Entanglement concentration for arbitrary four-particle linear cluster states

Cluster states, whose model are a remarkably rich structure in measurement-based quantum computation, hold high degree of entanglement, while entanglement is very fragile during the process of transmission because of the inevitable interaction with the environment. We propose two entanglement concentration protocols for four-particle linear cluster states which and are susceptible to the decoherence and the imperfect communication setups. In the first protocol, POVM operators are introduced to maximize the success probability, and the second protocol is based on cross-Kerr nonlinearity which is utilized to check the parity between the original particle and the ancillary particle. Both of the protocols have their own advantages. The first one can be easily realized in experiment by linear optics, while the one with cross-Kerr nonlinearity reach more than 90% success probability by iteration. Since the wide application of cluster states, the two protocols are efficient and valuable to different fields of quantum communication.

ECP with linear optics. In order to distill perfect cluster state by linear optics from the state in Eq. (2), local operations are needed. Any local POVM operations performed on particle i is as where a i , b i , c i and d i are all real number and satisfy |a i | + |c i | ≤ 1 and |b i | + |d i | ≤ 1. If Alice, Bob, Charlie and Daniel are only permitted to perform local operations on the particles hold by themselves, the state |Ψ〉 1234 will be changed into L 1 ⊗ L 2 ⊗ L 3 ⊗ L 4 |Ψ〉 1234 . After four local POVM operations, the ECP for four-particle LCS with linear optics is finished, which means that the whole system hold by four parties is hoped to be the cluster state in Eq. (1). Thus the coefficients of terms in the final state should satisfy the following basic conditions, 0, [1,16] \{1, 4, 13, 16}, where the coefficient of ith term is denoted as f i . According to the conditions, four parties can find the relationship between a i , b i , c i , d i (i = 1, 2, 3, 4) and λ i (i = 1, 2, 3, 4), then they would know the detailed operations performed by themselves to concentrate the imperfect cluster state. In order to describe the process of ECP with linear optics clearly, we consider a special case, when there is no operation on particle 4, i.e. L 4 = I = |H〉〈H| + |V〉〈V| which means a 4 = d 4 = 1 and b 4 = c 4 = 0. We show how to find the unknown parameters in the other three POVMs those can concentrate the entanglement in |Ψ〉 1234 of Eq. (2) with the maximum success probability. The final system without normalized is changed into |Ψ′〉 1234 = L 1 ⊗ L 2 ⊗ L 3 ⊗ I|Ψ〉 1234 , which also has the form of cluster state in Eq. (1) after normalized. If the coefficients are denoted as f′(i), i ∈ [1, 16] Z , they satisfy the conditions The success probability that four parties transform |Ψ〉 1234 of Eq. (2) into |ψ〉 1234 of Eq. (1) is P = |f′ 1 | 2 + |f′ 4 | 2 + |f′ 13 | 2 + |f′ 16 | 2 = 4|f′ 1 | 2 . Now our aim to concentrate the arbitrary four-particle cluster states can be divided into two steps. The first step is to solve the parameters a i , b i , c i , d i (i = 1, 2, 3) with respect to λ i (0 ≤ i ≤ 3) according to the conditions in Eq. (5). The second step is to maximize the success probability P. We solve them one by one as follows.
There are three kinds of relationship satisfying the conditions in Eq. (5), and the detailed process is shown in Supplementary Material. The first solution is λ 1 λ 2 − λ 0 λ 3 = 0. The second kind is Scientific RepoRts | 7: 1982 | DOI:10.1038/s41598-017-02146-9 1  1  2  2  3  3   1 1 2 2 3 3   2 3  3 3   2 1 2  0 1 2   1 2  0 3 and the third solution is Secondly, following the three solutions of the relationship between λ i and the coefficients of L i , we maximize the success probability of each solution to obtain the detailed POVMs operators. The maximization of the local probabilities implies that the constraints should be satisfied and that the state is transformed into the cluster state with the maximized success probability. The first solution with less constraints is hard to obtain the particular POVMs, so we take the second solution as an example to show how to maximize the success probability (Actually, the maximization of the success probability with the third solution is similar with that with the second solution.). Under the conditions in Eqs (6) and (9), the parameters with λ λ   with the success probability P = 4|λ 0 | 2 , if there exists λ 0 λ 3 = −λ 1 λ 2 . That means if Daniel doesn't perform any operations, Alice, Bob and Charlie can concentrate the entanglement of |Ψ〉 1234 into |ψ〉 1234 with the success probability P = 4|λ 0 | 2 by performing the local operations L 1 , L 2 and L 3 in Eq. (10) respectively. Furthermore, in the cluster state, particle 1 is symmetric with particle 2, so the solutions for maximizing success probability can be interchanged over particle 1 and particle 2. At the same time, particle 3 is symmetric with particle 4, thus the POVM operations over particle 3 and particle 4 can also be interchanged. Considering the symmetry over particle 1 and particle 2 (particle 3 and particle 4), only two parties from four perform local POVMs and distill the cluster state in Eq. (2) into |ψ〉 1234 . As the process that obtaining the parameters in three local POVMs, we can get one of solutions in the case is All the solutions of the ECP can be implemented by linear optics, polarization beam splitter and rotated operations, which is practical and economical. We take the solution in Eq. (11) as an example to show how to implement an ECP by linear optics. The schematic drawing is as Fig. 1.
According to the principle of our last entanglement concentration protocol with λ 1 λ 2 = −λ 0 λ 3 , only particle 2 and particle 3 are operated by some local POVM operations. After through PBS 1 (PBS 4 ), the vertical component in particle 2 (3) is rotated by R 2 (R 1 ). The wave plate R i is used to rotate |V〉 with an angle . After the vertical component of particle 2 and that of particle 3 pass through wave plates R 2 and R 1 , respectively, the state in Eq. (2) is changed into Then the vertical component of particle 2 and that of particle 3 in |Φ〉 1234 are reflected by PBS 2 and PBS 5 , respectively, while both of the horizontal components arrive the detectors. In theory, Alice can judge the protocol succeeds or not, according to the response of detectors. If particle 2 or particle 3 reaches detector D 2 or D 1 , the ECP protocol fails. When particle 2 and particle 3 pass through PBS 2 , PBS 3 , PBS 5 and PBS 6 , the whole state is transformed into |ψ〉 1234 in Eq. (1) with the success probability 4|λ 0 | 2 .
ECP with cross-Kerr nonlinearity. The section introduces the other way to concentrate the entanglement of the four-particle state in Eq. (2) with cross-Kerr nonlinearity 35,36 , which is based on the quantum Figure 1. Schematic drawing of ECP for a four-particle cluster state with linear optics. PBS represents a polarizing beam splitter, which transmits the particle in the horizontal polarization |H〉 and reflects the particle in the vertical polarization |V〉. R i represents a wave plate which can rotate the vertical polarization |V〉 with an angle θ i = arccos (λ 0 /λ i ). Symbol D 1 and D 2 are the single-photon detectors.

Figure 2.
Schematic drawing of ECP for a four-particle cluster state with ancillary particles. PCD means the "parity checking device" which distinguishes the parity between particle i and ancillary particle i′ using cross-Kerr nonlinearity. PBS i represents a polarizing beam splitter, which transmits the particle in the horizontal polarization |H〉 and reflects the particle in the vertical polarization |V〉. R i represents a wave plate which represents a Hadamard operation on the ancillary single particle.
Scientific RepoRts | 7: 1982 | DOI:10.1038/s41598-017-02146-9 non-demolition detection. The ECP for four-particle cluster state with cross-Kerr nonlinearity improves the success probability by iteration. The principle is shown in Fig. 2. Suppose Alice, Bob, Charlie and Daniel hold the particles 1, 2, 3 and 4 respectively. Firstly, Alice prepares an , so the whole system is  where λ 0 λ 3 = −λ 1 λ 2 is applied in the second equation. Based on the setups in Fig. 3 35 , Alice checks the parity on particle 1 and particle 1′, and measures the particle 1′ in the diagonal basis ± = ± H V ( ) 1 2 . If the measurement result is |+〉 (|−〉), Alice operates I (σ Z = |H〉 〈H| − |V〉〈V|) on particle 1. Then according to the output of PCD (parity checking device), the system is divided into two classes. After Alice's operations, the normalized system with even parity is The total system is in Alice makes particle 1′′ and particle 1 go through the PCD, measures particle 1′′ with the basis {|+〉, |−〉} and operates I or σ Z according to the measurement results of particle 1′′. If the output of PCD is even, the step is successful, otherwise the step fails, and Alice has to prepare the third ancillary particle and iterates above steps until the parity checking result is even. After two rounds, the probability of failure (i.e., the probability that the parity checking result is odd) is The success probability in the second round is  Figure 3. Schematic drawing of PCD operated on the original particle 1 and the ancillary particle 1′ 35 . ±θ represents that cross-Kerr nonlinearity makes |α〉 into |αe ±iθ 〉 when there is a particle passing. The even-parity states |HH〉 and |VV〉 will introduce phase shift ±θ to |α〉, while the odd-parity states |HV〉 and |VH〉 result in no phase shift. |χ〉〈χ| is the homodyne measurement that can distinguish different phase shifts.    14), here denote the system after Alice's successful operations as φ Ae 1234 , and Charlie does not need to know the number of rounds Alice operates.
After told that Alice's steps are successful, Charlie continues to do the concentration. At the beginning, Charlie prepares an ancillary particle 3′ in the state into two classes, with the probability If the parity checking result is even, the system is in a perfect four-particle cluster state |ψ〉 1234 . If the parity checking result is odd, Charlie has to do another round to obtain the perfect four-particle cluster state. The success probability in the second round is Scientific RepoRts | 7: 1982 | DOI:10.1038/s41598-017-02146-9 which depends on the coefficients of the initial state |Ψ〉 1234 and the numbers of iterations performed by Alice and Charlie.

Discussion
We introduce two ways to concentrate the entanglement from an arbitrary four-particle LCS |Ψ〉 1234 = λ 0 |H-HHH〉 1234 + λ 1 |HHVV〉 1234 + λ 2 |VVHH〉 1234 + λ 3 |VVVV〉 1234 . The first ECP is realized by a series of PBSs and two rotate operations, and the success probability is 4|λ 0 | 2 if the coefficients of |Ψ〉 1234 satisfy λ 0 λ 3 = −λ 1 λ 2 . The visible relationship between the success probability and the parameter |λ 0 | 2 is shown in Fig. 4(a). Apparently, the success probability is 4 times the parameter |λ 0 | 2 . Furthermore, the wave plates are imperfect in experiment, so we discuss the affection of accuracy of the wave plates on the concentration. Ignored the global phases, we consider the number of possible initial cluster states that can be concentrated by the ECP with linear optics, and simulate the probability distributions of the number with the parameter |λ 0 | 2 in Fig. 4(b) if the accuracies of the wave plate in Fig. 1 are 1/10 3 and 1/10 4 . In Fig. 4(b), the number of initial states that can be concentrated by the ECP with linear optics decreases with |λ 0 | 2 increasing. The higher the accuracy of the wave plates, the smoother the distribution of the number of the possible states. Besides only using linear optics, another advantage of the ECP is that the scheme doesn't need any ancillary particles. The only compromise is that the results need postselection. In Fig. 1, Bob and Charlie should observe that whether the detectors D 1 and D 2 click or not. Any detector clicks, the ECP with linear optics fails, else it succeeds. Thus the detection efficiency of the detectors in practice also affects that whether the ECP is successful or not. In Section II, we suppose the detection efficiency of both detectors D 1 and D 2 are 100%. However, the single-photon detectors are imperfect. The detection efficiency cannot reach 100%, and there exists dark counts in experiment. Therefore, more practical concentration of the entanglement for four-particle cluster states should be studied in the future. The second ECP for four-particle LCSs is realized via cross-Kerr nonlinearity which can check the parity between the original particle and the ancillary particle. Compared with the first ECP protocol, two particles of the original state in the second ECP with cross-Kerr nonlinearity is reentered the devices again and again until the whole system is in a perfect cluster state. The iteration increases the final success probability, which is related with four parameters, the number of Alice's iterations m, the number of Charlie's iterations n, and the coefficients |λ 0 | 2 and |λ 2 | 2 of |Ψ〉 1234 . No matter how many iterations Alice does, the whole systems before Charlie operates are in the same states. Thus the number of Charlie's iterations is independent with the number of Alice's iterations. When the number of Charlie's iterations is fixed as n = 1, the success probabilities as a function of the coefficients |λ 0 | 2 and |λ 2 | 2 are shown in Fig. 5. Figure 6(a-c) give the results when that of Alice's iterations is fixed as m = 1, and Fig. 6(d) shows the success probabilities when both of m and n are equal to 4. According to the simulation, we obtain the following conclusions: (i) With the parameters |λ 0 | 2 and |λ 2 | 2 increasing, the success probabilities increase. (ii) Both of Alice's iterations and Charlie's iterations can efficiently increase the success probabilities. (iii) The influence degree of Alice's iterations on the success probabilities is similar as that of Charlie's iterations. (iv) After 4 Alice's iterations and 4 Charlie's iterations, the success probabilities would reach more than 90%.
Compared with the first ECP with linear optics in Fig. 4(a), the second ECP with cross-Kerr nonlinearity in Fig. 6(d), though it is more difficult to be realized, would reach higher success probabilities for the same parameter |λ 0 | 2 . Besides the iteration increasing the success probabilities, the reason is that the success probability of the first ECP with linear optics is P = 4|λ 0 | 2 which should satisfy the hypotheses |λ 0 | ≤ |λ 1 | and |λ 0 | ≤ |λ 2 |. To sum up, we introduce two ECPs for four-particle LCSs, one with linear optics, the other with cross-Kerr nonlinearity. The first ECP is easily realized in experiment, while the success probability in the second one can reach more than 90% after 4 Alice's iterations and 4 Charlie's iterations. The wide application of cluster states makes our two ECPs play different important roles on quantum communication.