Abstract

We investigate a new approach for achieving hyperdistillation and hyperentanglement purification operations simultaneously on two-photon systems, whose state is described by nonlocal hyperentangled Bell states on both the spatial mode and polarization degree of freedoms. Exploiting linear optics and local entanglement resource, the quantum nondemolition (QND) parity-checking measurement and the heralded two-qubit amplification could are key steps in our scheme. With the QND parity-checking measurement and heralded qubit amplification operations, both the bit-flip (phase-flip) errors caused by decoherence in noisy channels and the vacuum errors caused by the transmission losses can be corrected. We show that the proposed scheme provides a new solution to overcome the problem of photon losses and decoherence simultaneously, which could be achieved with current technologies.

© 2015 Optical Society of America

1. Introduction

Quantum entanglement is a fundamental resource for quantum information processing, and also plays a significant role in various fascinating applications, such as quantum computation [1], quantum secret sharing [2, 3], quantum dense coding [4, 5], and quantum cryptography [611]. Entangled photons are generally considered as an ideal information carriers in quantum communication for its high-speed transmission and conspicuous low-noise properties. Hence, they can act as the flying qubits and are ordinarily used to connect distant quantum nodes in long distance communication. Theoretically, entanglement is realized under local operations and classical communications (LOCC), however, photon loss and decoherence caused by the interaction between the quantum system and the environment will inevitably occur during the practical entanglement distribution and long distance transmission.

The consequences of photon loss will cause the signal distortion. To overcome the signal distortion, one can increase the channel capacity to prevent signal from distorting or missing. One of the effective ways to increase capacity of channel is commonly known as hyperentanglement [12] which encodes information on one qubit in multiple degree of freedoms (DOFs). Hyperentanglment state refers to the entangled state of particle simultaneously on multiple DOFs. Besides improving the capacity of channel, hyperentanglement are also employed to increase the security of the communication [13, 14], eliminate the influence of the noise [1518], and entanglement analysis [19,20]. Moreover, the security of the quantum communication will be reduced by photon loss. For example, the violation of Bell inequalities can rigorously guarantee the presence of entanglement and the quantum nonlocal correlations, which is of fundamental importance in demonstrating entanglement-assisted reduction of communication complexity [21] and device-independent quantum communication. However, due to the low detection efficiency caused by photon losses in quantum communications (not all entangled photons are detected and the overall detection efficiency is below a certain threshold value), the optical loophole-free violation of Bell inequalities over long distance [22] can not be satisfied [23]. Therefore, signal losses in an optical Bell test can open the so-called detection efficiency loophole [24] which allows for local-hidden-variable theories that are able to reproduce the observed data in the experiments. Eliminating the vacuum component and improving the detection efficiency are the main requirements in defending the attacks based on the detection loophole [25, 26].

Meanwhile, decoherence will also decrease the entanglement of the quantum system. After transmitted over a noisy channel, the maximally (hyper)entangled photon states decay into less entangled pure states or mixed states, leading to a destruction of the fidelity and the security of long-distance quantum communication protocols. In order to eliminate the decoherence effect in entangled systems, two interesting quantum techniques could be exploited, entanglement purification and entanglement concentration, to obtain high-fidelity entangled photon systems. Specifically, entanglement purification is used to distill high-fidelity entangled photon systems from a less entangled states ensemble in a mixed state [2736] while entanglement concentration [3739] is to achieve the maximally entangled states from partially entangled pure states. Due to the fact that the maximally (hyper)entangled photon states usually decay into a mixed state after being transmitted over a noisy channel, entanglement purification is more general than entanglement concentration in practical applications. In 2013, Ren et al. [40] theoretically proposed a hyperentanglement concentration protocol for photons by performing parity-check operations on both spatial-mode and polarization DOFs independently resorting to linear optics, and they point out that it is impossible to accomplish hyperentanglement purification only using linear optics. However, they did not considered the influence of photon loss, and it has been proved that the relative weight of the vacuum component will increase largely after the linear-optical parity-check operations [23]. Therefore, the detection efficiency in the quantum communication protocols decreases sharply.

In this paper, we investigate the possibility of achieving hyperdistillation and hyperentanglement purification operations simultaneously on two-photon systems in nonlocal hyperentangled Bell states on both the spatial mode and polarization DOFs. The key operations of our scheme are the quantum nondemolition (QND) parity-checking measurement and the heralded two-qubit amplification which can be completed by one step exploiting linear optics and local entanglement resource. With these operations, the bit-flip errors and phase-flip errors caused by decoherence in noisy channels and the vacuum errors caused by the transmission losses could all be corrected, and also our scheme provides a new solution to overcome the problem of photon losses and decoherence simultaneously.

This paper is organized as follows: In Sec. II, we describe the approach for QND parity-checking operation which could accomplish parity-check measurement on both spatial-mode DOF and polarization DOF, and the heralded two-qubit amplification which can remove the vacuum components of the photons assisted by linear optical elements and local entanglement resources. In Sec. III, we present the hyperdistillation and hyperentanglement purification model in spatial mode and polarization DOFs. And Sec. IV is the discussion and summary.

2. Implementation of QND parity-check measurement and qubit amplification

The basic model of the present hyperdistillation and hyperentanglement purification scheme is depicted in Fig. 1 which consists of linear optical elements and single-photon detectors. The polarizing beam splitters (PBSs) transmit a horizontally polarized photon (|H〉) and reflect a vertically polarized photon (|V〉). There are four half-wave plates (HWPs) which can flip the polarization state of the photons, and eight quarter-wave plates (QWPs) which perform Hadamard operation [ |H12(|H+|V), |V12(|H|V)] on the polarizations DOF of the photons. To demonstrate the principle of the setup, we first describe the QND parity-check measurement process in both polarization DOF and spatial-mode DOF and implement qubit amplification that can distill expected information from a mixed entangled state ensemble.

 figure: Fig. 1

Fig. 1 The schematic diagram of the setup for QND parity-check measurement and qubit amplification. The inset shows the structure of detectors D. The setup consists of four input ports (marked with a1, a2, c1, c2), eight auxiliary input ports (marked with b1, b2, b3, b4, e1, e2, e3, e4), four output ports (marked with a′1, a′2, c′1, c′2), two polarizing beam splitters (PBS) which transmit a horizontally polarized photon |H〉 and reflect a vertically polarized photon |V〉. Successful operation of the distilling is heralded by eight-photon coincidence detection on the eight detectors D1, D2, D3, D4, D5, D6, D7 and D8.

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Assume that the initial state of the two photons A and C are

|ψ0AC=[(1P1)|vacAvac|+P1|ψAψA|][(1P2)|vacCvac|+P2|ψCψC|],
where
|ψA=(α1|a1+β1|a2)AS(γ1|H+ξ1|V)AP,|ψC=(α2|c1+β2|c2)CS(γ2|H+ξ2|V)CP.
Here, |vac〉 denotes vacuum state, and P1 (P2) represents the relative weight of the qubit components of the photon A(C). P denotes polarized state of the photons, S represents the photons’ spatial-mode state, and the coefficients satisfy |α1|2 + |β1|2 = |α2|2 + |β2|2 = |γ1|2 + |ξ1|2 = |γ2|2 + |ξ2|2 = 1. The even-parity component of the input photons AC is |ψACeven=χ(α1α2|a1c1+β1β2|a2c2)(γ1γ2|HH+ξ1ξ2|VV), where χ denotes the normalization coefficient. The locally prepared ancillary 8-photon state can be written as
|ω=12[(|HVHV+|VHVH)b1b2b3b4(|HHHHe1e2e3e4)+(|VVVV)b1b2b3b4(|HVHV+|VHVH)e1e2e3e4],
where b1, b2, b3, and b4 indicate the input-ports for 4 ancillary photons B1, B2, B3, and B4, respectively, and e1, e2, e3, e4 indicate the input-ports for the other 4 ancillary photons E1, E2, E3, E4, respectively. Successful operation of the distilling is heralded by 8-photon coincidence detection on the eight detectors D1, D2, D3, D4, D5, D6, D7, and D8. If the input state contains vacuum component, it will not cause the eight detectors response simultaneously, therefore, we can discard vacuum errors through the coincidence detection events.

After PBS1 and PBS2, the state of the ten-photon ACB1B2B3B4E1E2E3E4 system can be written as

|ψA|ψC|wPBS1,2(α1γ1|HD1+β1γ1|HD2+α1ξ1|VD8+β1ξ1|VD7)(α2γ2|HD3+β2γ2|HD4+α2ξ2|VD6+β2ξ2|VD5)12[(|Ha1VD2Hc1VD4+|VD1Ha2VD3Hc2)|HD5HD6HD7HD8+|VD1VD2VD3VD4(|HD5Vc1HD7Va1+|Vc2HD6Va2HD8)].
From Eq. (4), one can easily observes that there are only four circumstances which can cause the eight detectors response simultaneously. So utilizing eight-photon coincidence detection, the postselection process only picks up the case of the state
χ(α1α2γ1γ2|Ha1Hc1|HD1HD2HD3VD4HD5HD6HD7HD8+α1α2ξ1ξ2|Va1Vc1|VD1VD2VD3VD4HD5VD6HD7VD8+β1β2γ1γ2|Ha2Hc2|VD1HD2VD3HD4HD5HD6HD7HD8+β1β2ξ1ξ2|Va2Vc2|VD1VD2VD3VD4VD5HD6VD7HD8).
After the flipping operations of the HWPs and the rotation operations of the QWPs at output-ports, the single-photon detections on the eight detectors are performed on the bases {|H,|V〉}. Then, we can rewrite the system in the corresponding subspace as
18|ψACeven(|LD1D2D3D4|LD5D6D7D8|MD1D2D3D4|MD5D6D7D8)+χ8[(|A+|B|C+|D)(|LD1D2D3D4|MD5D6D7D8|MD1D2D3D4|LD5D6D7D8)+(|A|B+|C|D)(|ND1D2D3D4|ND5D6D7D8|KD1D2D3D4|KD5D6D7D8)+(|A|B|C+|D)(|ND1D2D3D4|KD5D6D7D8|KD1D2D3D4|ND5D6D7D8)+(|A+|B+|C|D)(|LD1D2D3D4|ND5D6D7D8|MD1D2D3D4|KD5D6D7D8)+(|A+|B|C+|D)(|LD1D2D3D4|KD5D6D7D8|MD1D2D3D4|ND5D6D7D8)+(|A|B+|C+|D)(|ND1D2D3D4|LD5D6D7D8|KD1D2D3D4|MD5D6D7D8)+(|A|B|C|D)(|ND1D2D3D4|MD5D6D7D8|KD1D2D3D4|LD5D6D7D8)],
where
|ψACeven=χ(α1α2|a1c1+β1β2|a2c2)AC(γ1γ2|HH+ξ1ξ2|VV)AC,|A=α1α2γ1γ2|HHAC|a1c1AC,|B=β1β2γ1γ2|HHAC|a2c2AC,|C=α1α2ξ1ξ2|VVAC|a1c1AC,|D=β1β2ξ1ξ2|VVAC|a2c2AC,|L=12(|HHHH+|HVHV+|VHVH+|VVVV),|M=12(|HHVV+|HVVH+|VHHV+|VVHH),|N=12(|HHVH+|HVVV+|VHHH+|VVHV),|K=12(|HHHV+|HVHH+|VHVV+|VVVH).
From Eq. (6), we can conclude that the state of output photons A′C′ have even parity in both polarization DOF and spatial-mode DOF. If the results on D1D2D3D4 (D5D6D7D8) is |HHHH〉, |VVVV〉, |HVHV〉 or |VHVH〉, one dose not need to do any operation on the spatial-mode DOF of the photon C′ when its polarized state is |HC′ (|VC′), that is, IH(V)S=(|c1c1|+|c2c2|)|H(V)CH(V)|. Meanwhile, if the measurement result on D1D2D3D4 (D5D6D7D8) is |HHVV〉, |VVHH〉, |HVVH〉 or |VHHV〉, a single-photon gate IH(V)S=(|c1c1|+|c2c2|)|H(V)CH(V)| should be used on the photon C′ at the output-port. If the results on D1D2D3D4 (D5D6D7D8) reveal |HHVH〉, |VVHV〉, |HVVV〉 or |VHHH〉, a single qubit operation σZ,H(V)S=(|c1c1||c2c2|)|H(V)CH(V)| on the photon C′ is needed to recover the same state as |ψACeven at output port. For the last case, a single qubit operation σZ,H(V)S on the photon C′ is required at output-port when the results of D1D2D3D4 (D5D6D7D8) is |HHHV〉, |VVVH〉, |HVHH〉 or |VHVV〉.

Based on the above discussion, the process of the QND parity-check measurement and qubit amplification is heralded by the detector clicks, simultaneously. The relation between the final measuring results at the detctors D1D2D3D4 (D5D6D7D8) and the single-photon operations on the photon C′ is shown in Table I. On condition that the detectors D1, D2, D3, D4, D5, D6, D7, and D8 are simultaneously registered, the relative probability of the qubit component in the output signal mode increases to 100%, which leads to a success probability Psuc=14P1P2(|α1α2|2+|β1β2|2)(|γ1γ2|2+|ξ1ξ2|2) to achieve the even-parity states in both polarization DOF and spatial-mode DOF in a heralded strategy.

Tables Icon

Table 1. Relation between the final states of D1D2D3D4(D5D6D7D8) and the corresponding single qubit operations on the photon C′.

3. Hyperentanglement purification in spatial-mode and polarization DOFs

So far, we have constructed the QND parity-check measurement and qubit amplification using linear optics. Based on the above results, the hyperdistillation and the hyperentanglement purification could work simultaneously. Suppose a hyperentangled photon source is located at the middle point between Alice and Bob, and the hyperentangled state can be written as 12(|HH+|VV)AB(|a1b1+|a2b2)AB. In practical transmission, the channel noise will inevitably induce the signal losses and decoherence, and the state becomes a mixture of the vacuum component with probability (1 − τ1) and a mixed hyperentangled component with probability τ1. The density matrix of the system AB is deteriorated into

ρAB=τ1[F1|ϕ+Pϕ+|+(1F1)|ψ+Pψ+|]AB[F2|ϕ+Sϕ+|+(1F1)|ψ+Sψ+|]AB+τ2|+AS+||+AP+||vacBvac|+τ3|vacAvac||+BS+||+BP+|+τ4|vacABvac|,
where |ϕ+ABP=12(|HH+|VV)AB, |ψ+ABP=12(|HV+|VH)AB, |ϕ+ABS=12(|a1b1+|a2b2)AB, and |ψ+ABS=12(|a1b2+|a2b1)AB. And F1 and F2 refer to the probabilities of |ϕ+P, and |ϕ+S in the mixed hyperentangled component, respectively. The coefficients satisfy the normalization relation as τ1 + τ2 + τ3 + τ4 = 1. Thus the fidelity of the two-photon system is defined as F = τ1F1F2.

Assume that there exists another identical two-photon system CD which has the same form as the photons AB,

ρCD=τ1[F1|ϕ+Pϕ+|+(1F1)|ψ+Pψ+|]CD[F2|ϕ+Sϕ+|+(1F2)|ψ+Sψ+|]CD+τ2|+CS+||+CP+||vacDvac|+τ3|vacCvac||+DS+||+DP+|+τ4|vacCDvac|,
where |ϕ+CDS=12(|c1d1+|c2c2)CD, |ψ+CDS=12(|c1d2+|c2d1)CD. The photons A and C, B and D belong to Alice and Bob, respectively, and the four-photon system can be described by density operators ρ0 = ρABρCD.

Exploiting the devices shown in Fig. 1, both Alice and Bob perform QND parity-checking measurement on spatial-mode DOF and on polarization DOF of their own photons simultaneously, and remove the vacuum component of the photons. The four-photon systems in the two DOFs can be amplified and purified independently. Alice and Bob perform qubit amplification and pick up the even-parity cases in two DOFs on the photon-pairs AC and CD, respectively. According to Eq. (6), when the is finished on both sides in the desired way, the density operator of the system ABCD becomes

ρABCD=[F1|Φ+PΦ+|+(1F1)|Ψ+PΨ+|]ABCD[F2|Φ+SΦ+|+(1F2)|Ψ+SΨ+|]ABCD,
where the entangled state in different DOFs could be described as
|Φ+ABCDP=12(|HHHH+|VVVV)ABCD,|Ψ+ABCDP=12(|HVHV+|VHVH)ABCD,|Φ+ABCDS=12(|a1b1c1d1+|a2b2c2d2)ABCD,|Ψ+ABCDS=12(|a1b2c1d2+|a2b1c2d1)ABCD.
Here, F1=F12F12+(1F1)2 and F2=F22F22+(1F2)2 refer to the probabilities of |Φ+P and |Φ+S in the mixed hyperentangled state, respectively.

Then Alice and Bob each measure the photons C and D in the polarization basis { |+P=12(|H+|V), |P=12(|H|V)} and in the spatial-mode basis { |+S=12(|c1(d1)+|c2(d2)), |S=12(|c1(d1)|c2(d2))}, respectively. The density operator of the system ABCD can be rewritten as

ρABCD=116{[F1|ϕ+Pϕ+|+(1F1)|ψ+Pψ+|]AB|++CDP+[F1|ϕ+Pϕ+|(1F1)|ψ+Pψ+|]AB|CDP+[F1|ϕPϕ|+(1F1)|ψPψ|]AB|+CDP+[F1|ϕPϕ|(1F1)|ψPψ|]AB|+CDP}{[F2|ϕ+Sϕ+|+(1F2)|ψ+Sψ+|]AB|++CDS+[F2|ϕ+Sϕ+|(1F2)|ψ+Sψ+|]AB|CDS+[F2|ϕSϕ|(1F2)|ψSψ|]AB|+CDS+[F2|ϕSϕ|(1F2)|ψSψ|]AB|+CDS},
where
|ϕABP=12(|HH|VV)AB,|ψABP=12(|HV|VH)AB,|ϕABS=12(|a1b1|a2b2)AB,|ψABS=12(|a1b2|a2b1)AB.

If the two clicked photon detectors of photons C and D are in the even-parity polarization(spatial) mode |++CDP(S), the two-photon system AB is projected into [F′1|ϕ+P(S)ϕ+| + (1 − F′1)|ψ+P(S)ψ+|]AB. However, if the two clicked photon detectors of photons C and D are in the other even-parity polarization(spatial) mode |CDP(S), the two-photon system AB is projected into [F′1|ϕ+P(S)ϕ+| − (1 − F′1)|ψ+P(S)ψ+|]AB. Similarly, if the outcome of two clicked detectors is in the odd-parity spatial mode |+CDP(S) ( |+CDP(S)), a phase-flip operation σzP(S) performed on the photon B is required to obtain the state [F′1|ϕ+P(S)ϕ+| − (1 − F′1)|ψ+P(S)ψ+|]AB([F′1|ϕ+P(S)ϕ+|+(1 − F′1)|ψ+〉|P(S)ψ+]AB). The fidelity of the two-photon system AB is F′ = F′1F′2. The condition for F′ > F is that F1 > 0.5 and F2 > 0.5. The fidelity of the final hyperentangled states in polarization DOF and spatial-mode DOF increases rapidly after the hyperentanglement purification operation (see in Fig. 2 for the cases with F1 = F2 = f).

 figure: Fig. 2

Fig. 2 Fidelity of the hyperentangled photon-pairs in polarization DOF and spatial-mode DOF for the cases with F1 = F2 = f. F’ and F represent the fidelity after and before hyperentanglement purification operation, respectively.

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According to the above discussion, by picking up the case that Alice and Bob have the same parity in both polarization DOF and spatial-mode DOF mentioned above, the bit-flip errors and the vacuum errors can be eliminated successfully. However, in a practical system, there exists not only bit-flip errors but also phase-flip errors. In principle, the phase-flip errors can be transformed into bit-flip errors with local Hadamard operations in both the spatial-mode and the polarization DOFs. Therefore, our scheme can also be used to purify the mixed hyperentangled Bell states with phase-flip errors in two DOFs.

4. Discussion and summary

In this section, we will prove that, without the hyperdistillation operation, the relative weight of the vacuum component Pvac largely increases after general entanglement purification protocols, and the detection efficiency for quantum communication schemes decreases sharply. Suppose a entangled-photon source is located at middle point between Alice and Bob, and the entangled state can be written as |ϕ+ABP. The photons A and B are sent to Alice and Bob through two lossy channels a1 and b1 with two independent transmission coefficients T1 and T2, respectively (1 > T1 > 0 and 1 > T2 > 0). These noisy channels would deteriorate the pure entangled state AB to

ρABP=T1T2[f|ϕ+Pϕ+|+(1f)|ψ+Pψ+|]AB+(1T1)T2|vacAvac||+BP+|+T1(1T2)|+AP+||vacBvac|+(1T1)(1T2)|vacAvac||vacBvac|.
Here, the relative weight of the vacuum components of the photons AB is Pvac0=1T1T2, and the fidelity of the qubit-components of the photons AB is f.

In order to increase the fidelity f of mixed entangled system nonlocally, Alice and Bob perform an entanglement purification operation on this resource. Suppose there are four identical two-photon systems AB, A1B1, CD and C1D1 in the mixed entangled Bell-class state as described in Eq. (13), Alice first performs the most simple parity-checking measurements described in [29] on the polarization DOF of the photon pairs AA1 and CC1, respectively, with only linear optics. Simultaneously, Bob perform the same parity-checking measurements directly on the photon pair BB1 and DD1, respectively. Without the distilling operation, after the entanglement purification, the relative weight of the vacuum component of the new mixed entangled four-qubit system is Pvac = 1 − ξ2, where the value ξ could be described as [41]

ξ=T1T2[f2+(1f)2]T1T2[f2+(1f)2+(1T1)T2+(1T2)T1+(1T1)(1T2)(2f+1)].

Figure 3 depicts the relative weight of the vacuum components Pvac0 and Pvac as a function of the transmission T in the case of T1 = T2 = T for different f. From Fig. 3, we see that without the distillation operation, the relative weight of the vacuum component Pvac increases largely, and the detection efficiency for Alice and Bob decreases sharply. The low detection efficiency will lead to the detection loophole of quantum commutation. However, by using the setup described in Fig. 1, the same improved four-qubit entangled component is distilled in a heralded way with a certain possibility. The heralding signal allows one party to introduce a new input in his detectors only when a hyperentangled state is shared between the other. Hence the overall detection efficiency required to close the detection loophole does not depend on the transmission efficiency, but reduces to the intrinsic detection efficiency of Alices and Bobs detectors. That is, the effective transmission efficiency of photons can been boosted up to 100% in theory even where channel loss exists.

 figure: Fig. 3

Fig. 3 Relative weight of the vacuum component of the hyperentangled photon-pairs (Pvac or P′vac) as a function of the transmission coefficient T in the case T1 = T2 = T for different initial real parameters f.

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To mirror a likely experimental scenario, we numerically simulated P′vac with the detector efficiency ηd = 95%, the dark count probability D = 1 dark count/s, and the coupling efficiency between the photon-pair source and the optical fibers ηc = 0.9. We assume that the photon sources are excited with a repetition rate of 10 GHz, and the initial source of entanglement is the spontaneous-parametric down-conversion process with photon pair generation rate 2 × 10−3 [23]. The final relative weight of the vacuum component can be calculated as P′vac = 1 − R/(R + D), where R denotes the final key rate (per second) of hyperentanglement-based device-independent quantum key distribution (DIQKD) [42] after entanglement distribution, hyperdistillation and hyperentanglement purification operations. From Fig. 3, the relative weight of the vacuum component of the final signals decreases sharply with the hyperdistillation operation, and when T > 0.1, P′vac is below 0.05. Thus, our scheme for hyerdistilling and hyperentanglement purification in a heralded strategy provides us a powerful tool to overcome the problem of losses and decoherence, and it can be applied to all hyperentanglement-based quantum communication protocols.

In this study, we have proposed a hyperdistillation and hyperentanglement purification protocol using linear optics. The parity-check measurement in two DOFs could be realized by only one-step operation. Compared with the previous works using linear optics, our protocol can overcome the difficulties posed by inherent channel losses during photon transmission, simplify the complex measurement procedure, and optimise quantum communication process. With the feasible technologies, this scheme may be widely used in quantum communication and quantum networks.

Acknowledgments

This work was supported by the National Natural Science Foundation of China through Grants (No. 11404031 and No. 61471050), Beijing Higher Education Young Elite Teacher Project (No. YETP0456), the Fundamental Research Funds for the Central Universities (No. 2014RC0903). The project was supported by Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P. R. China.

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21. H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010). [CrossRef]  

22. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

23. T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014). [CrossRef]  

24. P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418 (1970). [CrossRef]  

25. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969). [CrossRef]  

26. A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).

27. 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 (1996). [CrossRef]   [PubMed]  

28. 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(1996). [CrossRef]   [PubMed]  

29. J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001). [CrossRef]  

30. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

31. W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003). [CrossRef]   [PubMed]  

32. M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999). [CrossRef]  

33. A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005). [CrossRef]   [PubMed]  

34. Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007). [CrossRef]  

35. S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014). [CrossRef]   [PubMed]  

36. Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014). [CrossRef]  

37. 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]  

38. H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001). [CrossRef]  

39. T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069 (2013). [CrossRef]  

40. 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]  

41. After the linear optical entanglement purification operation [23] in polarization DOF on the photon-pairs AB and A1B1, the final state of the resived photon-pair A′B′ is ρAB=1T1T2[f2+(1f)2]+(1T1)T2+(1T2)T1+(1T1)(1T2)(2f+1){(1T1)(1T2)(2f+1)2|vac,vacABvac|+(1T1)T24|vacAvac|[f|HBH|+(1f)|VBV|]+T1(1T2)4|vacAvac|[f|VBV|+(1f)|HBH|]+(1T1)T24|vacBvac|[f|VAV|+(1f)|HAH|]+T1(1T2)4|vacBvac|[f|HAH|+(1f)|VAV|]+T1T2[f2+(1f)2]2[f|ϕ+ABϕ++(1f)|ψ+ABψ+||]}, where f=f2f2+(1f)2.

42. N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010). [CrossRef]   [PubMed]  

References

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  2. M. Hillery and V. Bužek, “Quantum secret sharing,” Phys. Rev. A 59, 1829 (1999).
    [Crossref]
  3. 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]
  4. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881 (1992).
    [Crossref] [PubMed]
  5. X. S. Liu, G. L. Long, D. M. Tong, and Feng Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
    [Crossref]
  6. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991).
    [Crossref] [PubMed]
  7. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys. Rev. Lett. 68, 557 (1992).
    [Crossref] [PubMed]
  8. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
    [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. X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
    [Crossref]
  11. W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
    [Crossref]
  12. R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015).
    [Crossref]
  13. D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
    [Crossref] [PubMed]
  14. N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
    [Crossref] [PubMed]
  15. M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
    [Crossref]
  16. 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]
  17. B. C. Ren and G. L. Long, “General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities, ” Optics express 22, 6547 (2014).
    [Crossref] [PubMed]
  18. Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
    [Crossref]
  19. C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
    [Crossref]
  20. 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]
  21. H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010).
    [Crossref]
  22. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).
  23. T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
    [Crossref]
  24. P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418 (1970).
    [Crossref]
  25. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
    [Crossref]
  26. A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).
  27. 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 (1996).
    [Crossref] [PubMed]
  28. 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(1996).
    [Crossref] [PubMed]
  29. J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
    [Crossref]
  30. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).
  31. W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003).
    [Crossref] [PubMed]
  32. M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999).
    [Crossref]
  33. A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005).
    [Crossref] [PubMed]
  34. Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
    [Crossref]
  35. S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014).
    [Crossref] [PubMed]
  36. Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014).
    [Crossref]
  37. 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]
  38. H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001).
    [Crossref]
  39. T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069 (2013).
    [Crossref]
  40. 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]
  41. After the linear optical entanglement purification operation [23] in polarization DOF on the photon-pairs AB and A1B1, the final state of the resived photon-pair A′B′ is ρA′B′=1T1T2[f2+(1−f)2]+(1−T1)T2+(1−T2)T1+(1−T1)(1−T2)(2f+1){(1−T1)(1−T2)(2f+1)2|vac,vac〉A′B′〈vac|+(1−T1)T24|vac〉A′〈vac|⊗[f|H〉B′〈H|+(1−f)|V〉B′〈V|]+T1(1−T2)4|vac〉A′〈vac|⊗[f|V〉B′〈V|+(1−f)|H〉B′〈H|]+(1−T1)T24|vac〉B′〈vac|⊗[f|V〉A′〈V|+(1−f)|H〉A′〈H|]+T1(1−T2)4|vac〉B′〈vac|⊗[f|H〉A′〈H|+(1−f)|V〉A′〈V|]+T1T2[f2+(1−f)2]2[f′|ϕ+〉A′B′〈ϕ++(1−f′)|ψ+〉A′B′〈ψ+||]}, where f′=f2f2+(1−f)2.
  42. N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
    [Crossref] [PubMed]

2015 (1)

R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015).
[Crossref]

2014 (4)

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

T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014).
[Crossref] [PubMed]

Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014).
[Crossref]

2013 (4)

T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069 (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]

Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
[Crossref]

C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
[Crossref]

2012 (2)

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]

A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).

2010 (3)

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]

N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
[Crossref] [PubMed]

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010).
[Crossref]

2009 (2)

W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
[Crossref]

M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
[Crossref]

2007 (1)

Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

2006 (1)

X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[Crossref]

2005 (1)

A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005).
[Crossref] [PubMed]

2004 (1)

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]

2003 (1)

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003).
[Crossref] [PubMed]

2002 (6)

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

X. S. Liu, G. L. Long, D. M. Tong, and Feng 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 (2002).
[Crossref]

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

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

2001 (2)

J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
[Crossref]

H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[Crossref]

1999 (2)

M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999).
[Crossref]

M. Hillery and V. Bužek, “Quantum secret sharing,” Phys. Rev. A 59, 1829 (1999).
[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 (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 (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(1996).
[Crossref] [PubMed]

1992 (2)

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

1991 (1)

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

1970 (1)

P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418 (1970).
[Crossref]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Aschauer, H.

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003).
[Crossref] [PubMed]

Bell, J. S.

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

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

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys. Rev. Lett. 68, 557 (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]

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

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 (1996).
[Crossref] [PubMed]

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

Briegel, H. J.

A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005).
[Crossref] [PubMed]

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003).
[Crossref] [PubMed]

Brukner, C.

J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
[Crossref]

Bruss, D.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

Buhrman, H.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010).
[Crossref]

Bužek, V.

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

Cabello, A.

A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).

Cao, C.

T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

Cerf, N. J.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Cheong, Y. W.

Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Cleve, R.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010).
[Crossref]

Cui, J. X.

Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014).
[Crossref]

de Wolf, R.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010).
[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]

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]

Deng, F. -G.

X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[Crossref]

Deng, F.-G.

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]

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(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]

Dür, W.

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003).
[Crossref] [PubMed]

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(1996).
[Crossref] [PubMed]

Ekert, A. K.

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

Feng, G. R.

S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014).
[Crossref] [PubMed]

Gisin, N.

N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
[Crossref] [PubMed]

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

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Gräfe, M.

R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015).
[Crossref]

He, L. Y.

C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
[Crossref]

Heilmann, R.

R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015).
[Crossref]

Hillery, M.

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

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Horodecki, M.

M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999).
[Crossref]

Horodecki, P.

M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999).
[Crossref]

Hou, S. Y.

S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014).
[Crossref] [PubMed]

Jozsa, R.

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(1996).
[Crossref] [PubMed]

Karlsson, A.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Lee, H. -W.

Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

Lee, J.

Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

Lee, S. W.

Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

Li, Feng

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

Li, X. -H.

X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[Crossref]

Liu, J.

Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
[Crossref]

Liu, X. S.

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 Feng Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[Crossref]

Liu, Y.

Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014).
[Crossref]

Lo, H. K.

H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[Crossref]

Long, G. L.

S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014).
[Crossref] [PubMed]

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

T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069 (2013).
[Crossref]

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]

W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
[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]

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

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

Lu, Y.

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]

Ma, H. Q.

C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
[Crossref]

Macchiavello, C.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[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(1996).
[Crossref] [PubMed]

Massar, S.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010).
[Crossref]

Mermin, N. D.

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

Miyake, A.

A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Nolte, S

R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015).
[Crossref]

Pan, J. W.

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
[Crossref]

Pan, J. -W.

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]

Pearle, P. M.

P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418 (1970).
[Crossref]

Pironio, S.

N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
[Crossref] [PubMed]

Popescu, S.

H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[Crossref]

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, 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 (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(1996).
[Crossref] [PubMed]

Ren, B. C.

B. C. Ren and G. L. Long, “General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities, ” Optics express 22, 6547 (2014).
[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]

Ribordy, G.

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

Sangouard, N.

N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
[Crossref] [PubMed]

Sanpera, 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(1996).
[Crossref] [PubMed]

Schumacher, B.

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 (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]

Sciarrino, F.

A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).

Sheng, Y. B.

S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014).
[Crossref] [PubMed]

Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
[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]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Simon, C.

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
[Crossref]

Smolin, J. A.

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 (1996).
[Crossref] [PubMed]

Szameit, A

R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015).
[Crossref]

Tittel, W.

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

Tong, D. M.

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

Uskov, D. B.

M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
[Crossref]

Wang, C.

T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
[Crossref]

W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
[Crossref]

Wang, T. J.

T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069 (2013).
[Crossref]

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]

Wang, W. Y.

W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
[Crossref]

Wiesner, S. J.

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

Wilde, M. M.

M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
[Crossref]

Wootters, W. K.

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 (1996).
[Crossref] [PubMed]

Xiao, L.

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]

Zbinden, H.

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

Zelinger, A.

J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
[Crossref]

Zhang, G. Y.

W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
[Crossref]

Zhang, R.

C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
[Crossref]

Zhang, Y.

C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013).
[Crossref]

Zhao, S. Y.

Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
[Crossref]

Zhou, H. -Y.

X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[Crossref]

Zhou, L.

Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
[Crossref]

Chin. Sci. Bull. (3)

W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009).
[Crossref]

Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013).
[Crossref]

Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature(London). (1)

J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001).
[Crossref]

Optics express (1)

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

Phys. Rev. A (14)

M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
[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]

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

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

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

X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[Crossref]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999).
[Crossref]

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, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[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]

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]

H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[Crossref]

Phys. Rev. A. (1)

Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

Phys. Rev. D (1)

P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418 (1970).
[Crossref]

Phys. Rev. Lett. (11)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005).
[Crossref] [PubMed]

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003).
[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 (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(1996).
[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 (1992).
[Crossref] [PubMed]

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

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

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

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After the linear optical entanglement purification operation [23] in polarization DOF on the photon-pairs AB and A1B1, the final state of the resived photon-pair A′B′ is ρA′B′=1T1T2[f2+(1−f)2]+(1−T1)T2+(1−T2)T1+(1−T1)(1−T2)(2f+1){(1−T1)(1−T2)(2f+1)2|vac,vac〉A′B′〈vac|+(1−T1)T24|vac〉A′〈vac|⊗[f|H〉B′〈H|+(1−f)|V〉B′〈V|]+T1(1−T2)4|vac〉A′〈vac|⊗[f|V〉B′〈V|+(1−f)|H〉B′〈H|]+(1−T1)T24|vac〉B′〈vac|⊗[f|V〉A′〈V|+(1−f)|H〉A′〈H|]+T1(1−T2)4|vac〉B′〈vac|⊗[f|H〉A′〈H|+(1−f)|V〉A′〈V|]+T1T2[f2+(1−f)2]2[f′|ϕ+〉A′B′〈ϕ++(1−f′)|ψ+〉A′B′〈ψ+||]}, where f′=f2f2+(1−f)2.

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Figures (3)

Fig. 1
Fig. 1 The schematic diagram of the setup for QND parity-check measurement and qubit amplification. The inset shows the structure of detectors D. The setup consists of four input ports (marked with a1, a2, c1, c2), eight auxiliary input ports (marked with b1, b2, b3, b4, e1, e2, e3, e4), four output ports (marked with a′1, a′2, c′1, c′2), two polarizing beam splitters (PBS) which transmit a horizontally polarized photon |H〉 and reflect a vertically polarized photon |V〉. Successful operation of the distilling is heralded by eight-photon coincidence detection on the eight detectors D1, D2, D3, D4, D5, D6, D7 and D8.
Fig. 2
Fig. 2 Fidelity of the hyperentangled photon-pairs in polarization DOF and spatial-mode DOF for the cases with F1 = F2 = f. F’ and F represent the fidelity after and before hyperentanglement purification operation, respectively.
Fig. 3
Fig. 3 Relative weight of the vacuum component of the hyperentangled photon-pairs (Pvac or P′vac) as a function of the transmission coefficient T in the case T1 = T2 = T for different initial real parameters f.

Tables (1)

Tables Icon

Table 1 Relation between the final states of D1D2D3D4(D5D6D7D8) and the corresponding single qubit operations on the photon C′.

Equations (15)

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| ψ 0 AC = [ ( 1 P 1 ) | vac A vac | + P 1 | ψ A ψ A | ] [ ( 1 P 2 ) | vac C vac | + P 2 | ψ C ψ C | ] ,
| ψ A = ( α 1 | a 1 + β 1 | a 2 ) A S ( γ 1 | H + ξ 1 | V ) A P , | ψ C = ( α 2 | c 1 + β 2 | c 2 ) C S ( γ 2 | H + ξ 2 | V ) C P .
| ω = 1 2 [ ( | H V H V + | V H V H ) b 1 b 2 b 3 b 4 ( | H H H H e 1 e 2 e 3 e 4 ) + ( | V V V V ) b 1 b 2 b 3 b 4 ( | H V H V + | V H V H ) e 1 e 2 e 3 e 4 ] ,
| ψ A | ψ C | w PBS 1 , 2 ( α 1 γ 1 | H D 1 + β 1 γ 1 | H D 2 + α 1 ξ 1 | V D 8 + β 1 ξ 1 | V D 7 ) ( α 2 γ 2 | H D 3 + β 2 γ 2 | H D 4 + α 2 ξ 2 | V D 6 + β 2 ξ 2 | V D 5 ) 1 2 [ ( | H a 1 V D 2 H c 1 V D 4 + | V D 1 H a 2 V D 3 H c 2 ) | H D 5 H D 6 H D 7 H D 8 + | V D 1 V D 2 V D 3 V D 4 ( | H D 5 V c 1 H D 7 V a 1 + | V c 2 H D 6 V a 2 H D 8 ) ] .
χ ( α 1 α 2 γ 1 γ 2 | H a 1 H c 1 | H D 1 H D 2 H D 3 V D 4 H D 5 H D 6 H D 7 H D 8 + α 1 α 2 ξ 1 ξ 2 | V a 1 V c 1 | V D 1 V D 2 V D 3 V D 4 H D 5 V D 6 H D 7 V D 8 + β 1 β 2 γ 1 γ 2 | H a 2 H c 2 | V D 1 H D 2 V D 3 H D 4 H D 5 H D 6 H D 7 H D 8 + β 1 β 2 ξ 1 ξ 2 | V a 2 V c 2 | V D 1 V D 2 V D 3 V D 4 V D 5 H D 6 V D 7 H D 8 ) .
1 8 | ψ A C even ( | L D 1 D 2 D 3 D 4 | L D 5 D 6 D 7 D 8 | M D 1 D 2 D 3 D 4 | M D 5 D 6 D 7 D 8 ) + χ 8 [ ( | A + | B | C + | D ) ( | L D 1 D 2 D 3 D 4 | M D 5 D 6 D 7 D 8 | M D 1 D 2 D 3 D 4 | L D 5 D 6 D 7 D 8 ) + ( | A | B + | C | D ) ( | N D 1 D 2 D 3 D 4 | N D 5 D 6 D 7 D 8 | K D 1 D 2 D 3 D 4 | K D 5 D 6 D 7 D 8 ) + ( | A | B | C + | D ) ( | N D 1 D 2 D 3 D 4 | K D 5 D 6 D 7 D 8 | K D 1 D 2 D 3 D 4 | N D 5 D 6 D 7 D 8 ) + ( | A + | B + | C | D ) ( | L D 1 D 2 D 3 D 4 | N D 5 D 6 D 7 D 8 | M D 1 D 2 D 3 D 4 | K D 5 D 6 D 7 D 8 ) + ( | A + | B | C + | D ) ( | L D 1 D 2 D 3 D 4 | K D 5 D 6 D 7 D 8 | M D 1 D 2 D 3 D 4 | N D 5 D 6 D 7 D 8 ) + ( | A | B + | C + | D ) ( | N D 1 D 2 D 3 D 4 | L D 5 D 6 D 7 D 8 | K D 1 D 2 D 3 D 4 | M D 5 D 6 D 7 D 8 ) + ( | A | B | C | D ) ( | N D 1 D 2 D 3 D 4 | M D 5 D 6 D 7 D 8 | K D 1 D 2 D 3 D 4 | L D 5 D 6 D 7 D 8 ) ] ,
| ψ A C even = χ ( α 1 α 2 | a 1 c 1 + β 1 β 2 | a 2 c 2 ) A C ( γ 1 γ 2 | H H + ξ 1 ξ 2 | V V ) A C , | A = α 1 α 2 γ 1 γ 2 | H H A C | a 1 c 1 A C , | B = β 1 β 2 γ 1 γ 2 | H H A C | a 2 c 2 A C , | C = α 1 α 2 ξ 1 ξ 2 | V V A C | a 1 c 1 A C , | D = β 1 β 2 ξ 1 ξ 2 | V V A C | a 2 c 2 A C , | L = 1 2 ( | H H H H + | H V H V + | V H V H + | V V V V ) , | M = 1 2 ( | H H V V + | H V V H + | V H H V + | V V H H ) , | N = 1 2 ( | H H V H + | H V V V + | V H H H + | V V H V ) , | K = 1 2 ( | H H H V + | H V H H + | V H V V + | V V V H ) .
ρ AB = τ 1 [ F 1 | ϕ + P ϕ + | + ( 1 F 1 ) | ψ + P ψ + | ] AB [ F 2 | ϕ + S ϕ + | + ( 1 F 1 ) | ψ + S ψ + | ] AB + τ 2 | + A S + | | + A P + | | vac B vac | + τ 3 | vac A vac | | + B S + | | + B P + | + τ 4 | vac AB vac | ,
ρ CD = τ 1 [ F 1 | ϕ + P ϕ + | + ( 1 F 1 ) | ψ + P ψ + | ] CD [ F 2 | ϕ + S ϕ + | + ( 1 F 2 ) | ψ + S ψ + | ] CD + τ 2 | + C S + | | + C P + | | vac D vac | + τ 3 | vac C vac | | + D S + | | + D P + | + τ 4 | vac CD vac | ,
ρ ABCD = [ F 1 | Φ + P Φ + | + ( 1 F 1 ) | Ψ + P Ψ + | ] ABCD [ F 2 | Φ + S Φ + | + ( 1 F 2 ) | Ψ + S Ψ + | ] ABCD ,
| Φ + ABCD P = 1 2 ( | H H H H + | V V V V ) ABCD , | Ψ + ABCD P = 1 2 ( | H V H V + | V H V H ) ABCD , | Φ + ABCD S = 1 2 ( | a 1 b 1 c 1 d 1 + | a 2 b 2 c 2 d 2 ) ABCD , | Ψ + ABCD S = 1 2 ( | a 1 b 2 c 1 d 2 + | a 2 b 1 c 2 d 1 ) ABCD .
ρ ABCD = 1 16 { [ F 1 | ϕ + P ϕ + | + ( 1 F 1 ) | ψ + P ψ + | ] AB | + + CD P + [ F 1 | ϕ + P ϕ + | ( 1 F 1 ) | ψ + P ψ + | ] AB | CD P + [ F 1 | ϕ P ϕ | + ( 1 F 1 ) | ψ P ψ | ] AB | + CD P + [ F 1 | ϕ P ϕ | ( 1 F 1 ) | ψ P ψ | ] AB | + CD P } { [ F 2 | ϕ + S ϕ + | + ( 1 F 2 ) | ψ + S ψ + | ] AB | + + CD S + [ F 2 | ϕ + S ϕ + | ( 1 F 2 ) | ψ + S ψ + | ] AB | CD S + [ F 2 | ϕ S ϕ | ( 1 F 2 ) | ψ S ψ | ] AB | + CD S + [ F 2 | ϕ S ϕ | ( 1 F 2 ) | ψ S ψ | ] AB | + CD S } ,
| ϕ AB P = 1 2 ( | H H | V V ) A B , | ψ AB P = 1 2 ( | H V | V H ) A B , | ϕ AB S = 1 2 ( | a 1 b 1 | a 2 b 2 ) A B , | ψ AB S = 1 2 ( | a 1 b 2 | a 2 b 1 ) AB .
ρ AB P = T 1 T 2 [ f | ϕ + P ϕ + | + ( 1 f ) | ψ + P ψ + | ] AB + ( 1 T 1 ) T 2 | v a c A v a c | | + B P + | + T 1 ( 1 T 2 ) | + A P + | | v a c B v a c | + ( 1 T 1 ) ( 1 T 2 ) | v a c A v a c | | v a c B v a c | .
ξ = T 1 T 2 [ f 2 + ( 1 f ) 2 ] T 1 T 2 [ f 2 + ( 1 f ) 2 + ( 1 T 1 ) T 2 + ( 1 T 2 ) T 1 + ( 1 T 1 ) ( 1 T 2 ) ( 2 f + 1 ) ] .

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