Abstract
We present two deterministic quantum entanglement distribution protocols for a fourphoton Dicke polarization entangled state resorting to the frequency and spatial degrees of freedom, which are immune to an arbitrary collectivenoise channel. Both of the protocols adopt the X homodyne measurement based on the crossKerr nonlinearity to complete the task of the singlephoton detection with nearly unit probability in principle. After the four receivers share the photons, they add some local unitary operations to obtain a standard fourphoton Dicke polarization entangled state.
Introduction
Entanglement^{1,2,3} plays an important role in quantum information processing, mainly including quantum computation^{4} and quantum communication. It is the information carrier in some interesting branches of quantum communication, such as quantum key distribution^{5}, quantum secret sharing^{6,7,8}, quantum secure direct communication^{9,10,11}, teleportation^{12}, quantum dense coding^{13,14}, and so on. In all of the above applications, two or multiqubit maximally entangled states must be shared as quantum channels by the parties at different locations. As photons are ideal carriers for longdistance transmission of quantum states, people always choose their entangled states in the polarization degree of freedom (DOF) to fulfill these tasks discussed previously. However, during a practical transmission, the polarization DOF of photons is easily influenced by the thermal fluctuation, vibration, and the imperfection of an optical fiber. That is, they suffer from the channel noise inevitably whether they are single photons or multiqubit entangled photons. Thus, various error correction and errorrejection processes are proposed. A method of distilling a maximally entangled state is entanglement purification, which can be used to decrease the influence of the channel noise and will be an efficient method in the case that the distance is not so long. For instance, Bennett et al.^{15} proposed an original entanglement purification protocol (EPP) for purifying a Werner state based on quantum controlledNOT gates in 1996. In 2001, Pan et al.^{16} proposed an EPP based on linear optics, without resorting to controlledNOT gates, which is feasible in experiment. Sheng et al.^{17} proposed an EPP based on crossKerr nonlinearity. However, entanglement purfication is essentially used to achieve a subset of maximally entangled states from lessentangled ones after infinite operations. All conventional EPPs cannot get perfect maximally entangled photons by far as they work probabilistically in principle. Thus, the faithful distribution of maximally pure entangled states between different and distant locations is valuable for the realization of longdistance quantum communication.
The polarization entanglement of photons^{18,19,20} is easily disturbed by the noise in quantum channel, so it is not an elegant way to directly transmit the polarization entanglement of photons over a noisy channel. Recently, some other DOFs attract much attention, such as the frequency DOF, the spatial mode, orbital angular momentum and so on. For example, since the bitflip error and phaseflip error can be corrected correspondingly as the frequency entanglement state is a maximally entangled pure state, and the frequency entanglement does not easily suffer from the channel noise in principle^{21}, Sheng and Deng^{22} proposed an EinsteinPodolskyRosen pair distribution protocol over an arbitrary collectivenoise channel exploiting conversion between polarization and frequency modes. The protocol can be generalized to the distribution of nqubit (n > 2) GreenbergerHorneZeilinger (GHZ) state. This protocol is very important because multiqubit entangled states have many advantages over the twoqubit entangled states in quantum information. Since then, in 2011, Lu et al.^{23} proposed an efficient W polarization entangled state distribution protocol over an arbitrary collectivenoise channel with the help of the crossKerr nonlinearity. In 2013, Dong et al.^{24} rendered a perfect entanglement distribution protocol of a fourphoton χtype polarizationentangled state exploiting spatial DOF to depress the effect of collective noise. For fourqubit entangled states, Verstraete et al.^{25} showed that there are nine families of states under stochastic local operations and classical communication, such as the above mentioned fourqubit χtype states, fourqubit GHZ states, fourqubit W states, and fourqubit cluster states. However, other nonequivalent classes of quantum states with interesting symmetries exist. For example, a novel fourqubit entangled state —the fourqubit Dicke state with two excitations that is symmetric under all permutations of qubits^{26}, has the form as
where H and V denote horizontal and vertical linear polarizations respectively. The fourphoton state , like W〉_{4}, is highly persistent against photon loss and projective measurements. In particular, Kiesel et al.^{27} showed that, in spite of the impossibility to transform a three photon GHZ type into a W state by local manipulation, both can be obtained via a projective measurement of the same photon in the state . Dicke states constitute a particularly relevant class of highly entangled, they have interesting applications in quantum information processing tasks, such as 1 → (N − 1) telecloning or opendestination teleportation^{28,29} and quantum games^{30,31}. Experimentally, highfidelity Dicke states with small particle numbers have been created with photons^{32,33}.
In this paper, we present two deterministic quantum entanglement distribution protocols for fourphoton Dicke state in polarization over an arbitrary collectivenoise channel with the help of the crossKerr nonlinearity. The two protocols exploit the frequency DOF and spatial DOF to against channel noise respectively, which are immune to an arbitrary collectivenoise channel. Both of these two protocols adopt the X homodyne measurement based on the crossKerr nonlinearity to complete the task of the singlephoton detection with nearly unit probability in principle. After the four receivers share the photons, they add some local unitary operations to obtain a standard fourphoton Dicke polarization entangled state deterministically. We describe the explicit distribution scheme of the Dicke polarization entangled state with frequency entanglement in Section 2. The deterministic quantum entanglement distribution with spatial entanglement is shown in Section 3. Finally, the discussion and conclusion are presented in Section 4.
Entanglement distribution of the Dicke state by frequency degree of freedom
For the sake of the clearness, let us first introduce the crossKerr nonlinearity, which was first used by Chuang and Yamamoto to realize the simple optical quantum computation^{34}. The interaction Hamiltonian has the form , here is the photonnumber operators of the signal (probe) mode, and κ is the strength of the nonlinearity. If the signal field contains n photons and the probe field is in an initial coherent state with amplitude α, the crossKerr nonlinearity interaction causes the combined signalprobe system to evolve as follows:
where θ = κt with t being the interaction time. It is easy to observe that the Fock state is unaffected by the interaction but the coherent state picks up a phase shift nθ directly proportional to the number of photons n in the signal mode. One can exactly obtain the information of photons in the Fock state but not destroy them by detecting the probe mode with a general homodyneheterodyne measurement. The crossKerr nonlinearity between photons offers an ideal playground for quantum state engineering, and a number of applications have been studied, such as constructing nondestructive quantum nondemoliton detectors (QND)^{35,36}, deterministic entanglement distillation^{37}, logicqubit entanglement^{38,39}, generation of multiphoton entangled state^{40,41,42,43,44}. In what follows, we explain the distribution process of the fourphoton Dicke state over an arbitrary collectivenoise channel with frequency DOF. On experiment, with present technology, the entanglement of photons in frequency DOF is not difficult to be prepared with spontaneous parametric downconversion^{45,46}. Frequency DOF has been used in a series of quantum information schemes because of its stability. We suppose that the center, say Susan wishes Alice, Bob, Charlie and David to share a polarization photon state as described in Eq. (1). By means of method in refs 45,46 she prepares a fourphoton Dicke state
where, the notions ω_{1}ω_{1}ω_{2}ω_{2}〉, ω_{1}ω_{2}ω_{1}ω_{2}〉, , ω_{2}ω_{2}ω_{1}ω_{1}〉 are six different frequency modes of the four photons. The subscripts A, B, C, and D mean that the four photons are distributed to Alice, Bob, Charlie and David, respectively. Suppose the collective noises in the four channels have the same form but different noise parameters which alter with time in principle, i.e.,
where α_{i}^{2} + β_{i}^{2} = 1 (i = 1, 2, 3, 4). The fourphoton entangled state in Eq. (3) suffering from collectivenoise channels is written as
After the noise channels, the resulting photons will pass through the polarization beam splitters (PBSs) which transmit the horizontal polarization mode H〉 and reflect the vertical polarization mode V〉. When Alice, Bob, Charlie and David combine their photons and their coherent probe beams with crossKerr nonlinearity media (shown in Fig. 1), the state with the four coherent states evolves as
If a photon appears in the mode 1 (3, 5 or 7), the crossKerr nonlinearity puts a phase shift θ on the coherent state α〉_{A} (α〉_{B}, α〉_{C} or α〉_{D}). Meanwhile, if a photon appears in the mode 2 (4, 6 or 8), the coherent state α〉_{A} (α〉_{B}, α〉_{C} or α〉_{D}) picks up no phase shift. After the X homodyne measurements on their coherent beams independently, Alice, Bob, Charlie and David will get some different phase shifts, and the four photons will collapse into different states with different phase shifts. For example, if Alice, Bob, Charlie and David have the same phase shift θ, the four photons will collapse into the state
which appears at output modes 1357 with the probability of α_{1}α_{2}α_{3}α_{4}^{2}. In a similar way, the other fifteen entangled states ϕ_{i}〉_{ABCD}: 2 ≤ i ≤ 16 can be distinguished. Here
The next step of our Dicke state disrtibution protocol is to convert the frequency DOF entangled states {ϕ_{i}〉_{ABCD}} to polarization DOF entangled ones. Without loss of generality, we take ϕ_{1}〉_{ABCD} as an example to describe the principle of the entanglement reconstructing process, shown in Fig. 2. Four polarization independent wavelength division multiplexers (WDMs) can be used to guide photons to different spatial modes according to their frequencies. That is, photons with the frequencies ω_{1} and ω_{2} will be guided to the corresponding spatial modes a_{1}(b_{1}, c_{1}, d_{1}) and a_{2}(b_{2}, c_{2}, d_{2}). Four halfwave plates (HWPs) are used to complete the transformation in suitable positions. After the four photons are coupled by the four PBSs, its state becomes
and will be in the output spatial modes a_{1}, b_{1}, c_{1} and d_{1}. The remaining states in {ϕ_{i}〉_{ABCD} : 2 ≤ i ≤ 16} are analogical with it. Following the similar way, Alice, Bob, Charlie and David can obtain 16 maximally entangled states in the polarization and frequency degrees of freedom. Finally, four participants can erase the distinguishability for the frequency of their photons with the help of quantum frequency upconversion^{47} and turn them into a standard polarization entangled Dicke state with local unitary operations. The collapsed states corresponding to the X homodyne measurement, together with the explicit output modes, the corresponding probabilities, and the corresponding local operations on photons can be seen in Table 1. The Dicke state distribution protocol has been successful and the success probability is . That is, this entanglement distribution process can be implemented with a unit probability in principle over collectivenoise channels since it is independent of the noise parameters {α_{1}, β_{1}, α_{2}, β_{2}, α_{3}, β_{3}, α_{4}, β_{4}}.
Entanglement distribution of the Dicke state by spatial degree of freedom
Besides the frequency DOF, the spatial DOF is also robust to the channel noise. As for the spatial entanglement, the bitflip error does not exist and the phaseflip error can be eliminated by controlling the lengths of channels exactly, so the spatial DOF can also be used to create the entanglement in polarization DOF. In this section, we demonstrate another scheme for distribution of entanglement with the spatial entanglement.
Suppose the center Susan has a polarization entangled Dicke state shown in Eq. (1), which can be generated by means of method in refs 32,33. First, she transforms the entangled mode of the Dicke state from the polarization DOF mode to the spatial DOF mode, by utilizing the combination of PBSs and HWPs, as shown in Fig. 3. Then the state to be transmitted in Susan’s is changed to
with local unitary operations. Here a, b, c, and d represent the four spatial modes of the entangled system. Subsequently, the four photons are transmitted through the collectivenoise channels. Here we suppose channels x_{1} and x_{2} (x = a, b, c, d) are so close that the noise affected on photons are identical, which is given as
where α_{i}^{2} + β_{i}^{2} = 1 (i = 1, 2, 3, 4). The state denoted as Eq. (10) suffered from collectivenoise channels is evolved to
Finally, the four photons arrive at receivers Alice, Bob, Charlie and David. In the receiving process, every receiver introduces a probe beam with crossKerr nonlinearity media (shown in Fig. 4) and then performs the X homodyne measurement independently. As a result, the polarization part of the fourphoton Dicke state collapses into HHHH〉, HHHV〉, with the probabilities of α_{1}α_{2}α_{3}α_{4}^{2}, α_{1}α_{2}α_{3}β_{4}^{2}, . In the end, the four photons pass through the combination of HWP and PBSs, and the transmitted state can be transformed to the original Dicke state with some local unitary operations. All X measurement outcomes (polarization states), together with the explicit output and the corresponding local operations on photons can be seen in Table 2. The successful distribution of the fourphoton polarization entangled Dicke state is confirmed and the total probability is P = α_{1}α_{2}α_{3}α_{4}^{2} + α_{1}α_{2}α_{3}β_{4}^{2} + + β_{1}β_{2}β_{3}β_{4}^{2} = 1.
We can generalize the above scheme to the case for distribution of 2nqubit system (n > 2) in a highly entangled and symmetric Dicke state over an arbitrary collectivenoise channel. The highly symmetric Dicke state is given by
where is a normalization factor with as binomial coefficient, and ∑_{σ} P_{σ}(…) means the sum over all permutations of the photonic qubits. Through the setup shown in Fig. 3, the state to be transmitted is transformed into
where i, j, k, … ∈ 1, 2. After passing through the collectivenoise channels, the 2nqubit system evolves as
Similar to Fig. 4, every receiver use their QNDs with crossKerr nonlinearity to check the polarization part of the 2nphoton Dicke state, and then, the 2n photons pass through the combination of HWP and PBSs. Finally the transmitted state can be transformed to the original Dicke state with some bitflip operations on part of the 2n photons.
Discussion and Conclusion
In the process of describing the principle of our protocols, we exploit the crossKerr nonlinearity interaction between photons and the coherent states. Although a lot of works have been studied in the area of crossKerr nonlinearities, we should acknowledge that it is still a quite controversial concept to have a clean crossKerr nonlinearity in the optical singlephoton regime with present science and technology. In nature, crossKerr nonlinearity is extremely small and unsuitable for single photon interaction. Fortunately, it was suggested that the nonlinearity magnitude could be θ ~ 10^{−2} with the help of electromagnetically induced transparency^{48}. Moreover, in the regime of weak crossKerr nonlinearity, the demanded strength of nonlinearity can be compensated by using a probe coherent state with very large amplitude. For a realistic system, however, the intensity of the coherent beam cannot be boundlessly large because a laser beam with too strong intensity will bring about other effects in Kerr medium due to the effects of decoherence. Recently, as pointed out by GeaBanacloche^{49}, the large phase shifts via the giant Kerr effect with singlephoton wave packets is impossible at present. A proper candidate for weak crossKerr nonlinearity should be atomic ensemble, and the fundamental problem with the crossKerr nonlinearity in atomic ensemble was discussed by GeaBanacloche^{49}, and He and Scherer^{50}.
Besides the influence of crossKerr medium, the experiment feasibility of the present protocols also depends on the veracity of the X homodyne measurement. For the X homodyne measurement, we only consider the error chiefly coming from the overlap adjacent curves because of the fact that the coherent states of the probe beam with different phase shifts are not completely orthogonal. In fact, it is only one type of detection error in homodyne, other errors, such as the noises in detection, the reduced fidelity to the process in Eq. (2) due to multimode effect and decoherence, etc., also exist in a realistic implementation. Exploiting the appropriate measurement methods, the disadvantageous influence can be overcome or alleviated and the error probability will be decreased. In 2010, Wittmann et al. investigated quantum measurement strategies capable of discriminating two coherent states using a homodyne detector and a photon number resolving (PNR) detector^{51}. In order to lower the error probability, the postselection strategy is applied to the measurement data of homodyne detector as well as a PNR detector. They indicated that the performance of the new displacement controlled PNR is better than homodyne receiver.
To summarize, we have proposed two quantum entanglement distribution protocols of a fourphoton Dicke polarization entangled state over collectivenoise channel with the assistance of the crossKerr nonlinearity. During transmission, the polarization DOF is easily disturbed over the collectivenoise channel, we perform the entanglement conversion between polarization DOF and frequency DOF to eliminate the effect collective noise in the first quantum entanglement distribution protocol. After transmission, four receivers exploit the polarization independent WDMs to guide photons to different spatial modes according to their frequencies. At the same time, we fulfill another quantum entanglement distribution with the application of the spatial entanglement, which affords facilities for the experimental implementation and application because of releasing the polarization independent WDMs. These two protocols can be implemented deterministically in principle. By virtue of the availability of optical elements and techniques involved, we hope the present schemes for the distribution of entangled state in the polarization DOF can be experimentally implemented in the longdistance quantum communications.
Additional Information
How to cite this article: Wang, M.Y. et al. Deterministic distribution of fourphoton Dicke state over an arbitrary collectivenoise channel with crossKerr nonlinearity. Sci. Rep. 6, 29853; doi: 10.1038/srep29853 (2016).
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos: 11475054, 11371005, and Hebei Natural Science Foundation of China under Grant Nos: A2014205060, A2014205064 and A2016205145.
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College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China
 MeiYu Wang
 & FengLi Yan
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China
 Ting Gao
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M.Y.W., F.L.Y. and T.G. contributed equally to this work. All authors wrote the main manuscript text and reviewed the manuscript.
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The authors declare no competing financial interests.
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Correspondence to FengLi Yan or Ting Gao.
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