from both the spatial modes |a1〉 and |a2〉 passes through Z1 and X2, respectively, the state of the system composed of QD1 and the photon A is transformed into
    |+1|χA[|1(α0|R+β0|L)A+|1(α0|R+β0|L)A](γ0|a1+δ0|a2).
    If Alice has a photon pair AC in the hyperentangled Bell state |ϕ±AC = (α0|RR〉 ± β0|LL〉)AC ⊗ (γ0|a1c1〉 + δ0|a2c2〉) (|ϕ1±AC=(α0|RL±β0|LR)AC(γ0|a1c1+δ0|a2c2)), after they interact with QD1, CPBSs, X1, X2, Z1, X3, X4, and Z2 in sequence as shown in Fig. 2, the state of the system composed of QD1 and the photon pair AC is transformed into
    |+1|ϕ±AC|+1|ϕ±AC,|+1|ϕ1±AC|1|ϕ1±AC.
    By measuring the excess electron spin in the orthogonal basis {|+〉1, |−〉1} [18], one can distinguish the even-parity polarization mode (|ϕ±AC) of the photon pair AC from the odd-parity polarization mode ( |ϕ1±AC) without disturbing its state in the spatial-mode DOF. That is, if the excess electron spin is in the state |+〉1, the polarization DOF of the hyperentangled state is in the even-parity mode. Otherwise, the polarization DOF of the hyperentangled state is in the odd-parity mode. If the spatial-mode DOF of the two-photon system AC is in any other state, the outcome of our polarization parity-check QND is exactly the same as that in Eq. (6).
  • (2) Spatial-mode parity-check QND (S-QND). Suppose that the initial state of QD2 is |+〉2 and the photon B is in the state |χB = (α0|R〉 + β0|L〉)B(γ0|b1〉 +δ0|b2〉). Bob lets the photon from the spatial mode |b2〉 interact with QD2 and then pass through Z3. The state of the system composed of QD2 and the photon B is transformed into
    |+2|χB[|2(γ0|b1+δ0|b2)+|2(γ0|b1δ0|b2)](α0|R+β0|L)B.
    If Bob has two photons BD in the hyperentangled Bell state |ϕ±BD = (α0|RR〉 + β0|LL〉)BD ⊗ (γ0|b1d1〉 ± δ0|b2d2〉) (|ϕ1±BD=(α0|RR+β0|LL)BD(γ0|b1d2±δ0|b2d1)), after they interact with QD2, and pass though Z3 and Z4 in sequence as shown in Fig. 2, the state of the system composed of QD2 and the photon pair BD is transformed into
    |+2|ϕ±BD|+2|ϕ±BD,|+2|ϕ1±BD|2|ϕ1±BD.
    By measuring the excess electron spin in the orthogonal basis {|+〉2, |−〉2} [18], one can distinguish the even-parity spatial mode (|ϕ±BD) from the odd-parity spatial mode ( |ϕ1±BD) without disturbing the state of the polarization DOF of the photon pair BD. That is, if the excess electron spin is in the state |+〉2, the spatial-mode DOF of the hyperentangled state is in the even-parity mode. Otherwise, the spatial-mode DOF of the hyperentangled state is in the odd-parity mode. If the polarization DOF of the two-photon system BD is in any other state, the outcome of our spatial-mode parity-check QND is exactly the same as that in Eq. (8).

2.2. Hyperentanglement concentration for arbitrary partially hyperentangled Bell states

In this section, we investigate the general hyperentanglement concentration for two-photon systems in an arbitrary partially hyperentangled Bell state |ϕAB that decays with the interrelationship between the polarization and the spatial-mode DOFs.

I. The first round of the hyperentanglement concentration process

As there are three independent unknown parameters in the arbitrary partially hyperentangled Bell state, four copies of the two-photon systems in the state |ϕAB are required to implement the hyper-ECP with the Schmidt projection method. That is,

|ϕAB=(α|RR|a1b1+β|LL|a1b1+γ|RR|a2b2+δ|LL|a2b2)AB,|ϕCD=(α|RR|c1d1+β|LL|c1d1+γ|RR|c2d2+δ|LL|c2d2)CD,|ϕAB=(α|RR|a1b1+β|LL|a1b1+γ|RR|a2b2+δ|LL|a2b2)AB,|ϕCD=(α|RR|c1d1+β|LL|c1d1+γ|RR|c2d2+δ|LL|c2d2)CD.
Here, the subscripts AB, CD, A′B′, and C′D′ represent the four copies of the two-photon pairs. The four photons A, C, A′, and C′ belong to Alice, and the four photons B, D, B′, and D′ belong to Bob. Alice and Bob divide their four photons into two groups. That is, Alice has two photons A and C in one group, and the other two photons A′ and C′ in the other group. Bob has two photons B and D in one group, and the other two photons B′ and D′ in the other group.

  • (1) The first step. The principle of our hyper-ECP for the photon pairs AB and CD is shown in Fig. 2, and the same operations are performed on the photon pairs A′B′ and C′D′. The initial state of the four-photon system ABCD is described as |Φ〉ABCD = |ϕAB ⊗ |ϕCD. Alice and Bob perform the P-QND and S-QND on the photon pairs AC and BD, respectively. The outcomes can be divided into four groups, and they are discussed in detail as follows.
    • (1.1) If the outcomes of the P-QND and S-QND are in an even-parity polarization mode and an odd-parity spatial mode, respectively, the four-photon system is projected into the state |Φ1ABCD1 with the probability of p(1)1 = 2(|αγ|2 + |βδ|2). Here
      |Φ1ABCD1=1p(1)1(αγ|RRRR+βδ|LLLL)ABCD(|a1b1c2d2+|a2b2c1d1).
      Alice and Bob perform Hadamard operations on the polarization and the spatial-mode DOFs of the photons C and D, respectively, and then the state |Φ1ABCD1 is transformed into the state |Φ1ABCD1. Here
      |Φ1ABCD1=14p(1)1[(αγ|RR+βδ|LL)(|RR+|LL)+(αγ|RRβδ|LL)(|RL+|LR)]ABCD[(|a1b1+|a2b2)(|c1d1+|c2d2)+(|a1b1+|a2b2)(|c1d2+|c2d1)].
      At last, the two photons C and D are detected by the single-photon detectors shown in Fig. 2. If the outcome of the two clicked detectors is in an even-parity polarization mode and an even-parity spatial mode, the two-photon system AB is projected into the state |ϕ1AB1=1p(1)1(αγ|RR+βδ|LL)AB(|a1b1+|a2b2). This is a partially hyperentangled Bell-type state with the spatial-mode DOF in a maximally entangled Bell state. If the outcome of the two clicked detectors is in an odd-parity polarization mode (an odd-parity spatial mode), a phase-flip operation σzp=|RR||LL|(σzs=|b1b1|+|b2b2|) on the photon B is required to obtain the partially hyperentangled Bell-type state |ϕ1AB1.
    • (1.2) If the outcomes of the P-QND and S-QND are in an odd-parity polarization mode and an even-parity spatial mode, respectively, the four-photon system is projected into the state |Φ2ABCD1 with the probability of p(1)2 = 2(|αβ|2 + |γδ|2). Here
      |Φ2ABCD1=1p(1)2(|RRLL+|LLRR)ABCD(αβ|a1b1c1d1+γδ|a2b2c2d2).
      After performing Hadamard operations and detections on the photons C and D and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the state of the two-photon system AB is transformed into |ϕ2AB1=1p(1)2(|RR+|LL)AB(αβ|a1b1+γδ|a2b2), which is a partially hyperentangled Bell-type state with the polarization DOF in a maximally entangled Bell state.
    • (1.3) If the outcomes of the P-QND and S-QND are in an odd-parity polarization mode and an odd-parity spatial mode, respectively, the four-photon system is projected into the state |Φ3ABCD1with the probability of p(1)3 = 2(|αδ|2 + |βγ|2). Here
      |Φ3ABCD1=1p(1)3[(αδ|RRLL+βγ|LLRR)ABCD|a1b1c2d2+(αδ|LLRR+βγ|RRLL)ABCD|a2b2c1d1)].
      Alice and Bob perform the polarization bit-flip operations σxp=|RL|+|LR| on the spatial modes | a2〉, |b2〉, and |c1〉, |d1〉, respectively, and then the state |Φ3ABCD1 is transformed into |Φ3ABCD1. Here
      |Φ3ABCD1=1p(1)3[(αδ|RRLL+βγ|LLRR)ABCD(|a1b1c2d2+|a2b2c1d1).
      After performing Hadamard operations and detections on the photons C and D and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the state of the two-photon system AB is transformed into |ϕ3AB1=1p(1)3(αδ|RR+βγ|LL)AB(|a1b1+|a2b2), which is similar to |ϕ1AB1 by replacing αγ and βδ with αδ and βγ, respectively.
    • (1.4) If the outcomes of the P-QND and S-QND are in an even-parity polarization mode and an even-parity spatial mode, respectively, the four-photon system is projected into the state |Φ4ABCD1 with the probability of p(1)4 = |α2|2 + |β2|2 + |γ2|2 + |δ2|2. Here
      |Φ4ABCD1=1p(1)4(α2|RRRR|a1b1c1d1+β2|LLLL|a1b1c1d1+γ2|RRRR|a2b2c2d2+δ2|LLLL|a2b2c2d2)ABCD.
      After performing Hadamard operations and detections on the photons C and D and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the state of the two-photon system AB is transformed into |ϕ4AB1=1p(1)4(α2|RR|a1b1+β2|LLa1b1+γ2|RR|a2b2+δ2|LL|a2b2)AB, which is an arbitrary partially hyperentangled Bell state with less entanglement. In this condition, another round of the hyper-ECP process is required.

      Alice and Bob perform the same operations on the photon pairs A′B′ and C′D′ as those on the photon pairs AB and CD, and the same results can be obtained. That is, the four cases in Eqs. (10), (12), (14), and (15) are obtained with the probabilities p(1)1, p(1)2, p(1)3, and p(1)4, respectively, by replacing the four photons ABCD with A′B′C′D′.

  • (2) The second step. In this step, we are going to distill a maximally hyperentangled Bell state from the partially hyperentangled Bell-type states obtained in the first three cases of the first step. The principle of this step is the same as the first one (shown in Fig. 2) by replacing the photons ABCD with ABA′B′. That is, Alice and Bob perform the P-QND and S-QND on the photon pairs AA′ and BB′, respectively.
    • (2.1) For the case in (1.1), the state of the four-photon system ABA′B′ is |Φ1ABAB1=|ϕ1AB1|ϕ1AB1. Alice and Bob pick up the case when the outcome of the P-QND is in an odd-parity polarization mode, and a maximally hyperentangled Bell state can be obtained whether the outcome of the S-QND is in an odd-parity spatial mode or in an even-parity spatial mode. That is, the four-photon system is projected into the state |Φ11ABAB1 or |Φ12ABAB1 with the same probability P(1)1 = 4|αβγδ|2/p(1)1. Here,
      |Φ11ABAB1=12(|RRLL+|LLRR)ABAB(|a1b1a1b1+|a2b2a2b2),|Φ12ABAB1=12(|RRLL+|LLRR)ABAB(|a1b1a2b2+|a2b2a1b1).
      After performing Hadamard operations and detections on the photons A′ and B′ and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the two-photon system AB is projected into the maximally hyperentangled Bell state |ψAB.

      If the outcome of the P-QND is in an even-parity polarization mode, the four-photon system is projected into the state |Φ13ABAB1 or |Φ14ABAB1 with the same probability P(1)1=y12/p(1)1. Here

      |Φ13ABAB1=1y1(α2γ2|RRRR+β2δ2|LLLL)ABAB(|a1b1a1b1+|a2b2a2b2),|Φ14ABAB1=1y1(α2γ2|RRRR+β2δ2|LLLL)ABAB(|a1b1a2b2+|a2b2a1b1),
      where y1=2(|α2γ2|2+|β2δ2|2). After performing Hadamard operations and detections on the photons A′ and B′ and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the two-photon system AB is projected into the partially hyperentangled Bell-type state |Φ13AB1=1y1(α2γ2|RR+β2δ2|LL)AB(|a1b1+|a2b2). In this condition, another round of the hyper-ECP process (only the second step) is required.

    • (2.2) For the case in (1.2), the state of the four-photon system ABA′B′ is |Φ2ABAB1=|ϕ2AB1|ϕ2AB1. Alice and Bob pick up the case when the outcome of the S-QND is in an odd-parity spatial mode, and a maximally hyperentangled Bell state can be obtained whether the outcome of the P-QND is in an odd-parity polarization mode or in an even-parity polarization mode. That is, the four-photon system is projected into the state |Φ21ABAB1 or |Φ22ABAB1 with the same probability P(1)2 = 4|αβγδ|2/p(1)2. Here
      |Φ21ABAB1=12(|RRRR+|LLLL)ABAB(|a1b1a2b2+|a2b2a1b1),|Φ22ABAB1=12(|RRLL+|LLRR)ABAB(|a1b1a2b2+|a2b2a1b1).
      After performing Hadamard operations and detections on the photons A′ and B′ and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the two-photon system AB is projected into the maximally hyperentangled Bell state |ψAB.

      If the outcome of the S-QND is in an even-parity spatial mode, the four-photon system is projected into the state |Φ23ABAB1 or |Φ24ABAB1 with the same probability P(1)2=y22/p(1)2. Here

      |Φ23ABAB1=1y2(|RRRR+|LLLL)ABAB(α2β2|a1b1a1b1+γ2δ2|a2b2a2b2),|Φ24ABAB1=1y2(|RRLL+|LLRR)ABAB(α2β2|a1b1a1b1+γ2δ2|a2b2a2b2),
      where y2=2(|α2β2|2+|γ2δ2|2). After performing Hadamard operations and detections on the photons A′ and B′ and performing the conditional phase-flip operation σzp ( σzs) on the photon B, the two-photon system AB is projected into the partially hyperentangled Bell-type state |Φ23AB1=1y2(|RR+|LL)AB(α2β2|a1b1+γ2δ2|a2b2). In this condition, another round of the hyper-ECP process (only the second step) is required.

    • (2.3) For the case in (1.3), the same operations are performed on the state |Φ3ABAB1=|ϕ3AB1|ϕ3AB1 as those on the state |Φ1ABAB1 in case (2.1). The maximally hyperentangled Bell state |ψAB can be obtained with the probability of 2P(1)3 = 8|αβγδ|2/p(1)3. And the state |Φ33AB1=1y3(α2δ2|RR+β2γ2|LLAB|a1b1+|a2b2) can be obtained with the probability of 2P(1)3=2y32/p(1)3, which requires another round of the hyper-ECP process (only the second step). Here y3=2(|α2δ2|2+|β2γ2|2).

II. The iteration of the hyper-ECP process

For the arbitrary partially hyperentangled Bell state |ϕ4AB1 preserved in the case (1.4), four copies of the two-photon systems in this state are required to complete the two steps in the second round of the hyper-ECP process. While for the partially hyperentangled Bell-type states |Φ13AB1, |Φ23AB1, and |Φ33AB1 preserved in the second step, two copies of the photon systems in each state are required to complete the second step in the second round of the hyper-ECP process. Therefore, the partially hyperentangled Bell states preserved in the n-th round of the hyper-ECP process can be used to distill the maximally hyperentangled Bell state in the (n+1)-th round. The success probability of the hyperentanglement concentration process in each round can be described as follows:

P(1)=8|αβγδ|2[1p(1)1+1p(1)2+1p(1)3],P(2)=16|α2β2γ2δ2|2[1y12p(1)1+1y22p(1)2+1y32p(1)3]+8p(1)4|α2β2γ2δ2|2[1y12+1y22+1y32],,P(n)=2n+2[|α2(n1)β2(n1)γ2(n1)δ2(n1)|22(|α2(n1)γ2(n1)|2+|β2(n1)δ2(n1)|2)p(1)1+]+2n+1p(1)4[|α2(n1)β2(n1)γ2(n1)δ2(n1)|22(|α2(n1)γ2(n1)|2+|β2(n1)δ2(n1)|2)y12+]++8(|α2(n1)|2+|β2(n1)|2+|γ2(n1)|2+|δ2(n1)|2)p(1)4×[|α2(n1)β2(n1)γ2(n1)δ2(n1)|22(|α2(n1)γ2(n1)|2+|β2(n1)δ2(n1)|2)+].
After the n-th round iteration of the hyper-ECP process, the total success probability is
Pn=i=1nP(i).

In Fig. 3, the success probability P of our hyper-ECP for the case |β| = |δ| is shown with the parameters |α| and |γ|. In the hyper-ECP for the two-photon systems in an arbitrary partially hyperentangled Bell state with linear optics, only one of the three cases (1.1), (1.2), and (1.3) can be preserved, and only one of the two spatial-mode (polarization) parity cases is preserved in the second step. These six cases can all be preserved in our hyper-ECP with nonlinear optical elements, so the success probability P1 of the first round of the hyper-ECP process is almost five times larger than that in the hyper-ECP with linear optical elements. With the iteration of our hyper-ECP process, the success probability P will be increased largely.

 figure: Fig. 3

Fig. 3 The success probability P of our hyper-ECP for two-photon systems in an arbitrary partially hyperentangled Bell state with n round iteration of the hyper-ECP process. Here we use the examples n = 1 and n = 5 to show the success probabilities of our hyper-ECP P1 and P5, respectively. The parameters of the arbitrary partially hyperentangled Bell states are chosen as |β| = |δ|.

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3. Discussion and summary

It is obvious that our hyper-ECP can be generalized to distill some multi-photon systems in a maximally hyperentangled N-photon GHZ state from those in an arbitrary partially hyper-entangled GHZ-class state that decays with the interrelationship between the polarization and the spatial-mode DOFs. Suppose an arbitrary partially hyperentangled N-photon GHZ state is described as

|ϕN=(α|RRR|a1b1z1+β|LLL|a1b1z1+γ|RRR|a2b2z2+δ|LLL|a2b2z2)ABZ.
Here α, β, γ, and δ are the four parameters unknown to the N remote users in quantum communication, and they satisfy the relation |α|2 + |β|2 + |γ|2 + |δ|2 = 1. The subscripts A, B, ···, and Z represent the photons that are kept by the remote users, called Alice, Bob, ···, and Zach, respectively. As there are also three independent parameters in the state |ϕN〉, which is similar to the state |ϕAB, our hyper-ECP can be straightforwardly used to distill maximally hyperentangled N-photon GHZ states with any two remote users of the N parities performing the S-QND and P-QND on their photon pairs, respectively. The success probability of our hyper-ECP for N-photon systems in an arbitrary partially hyperentangled GHZ state is the same as the one for two-photon systems in an arbitrary partially hyperentangled Bell state.

Hyperentanglement is a promising resource in quantum information processing, and several hyper-ECPs were proposed for partially hyperentangled states with the independent decoher-ence of the entanglement in the polarization and the spatial-mode (angular-momentum) DOFs of photon pairs [40, 45, 46]. The present hyper-ECP is proposed to concentrate the photon systems in an arbitrary partially hyperentangled state with unknown parameters, resorting to the P-QNDs and S-QNDs that are constructed with the optical property of QD-cavity systems.

In experiment, the input-output rule of a QD-cavity system is not perfect, which could reduce the fidelities and efficiencies of the P-QND and S-QND. The heavy hole mixing can decrease the fidelity by a few percent [54], but it can be reduced by engineering the shape, size, and type of QDs. The fine structure splitting is immune in this scheme with charged excitons [55, 56]. The optical dephasing can slightly reduce the fidelity due to its several hundred picosecond coherence time [5761]. The spin dephasing of X comes mainly from the hole-spin dephasing and it can be safely neglected with the hole spin coherence time three orders of magnitude longer than the cavity photon life time (around tens of picoseconds in the strong coupling regime) [6264]. The electron spin coherence time, which is reduced by the hyperfine interaction, could be extended to μs using spin echo techniques (using single-photon pluses to play the role of the π pulse) [65]. The weak excitation condition in the QD-cavity system requires the intracavity photon number to be less than the critical photon number (n0 = γ2/2g2) [66,67]. That is, the time interval of the input photons is required to be Δt = τ/n0, where τ is the cavity photon life time. In the strong coupling regime for the cavity Q-factor of 104 − 105, the critical photon number is n0 ∼ 10−3, and the time interval of the input photons is Δtns [65], which is shorter than the electron spin coherence time. The speed of the photon-spin interaction is determined by the cavity photon life time (ps) [59]. And the optical spin manipulation method at single-photon levels can be used in our scheme, instead of the other optical spin manipulation [65].

The fidelity and efficiency of our scheme are mainly reduced by the coupling strength g and the cavity side leakage κs. In the case κs < 1.3κ, a phase shift of ±π2 can be achieved [65], and it is not very sensitive to the cavity side leakage when κs < κ [68]. In 2011, Young et al. [69] confirmed a QD-induced phase shift of 0.2 rad in d = 2.5μm pillar microcavity (g > (κ + κs + γ)/4). The fidelities and efficiencies of the P-QND (Fp, ηp) and S-QND (Fs, ηs) can be obtained (for the even parity states |ϕ±AC and |ϕ±BD),

Fp=Fs=|ψf|ψideal|2=(2|r|4+2|r0|4+4)2(|r|+|r0|2)16(2|r|8+2|r0|8+4)(|r|2+|r0|2),ηp=ηs=[12+14(|r|4+|r0|4)]2.
Here, |ψideal〉 and |ψf〉 are the final states in the ideal condition and in the experimental condition, respectively. The coupling strength g ≅ 0.5(κ + κs) [70] was reported in d = 1.5μm micropillar microcavities (Q ∼ 8800), and the coupling strength can be enhanced to g ≅ 2.4(κ + κs) (Q ∼ 40000) [71] by improving the sample designs, growth, and fabrication [72]. Here the quality factor is dominated by the side leakage and the cavity loss rate (κs), and κs can be reduced by thinning down the top mirrors, which may decrease the quality factor, increase κ, and keep κs nearly unchanged [65]. Using this method, the quality factor Q ∼ 17000 (gκ + κs) was achieved with κs ∼ 0.7 [65]. In the condition g ≅ 2.4(κ + κs), κs ∼ 0, and γ ∼ 0.1κ, the fidelities and efficiencies of the two QNDs are Fp = Fs = 100% and ηp = ηs = 98.8%, respectively. In the case g ≅ 1.3(κ + κs) and κs ∼ 0.3, the fidelities and efficiencies of the two QNDs are Fp = Fs = 87% and ηp = ηs = 67%, respectively. While for the case gκ + κs and κs ∼ 0.7, the fidelities and efficiencies of the two QNDs are Fp = Fs = 72.5% and ηp = ηs = 58%, respectively. The strong coupling and low cavity side leakage are required in this scheme. Our scheme is implemented with a QD-induced phase shift of ±π2, which requires the photons to interact with the QD-cavity system for twice to achieve a phase shift of ±π. If the QD-induced phase shift of ±π can be achieved, the photons only have to interact with the QD-cavity system for once, which will increase the fidelity and efficiency of our scheme largely.

The P-QND introduced here is more convenient to complete the interaction between the QD-cavity system and the |R〉 polarization of the two spatial modes |k1〉 and |k2〉 (k = a, c), compared with the P-QND constructed in [40]. Moreover, our hyper-ECP presented here is used to concentrate the two-photon systems in an arbitrary partially hyperentangled Bell state with the interrelationship between the decoherence of the entanglement in the two DOFs, which is not taken into account in the other hyper-ECPs [40,45,46]. With nonlinear optical elements, the success probability of the first round of the hyper-ECP process is almost five times larger than that in the hyper-ECP with linear optical elements [46], which is caused by preserving the states that are discarded in the latter. With the iteration of the hyper-ECP process, the success probability P can be increased largely.

In summary, we have proposed a general hyper-ECP for improving the entanglement of the two-photon systems in an arbitrary partially hyperentangled Bell state that decays with the interrelationship between the polarization and the spatial-mode DOFs, resorting to the P-QNDs and S-QNDs that are constructed by the optical property of QD-cavity systems. This hyper-ECP can be straightforwardly generalized for arbitrary partially hyperentangled N-photon GHZ-class states, and it is useful in the long-distance quantum communication proposals based on hyperentanglement, especially for the case with the interrelationship between the two DOFs of multi-photon systems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11175094 and 91221205, and the National Basic Research Program of China under Grants No. 2009CB929402 and No. 2011CB9216002. GLL is a member of the Center of Atomic and Molecular Nanosciences, Tsinghua University.

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References

  • View by:

  1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
    [Crossref]
  2. A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
    [Crossref] [PubMed]
  3. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
    [Crossref] [PubMed]
  4. 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]
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2014 (1)

2013 (4)

H. R. Wei and F. G. Deng, “Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity,” Opt. Express 21, 17671–17685 (2013).
[Crossref] [PubMed]

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]

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, H. R. Wei, and F. G. Deng, “Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by a quantum dot inside a one-side optical microcavity,” Laser phys. Lett. 10, 095202 (2013).
[Crossref]

2012 (6)

T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
[Crossref]

T. J. Wang, S. Y. Song, and G. L. Long, “Quantum repeater based on spatial entanglement of photons and quantum-dot spins in optical microcavities,” Phys. Rev. A 85, 062311 (2012).
[Crossref]

B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012).
[Crossref] [PubMed]

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]

F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
[Crossref]

L. Chen, “Comblike entangled spectrum for composite spin-orbit modes from hyperconcentration,” Phys. Rev. A 85, 012311 (2012).
[Crossref]

2011 (6)

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19, 25685–25695 (2011).
[Crossref]

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011).
[Crossref]

C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B 83, 115303 (2011).
[Crossref]

A. B. Young, R. Oulton, C. Y. Hu, A. C. T. Thijssen, C. Schneider, S. Reitzenstein, M. Kamp, S. Höfling, L. Worschech, A. Forchel, and J. G. Rarity, “Quantum-dot-induced phase shift in a pillar microcavity,” Phys. Rev. A 84, 011803 (2011).
[Crossref]

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

N. Pisenti, C. P. E. Gaebler, and T. W. Lynn, “Distinguishability of hyperentangled Bell states by linear evolution and local projective measurement,” Phys. Rev. A 84, 022340 (2011).
[Crossref]

2010 (4)

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (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 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]

2009 (3)

G. Vallone, R. Ceccarelli, F. De Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301 (2009).
[Crossref]

C. Y. Hu, W. J. Munro, J. L. OBrien, and J. G. Rarity, Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity, Phys. Rev. B 80, 205326 (2009).
[Crossref]

D. Brunner, B. D. Gerardot, P. A. Dalgarno, G. Wüst, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “A coherent single-hole spin in a semiconductor,” Science 325, 70–72 (2009).
[Crossref] [PubMed]

2008 (7)

B. D. Gerardot, D. Brunner, P. A. Dalgarno, P. Öhberg, S. Seidl, M. Kroner, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “Optical pumping of a single hole spin in a quantum dot,” Nature (London) 451, 441–444 (2008).
[Crossref]

C. Y. Hu, W. J. Munro, and J. G. Rarity, “Deterministic photon entangler using a charged quantum dot inside a microcavity,” Phys. Rev. B 78, 125318 (2008).
[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]

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]

C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: applications to entangling remote spins via a single photon,” Phys. Rev. B 78, 085307 (2008).
[Crossref]

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nature Phys. 4, 282–286 (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 (4)

T. C. Wei, J. T. Barreiro, and P. G. Kwiat, “Hyperentangled Bell-state analysis,” Phys. Rev. A 75, 060305 (2007).
[Crossref]

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauß, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” Appl. Phys. Lett. 90, 251109 (2007).
[Crossref]

D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter, J. J. Finley, D. V. Bulaev, and D. Loss, “Observation of extremely slow hole spin relaxation in self-assembled quantum dots,” Phys. Rev. B 76, 241306 (2007).
[Crossref]

2006 (1)

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
[Crossref] [PubMed]

2005 (3)

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense codi,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[Crossref]

M. Barbieri, C. Cinelli, P. Mataloni, and F. De Martini, “Polarization-momentum hyperentangled states: realization and characterization,” Phys. Rev. A 72, 052110 (2005).
[Crossref]

2004 (4)

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[Crossref]

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Radiatively limited dephasing in InAs quantum dots,” Phys. Rev. B 70, 033301 (2004).
[Crossref]

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature (London) 432, 197–200 (2004).
[Crossref]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) 432, 200–203 (2004).
[Crossref]

2003 (4)

G. Bester, S. Nair, and A. Zunger, “Pseudopotential calculation of the excitonic fine structure of million-atom self-assembled In1−x Gax As/GaAs quantum dots,” Phys. Rev. B 67, 161306 (2003).
[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]

S. P. Walborn, S. Pádua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[Crossref]

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

2002 (6)

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. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[Crossref]

J. J. Finley, D. J. Mowbray, M. S. Skolnick, A. D. Ashmore, C. Baker, A. F. G. Monte, and M. Hopkinson, “Fine structure of charged and neutral excitons in InAs-Al0.6Ga0.4As quantum dots,” Phys. Rev. B 66, 153316 (2002).
[Crossref]

2001 (5)

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

D. Birkedal, K. Leosson, and J. M. Hvam, “Long lived coherence in self-assembled quantum dots,” Phys. Rev. Lett. 87, 227401 (2001).
[Crossref] [PubMed]

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

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

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

1999 (2)

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

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

1998 (2)

P. G. Kwiat and H. Weinfurter, “Embedded Bell-state analysis,” Phys. Rev. A 58, R2623–R2626 (1998).
[Crossref]

C. Y. Hu, W. Ossau, D. R. Yakovlev, G. Landwehr, T. Wojtowicz, G. Karczewski, and J. Kossut, “Optically detected magnetic resonance of excess electrons in type-I quantum wells with a low-density electron gas,” Phys. Rev. B 58, R1766–R1769 (1998).
[Crossref]

1997 (1)

R. J. Warburton, C. S. Dürr, K. Karrai, J. P. Kotthaus, G. Medeiros-Ribeiro, and P. M. Petroff, “Charged excitons in self-assembled semiconductor quantum dots,” Phys. Rev. Lett. 79, 5282–5285 (1997).
[Crossref]

1996 (3)

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[Crossref] [PubMed]

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]

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 Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref] [PubMed]

Abstreiter, G.

D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter, J. J. Finley, D. V. Bulaev, and D. Loss, “Observation of extremely slow hole spin relaxation in self-assembled quantum dots,” Phys. Rev. B 76, 241306 (2007).
[Crossref]

Ashmore, A. D.

J. J. Finley, D. J. Mowbray, M. S. Skolnick, A. D. Ashmore, C. Baker, A. F. G. Monte, and M. Hopkinson, “Fine structure of charged and neutral excitons in InAs-Al0.6Ga0.4As quantum dots,” Phys. Rev. B 66, 153316 (2002).
[Crossref]

Baker, C.

J. J. Finley, D. J. Mowbray, M. S. Skolnick, A. D. Ashmore, C. Baker, A. F. G. Monte, and M. Hopkinson, “Fine structure of charged and neutral excitons in InAs-Al0.6Ga0.4As quantum dots,” Phys. Rev. B 66, 153316 (2002).
[Crossref]

Barbieri, M.

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

M. Barbieri, C. Cinelli, P. Mataloni, and F. De Martini, “Polarization-momentum hyperentangled states: realization and characterization,” Phys. Rev. A 72, 052110 (2005).
[Crossref]

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nature Phys. 4, 282–286 (2008).
[Crossref]

T. C. Wei, J. T. Barreiro, and P. G. Kwiat, “Hyperentangled Bell-state analysis,” Phys. Rev. A 75, 060305 (2007).
[Crossref]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[Crossref]

Bayer, M.

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[Crossref]

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, 2046–2052 (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, 2046–2052 (1996).
[Crossref] [PubMed]

Berthiaume, A.

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

Bester, G.

G. Bester, S. Nair, and A. Zunger, “Pseudopotential calculation of the excitonic fine structure of million-atom self-assembled In1−x Gax As/GaAs quantum dots,” Phys. Rev. B 67, 161306 (2003).
[Crossref]

Bichler, M.

D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter, J. J. Finley, D. V. Bulaev, and D. Loss, “Observation of extremely slow hole spin relaxation in self-assembled quantum dots,” Phys. Rev. B 76, 241306 (2007).
[Crossref]

Bimberg, D.

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Birkedal, D.

D. Birkedal, K. Leosson, and J. M. Hvam, “Long lived coherence in self-assembled quantum dots,” Phys. Rev. Lett. 87, 227401 (2001).
[Crossref] [PubMed]

Borri, P.

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Radiatively limited dephasing in InAs quantum dots,” Phys. Rev. B 70, 033301 (2004).
[Crossref]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

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]

Brukner, C.

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

Brunner, D.

D. Brunner, B. D. Gerardot, P. A. Dalgarno, G. Wüst, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “A coherent single-hole spin in a semiconductor,” Science 325, 70–72 (2009).
[Crossref] [PubMed]

B. D. Gerardot, D. Brunner, P. A. Dalgarno, P. Öhberg, S. Seidl, M. Kroner, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “Optical pumping of a single hole spin in a quantum dot,” Nature (London) 451, 441–444 (2008).
[Crossref]

Bulaev, D. V.

D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter, J. J. Finley, D. V. Bulaev, and D. Loss, “Observation of extremely slow hole spin relaxation in self-assembled quantum dots,” Phys. Rev. B 76, 241306 (2007).
[Crossref]

Bužek, V.

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

Ceccarelli, R.

G. Vallone, R. Ceccarelli, F. De Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301 (2009).
[Crossref]

Chen, L.

L. Chen, “Comblike entangled spectrum for composite spin-orbit modes from hyperconcentration,” Phys. Rev. A 85, 012311 (2012).
[Crossref]

Cinelli, C.

M. Barbieri, C. Cinelli, P. Mataloni, and F. De Martini, “Polarization-momentum hyperentangled states: realization and characterization,” Phys. Rev. A 72, 052110 (2005).
[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]

Dalgarno, P. A.

D. Brunner, B. D. Gerardot, P. A. Dalgarno, G. Wüst, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “A coherent single-hole spin in a semiconductor,” Science 325, 70–72 (2009).
[Crossref] [PubMed]

B. D. Gerardot, D. Brunner, P. A. Dalgarno, P. Öhberg, S. Seidl, M. Kroner, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “Optical pumping of a single hole spin in a quantum dot,” Nature (London) 451, 441–444 (2008).
[Crossref]

De Martini, F.

G. Vallone, R. Ceccarelli, F. De Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301 (2009).
[Crossref]

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

M. Barbieri, C. Cinelli, P. Mataloni, and F. De Martini, “Polarization-momentum hyperentangled states: realization and characterization,” Phys. Rev. A 72, 052110 (2005).
[Crossref]

Deng, F. G.

H. R. Wei and F. G. Deng, “Universal quantum gates on electron-spin qubits with quantum dots inside single-side optical microcavities,” Opt. Express 22, 593–607 (2014).
[Crossref] [PubMed]

H. R. Wei and F. G. Deng, “Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity,” Opt. Express 21, 17671–17685 (2013).
[Crossref] [PubMed]

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, H. R. Wei, and F. G. Deng, “Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by a quantum dot inside a one-side optical microcavity,” Laser phys. Lett. 10, 095202 (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, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
[Crossref]

B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012).
[Crossref] [PubMed]

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, “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]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[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, “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]

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense codi,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[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]

Deppe, D. G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) 432, 200–203 (2004).
[Crossref]

Deutsch, D.

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]

Du, F. F.

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]

Dürr, C. S.

R. J. Warburton, C. S. Dürr, K. Karrai, J. P. Kotthaus, G. Medeiros-Ribeiro, and P. M. Petroff, “Charged excitons in self-assembled semiconductor quantum dots,” Phys. Rev. Lett. 79, 5282–5285 (1997).
[Crossref]

Ekert, A.

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]

Ekert, A. K.

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

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) 432, 200–203 (2004).
[Crossref]

Fafard, S.

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[Crossref]

Finley, J. J.

D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter, J. J. Finley, D. V. Bulaev, and D. Loss, “Observation of extremely slow hole spin relaxation in self-assembled quantum dots,” Phys. Rev. B 76, 241306 (2007).
[Crossref]

J. J. Finley, D. J. Mowbray, M. S. Skolnick, A. D. Ashmore, C. Baker, A. F. G. Monte, and M. Hopkinson, “Fine structure of charged and neutral excitons in InAs-Al0.6Ga0.4As quantum dots,” Phys. Rev. B 66, 153316 (2002).
[Crossref]

Forchel, A.

A. B. Young, R. Oulton, C. Y. Hu, A. C. T. Thijssen, C. Schneider, S. Reitzenstein, M. Kamp, S. Höfling, L. Worschech, A. Forchel, and J. G. Rarity, “Quantum-dot-induced phase shift in a pillar microcavity,” Phys. Rev. A 84, 011803 (2011).
[Crossref]

S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauß, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” Appl. Phys. Lett. 90, 251109 (2007).
[Crossref]

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature (London) 432, 197–200 (2004).
[Crossref]

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[Crossref]

Gaebler, C. P. E.

N. Pisenti, C. P. E. Gaebler, and T. W. Lynn, “Distinguishability of hyperentangled Bell states by linear evolution and local projective measurement,” Phys. Rev. A 84, 022340 (2011).
[Crossref]

Gasparoni, S.

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

Gerardot, B. D.

D. Brunner, B. D. Gerardot, P. A. Dalgarno, G. Wüst, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “A coherent single-hole spin in a semiconductor,” Science 325, 70–72 (2009).
[Crossref] [PubMed]

B. D. Gerardot, D. Brunner, P. A. Dalgarno, P. Öhberg, S. Seidl, M. Kroner, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, “Optical pumping of a single hole spin in a quantum dot,” Nature (London) 451, 441–444 (2008).
[Crossref]

Gibbs, H. M.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) 432, 200–203 (2004).
[Crossref]

Gisin, N.

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

Gorbunov, A.

S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauß, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” Appl. Phys. Lett. 90, 251109 (2007).
[Crossref]

Gorbunov, A. A.

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[Crossref]

Hawrylak, P.

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[Crossref]

Heiss, D.

D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter, J. J. Finley, D. V. Bulaev, and D. Loss, “Observation of extremely slow hole spin relaxation in self-assembled quantum dots,” Phys. Rev. B 76, 241306 (2007).
[Crossref]

Hendrickson, J.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) 432, 200–203 (2004).
[Crossref]

Hillery, M.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
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Figures (3)

Fig. 1
Fig. 1 The optical transitions for a negatively charged exciton X with circularly polarized photons. (a) A charged QD embedded in a one-side micropillar microcavity with a circular cross section. (b) The spin-dependent transition rules of a negatively charged exciton Xaccording to Pauli’s exclusion principle. L and R represent the left and the right circularly polarized photons, respectively. ↑↓⇑ (↓↑⇓) represents the negatively charged exciton X in the spin state | + 3 2 ( | 3 2 ). ↑ and ↓ represent the excess electron spin states | + 1 2 and | 1 2 , respectively.
Fig. 2
Fig. 2 Schematic diagram of the general hyper-ECP for two-photon systems in an arbitrary partially hyperentangled Bell state. The optical elements in the blue dashed box are used to perform the polarization parity-check QND on a two-photon system, and the optical elements in the pink dashed box are used to perform the spatial-mode parity-check QND on a two-photon system. The small mirror is used to reflect the photon, which makes the photon interact with the cavity twice. Zi (i = 1, 2, 3, 4) represents a half-wave plate which is used to perform a polarization phase-flip operation σ z p = | R R | | L L |. Xj (j = 1, 2, 3, 4) represents a half-wave plate which is used to perform a polarization bit-flip operation σ x p = | R L | + | L R |. R45 represents a half-wave plate which is used to perform the polarization Hadamard operation. CPBS represents a polarizing beam splitter in the circular basis, which transmits the photon in the right-circular polarization |R〉 and reflects the photon in the left-circular polarization |L〉, respectively. BS represents a 50:50 beam splitter which is used to perform the spatial-mode Hadamard operation. DL represents a time-delay device which makes the two wave packets reach the last CPBS in each Mach-Zehnder interferometer simultaneously. Dk (k = L1, R1, R2, L2) represents a single-photon detector. D represents the same operations as the ones performed by Alice in the green dotted box.
Fig. 3
Fig. 3 The success probability P of our hyper-ECP for two-photon systems in an arbitrary partially hyperentangled Bell state with n round iteration of the hyper-ECP process. Here we use the examples n = 1 and n = 5 to show the success probabilities of our hyper-ECP P1 and P5, respectively. The parameters of the arbitrary partially hyperentangled Bell states are chosen as |β| = |δ|.

Equations (23)

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| ψ A B = 1 2 ( | R R + | L L ) A B ( | a 1 b 1 + | a 2 b 2 ) , | ϕ 0 A B = ( α 0 | R R + β 0 | L L ) A B ( γ 0 | a 1 b 1 + δ 0 | a 2 b 2 ) , | ϕ A B = ( α | R R | a 1 b 1 + β | L L | a 1 b 1 + γ | R R | a 2 b 2 + δ | L L | a 2 b 2 ) A B .
d a ^ d t = [ i ( ω c ω ) + κ 2 + κ s 2 ] a ^ g σ ^ κ a ^ in , d σ ^ d t = [ i ( ω X ω ) + γ 2 ] σ ^ g σ ^ z a ^ , a ^ out = a ^ in + κ a ^ ,
r ( ω ) = a ^ in a ^ = 1 κ [ i ( ω X ω ) + γ 2 ] [ i ( ω X ω ) + γ 2 ] [ i ( ω c ω ) + κ 2 + κ s 2 ] + g 2 .
| L , | L , , | L , | L , , | R , | R , , | R , | R , .
| + 1 | χ A [ | 1 ( α 0 | R + β 0 | L ) A + | 1 ( α 0 | R + β 0 | L ) A ] ( γ 0 | a 1 + δ 0 | a 2 ) .
| + 1 | ϕ ± A C | + 1 | ϕ ± A C , | + 1 | ϕ 1 ± A C | 1 | ϕ 1 ± A C .
| + 2 | χ B [ | 2 ( γ 0 | b 1 + δ 0 | b 2 ) + | 2 ( γ 0 | b 1 δ 0 | b 2 ) ] ( α 0 | R + β 0 | L ) B .
| + 2 | ϕ ± B D | + 2 | ϕ ± B D , | + 2 | ϕ 1 ± B D | 2 | ϕ 1 ± B D .
| ϕ A B = ( α | R R | a 1 b 1 + β | L L | a 1 b 1 + γ | R R | a 2 b 2 + δ | L L | a 2 b 2 ) A B , | ϕ C D = ( α | R R | c 1 d 1 + β | L L | c 1 d 1 + γ | R R | c 2 d 2 + δ | L L | c 2 d 2 ) C D , | ϕ A B = ( α | R R | a 1 b 1 + β | L L | a 1 b 1 + γ | R R | a 2 b 2 + δ | L L | a 2 b 2 ) A B , | ϕ C D = ( α | R R | c 1 d 1 + β | L L | c 1 d 1 + γ | R R | c 2 d 2 + δ | L L | c 2 d 2 ) C D .
| Φ 1 A B C D 1 = 1 p ( 1 ) 1 ( α γ | R R R R + β δ | L L L L ) A B C D ( | a 1 b 1 c 2 d 2 + | a 2 b 2 c 1 d 1 ) .
| Φ 1 A B C D 1 = 1 4 p ( 1 ) 1 [ ( α γ | R R + β δ | L L ) ( | R R + | L L ) + ( α γ | R R β δ | L L ) ( | R L + | L R ) ] A B C D [ ( | a 1 b 1 + | a 2 b 2 ) ( | c 1 d 1 + | c 2 d 2 ) + ( | a 1 b 1 + | a 2 b 2 ) ( | c 1 d 2 + | c 2 d 1 ) ] .
| Φ 2 A B C D 1 = 1 p ( 1 ) 2 ( | R R L L + | L L R R ) A B C D ( α β | a 1 b 1 c 1 d 1 + γ δ | a 2 b 2 c 2 d 2 ) .
| Φ 3 A B C D 1 = 1 p ( 1 ) 3 [ ( α δ | R R L L + β γ | L L R R ) A B C D | a 1 b 1 c 2 d 2 + ( α δ | L L R R + β γ | R R L L ) A B C D | a 2 b 2 c 1 d 1 ) ] .
| Φ 3 A B C D 1 = 1 p ( 1 ) 3 [ ( α δ | R R L L + β γ | L L R R ) A B C D ( | a 1 b 1 c 2 d 2 + | a 2 b 2 c 1 d 1 ) .
| Φ 4 A B C D 1 = 1 p ( 1 ) 4 ( α 2 | R R R R | a 1 b 1 c 1 d 1 + β 2 | L L L L | a 1 b 1 c 1 d 1 + γ 2 | R R R R | a 2 b 2 c 2 d 2 + δ 2 | L L L L | a 2 b 2 c 2 d 2 ) A B C D .
| Φ 1 1 A B A B 1 = 1 2 ( | R R L L + | L L R R ) A B A B ( | a 1 b 1 a 1 b 1 + | a 2 b 2 a 2 b 2 ) , | Φ 1 2 A B A B 1 = 1 2 ( | R R L L + | L L R R ) A B A B ( | a 1 b 1 a 2 b 2 + | a 2 b 2 a 1 b 1 ) .
| Φ 1 3 A B A B 1 = 1 y 1 ( α 2 γ 2 | R R R R + β 2 δ 2 | L L L L ) A B A B ( | a 1 b 1 a 1 b 1 + | a 2 b 2 a 2 b 2 ) , | Φ 1 4 A B A B 1 = 1 y 1 ( α 2 γ 2 | R R R R + β 2 δ 2 | L L L L ) A B A B ( | a 1 b 1 a 2 b 2 + | a 2 b 2 a 1 b 1 ) ,
| Φ 2 1 A B A B 1 = 1 2 ( | R R R R + | L L L L ) A B A B ( | a 1 b 1 a 2 b 2 + | a 2 b 2 a 1 b 1 ) , | Φ 2 2 A B A B 1 = 1 2 ( | R R L L + | L L R R ) A B A B ( | a 1 b 1 a 2 b 2 + | a 2 b 2 a 1 b 1 ) .
| Φ 2 3 A B A B 1 = 1 y 2 ( | R R R R + | L L L L ) A B A B ( α 2 β 2 | a 1 b 1 a 1 b 1 + γ 2 δ 2 | a 2 b 2 a 2 b 2 ) , | Φ 2 4 A B A B 1 = 1 y 2 ( | R R L L + | L L R R ) A B A B ( α 2 β 2 | a 1 b 1 a 1 b 1 + γ 2 δ 2 | a 2 b 2 a 2 b 2 ) ,
P ( 1 ) = 8 | α β γ δ | 2 [ 1 p ( 1 ) 1 + 1 p ( 1 ) 2 + 1 p ( 1 ) 3 ] , P ( 2 ) = 16 | α 2 β 2 γ 2 δ 2 | 2 [ 1 y 1 2 p ( 1 ) 1 + 1 y 2 2 p ( 1 ) 2 + 1 y 3 2 p ( 1 ) 3 ] + 8 p ( 1 ) 4 | α 2 β 2 γ 2 δ 2 | 2 [ 1 y 1 2 + 1 y 2 2 + 1 y 3 2 ] , , P ( n ) = 2 n + 2 [ | α 2 ( n 1 ) β 2 ( n 1 ) γ 2 ( n 1 ) δ 2 ( n 1 ) | 2 2 ( | α 2 ( n 1 ) γ 2 ( n 1 ) | 2 + | β 2 ( n 1 ) δ 2 ( n 1 ) | 2 ) p ( 1 ) 1 + ] + 2 n + 1 p ( 1 ) 4 [ | α 2 ( n 1 ) β 2 ( n 1 ) γ 2 ( n 1 ) δ 2 ( n 1 ) | 2 2 ( | α 2 ( n 1 ) γ 2 ( n 1 ) | 2 + | β 2 ( n 1 ) δ 2 ( n 1 ) | 2 ) y 1 2 + ] + + 8 ( | α 2 ( n 1 ) | 2 + | β 2 ( n 1 ) | 2 + | γ 2 ( n 1 ) | 2 + | δ 2 ( n 1 ) | 2 ) p ( 1 ) 4 × [ | α 2 ( n 1 ) β 2 ( n 1 ) γ 2 ( n 1 ) δ 2 ( n 1 ) | 2 2 ( | α 2 ( n 1 ) γ 2 ( n 1 ) | 2 + | β 2 ( n 1 ) δ 2 ( n 1 ) | 2 ) + ] .
P n = i = 1 n P ( i ) .
| ϕ N = ( α | R R R | a 1 b 1 z 1 + β | L L L | a 1 b 1 z 1 + γ | R R R | a 2 b 2 z 2 + δ | L L L | a 2 b 2 z 2 ) A B Z .
F p = F s = | ψ f | ψ ideal | 2 = ( 2 | r | 4 + 2 | r 0 | 4 + 4 ) 2 ( | r | + | r 0 | 2 ) 16 ( 2 | r | 8 + 2 | r 0 | 8 + 4 ) ( | r | 2 + | r 0 | 2 ) , η p = η s = [ 1 2 + 1 4 ( | r | 4 + | r 0 | 4 ) ] 2 .

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