Abstract
We propose a theoretical protocol for preparing fourphoton polarization entangled decoherencefree states, which are immune to the collective noise. With the assistance of the crossKerr nonlinearities, a twophoton spatial entanglement gate, two controlledNOT gates, a fourphoton polarization entanglement gate are inserted into the circuit, where X homodyne measurements are aptly applied. Combined with some swap gates and simple linear optical elements, fourphoton polarization entangled decoherencefree states which can be utilized to represent two logical qubits, 0〉_{L} and 1〉_{L} are achieved at the output ports of the circuit. This generation scheme may be implemented with current experimental techniques.
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. Most of the above applications require maximally entangled states or noiseless quantum channels. However, in a realistic situation, decoherence, induced by uncontrolled coupling between a quantum system and the environment, is inevitable. When qubits are coupled to the environment, the quantum superposition and coherence are easily destructed, and as a result the maximally entangled state collapses into a nonmaximally entangled one or even a mixed state. This will degrade the fidelity and security of quantum communication. To overcome this flaw, some specific entangled states, which are called decoherencefree states^{15,16,17}, are proposed. Decoherencefree states, no matter how strong the qubitenvironment interaction, exhibit some symmetry, so the quantum states are invariant under this interaction. Therefore, the decoherencefree states are very useful for longdistance quantum information transmission and storage.
Due to the fact that photons have the merits of higher speed, lower decoherence, easier manipulation, and lower energy cost compared with more massive qubits, polarization photons are destined to have a central role in longdistance communication. Recently, the encoding in decoherencefree states of polarization photons to overcome collective decoherence attracts the extensively attention. An optical experiment has been reported to overcome collective noise by encoding quantum information into the decoherencefree state^{18}. For two qubits, there is only one decoherencefree singlet state, i.e., , where H and V denote horizontal and vertical linear polarizations respectively. Therefore, it is not sufficient to fully protect the quantum information of an arbitrary logical qubit against collective noise. Another nontrivial example is the fourphoton polarization entangled decoherencefree state
with
The dimension of the above fourqubit decoherencefree state in Eq. (1) is 2, and thus it is sufficient to fully protect an arbitrary logical qubit against collective noise in contrast to the twoqubit state. With its interesting applications, Bourennane et al.^{19} have generated fourphoton polarization entangled decoherencefree states via a spontaneous parametric downconversion source. Recently, Zou et al.^{20} and Gong et al.^{21} proposed two different schemes to generate fourphoton polarization entangled decoherencefree states based on linear optical elements and postselection strategy. Subsequently, Xia et al.^{22} presented a protocol for the controlled generation of the fourphoton polarization entangled decoherencefree state with conventional photon detectors. In 2010, Wang et al.^{23} proposed a probabilistic linearopticsbased scheme for local conversion of four EinsteinPodolskyRosen photon pairs into fourphoton polarization entangled decoherencefree states. In 2013, Xia et al.^{24} also put forward a probabilistic protocol for preparation of fourphoton polarization entangled decoherencefree states with the help of the crossKerr nonlinearity medium.
In this paper, we present an alternative scheme to generate the fourphoton polarization entangled decoherencefree states with the assistance of the crossKerr nonlinearities. The states representing the logical qubits 0〉_{L} and 1〉_{L} can be achieved at different output ports of two beam splitters, combined with the output ports of the photon 3 and the photon 4. The rest of the paper is organized as follows. In Sec. II, we show how to generate these two logical qubits in the fourphoton polarization entangled decoherencefree states based on the weak crossKerr nonlinearities. The discussion and conclusion are presented in Sec. III.
Generations of fourqubit entangled decoherencefree states
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^{25}. The interaction Hamiltonian has the form , where 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)^{26,27}, deterministic entanglement distillation^{28}, logicqubit entanglement^{29,30}, generation of multiphoton entangled states and decoherencefree states^{24,31,32,33,34,35,36}.
In what follows, we explain the detailed procedures for generating the fourphoton polarization entangled docoherencefree states abided by the following processes, which is also illustrated in Fig. 1.
Assume the four single photons are initially prepared in the state , and let them enter into the circuit shown in Fig. 1 from the input ports. The first step is to create the spatial entanglement of the photons 1 and 2. Passing through beam splitters, BS_{1} and BS_{2}, which have the following function between two input modes (a,b) and two output modes (c,d): , , the photons (1, 2) enter into the paths (S_{11}, S_{12}) and the paths (S_{21}, S_{22}) respectively. Accompanying with the coherent state, the photons (1, 2) enter into Kerr media. Then, the state of photons (1, 2) with the coherent state α〉 evolves as
Performing an X homodyne measurement on the coherent state with α real, there are two measurement outcomes corresponding to scenarios of phase shift (0, ±θ). For the convenience of analysis, we expand the state in terms of the eigenstates of the X operator:
where the coefficients^{37} are
Explicitly, if zero phase shift occurs, the spatial entangled state of the photons (1, 2) is created and can be written as
Otherwise, another measurement outcome (nonzero phase shift) is obtained, a phase shift operation 2ϕ_{1}(x) should be performed on the photon 2 passing through the path S_{21} to erase the phase difference between two terms of and . By omitting a global phase ϕ_{1}(x), the photons (1, 2) are in the following spatial entangled state
Without considering other conditions, there is a small probability of error to distinguish the state in Eq. (8) and the state in Eq. (9) from each other due to the overlap of the measurement functions f(x, α) and f(x, α cos θ), which is given by , x_{d} = 2α(1 − cos θ). It is less than 10^{−5} when the distance x_{d} ~ αθ^{2} > 9^{26}.
For simplifying description in the later processes, we take an example as the representative of two different scenarios of phase shift. If zero phase shift is witnessed by the X homodyne measurement, half wave plates, HWP22.5°s, are inserted into the paths S_{11}, S_{22} at first, which function as Hadamard transformation operations to transform the state of the photons (1, 2) from and to and . Then two controlledNOT (CNOT) gates are performed on two paths (S_{11}, S_{21}) (photon 1 as control photon and photon 2 as target photon), and paths (S_{22}, S_{12}) (photon 2 as control photon and photon 1 as target photon) respectively. The CNOT gate is important in the experimental realization. Knill et al.^{38} firstly proposed a probabilistic CNOT gate on two photonic qubits by using linear optical elements and postselection. The crossKerr nonlinearity has also been used to implement the CNOT gate^{26,39,40}. After two CNOT gates, the fourphoton system will evolve into
A polarization entanglement gate illustrated in Fig. 2 is put into the paths (S_{11}, S_{12}, S_{21}, S_{22}, S_{3}, S_{4}) of the photons (1, 2, 3, 4) to entangle them with the polarization degree of freedom. Affected by crossKerr nonlinearities, the horizontal polarization mode of photons (1, 2) via the paths S_{11}, S_{12}, S_{21}, S_{22} will accumulate the phase shift θ, −θ respectively while the vertical polarization mode of photons (3, 4) will accumulate the phase shift θ, −θ respectively on the coherent state α〉. As the consequence of the nonlinear interaction between photons and the coherent state, the state of the whole system can be expressed as
After the photons leave Kerr media, the X homodyne measurement is performed on the coherent state. If zero phase shift occurs, no phase modulation is necessary. Otherwise, if nonzero phase shift of the coherent state presents on the measurement setup, a phase shift 2ϕ(x) operation should be performed on the photon 3 in the path S_{31}. Moreover, a HWP45° should be inserted into the path S_{3} to perform σ_{x} operation on the photon 3. So the fourphoton state can be denoted as
Here, two swap gates need to be performed on two photons in the paths (S_{21}, S_{3}) and (S_{22}, S_{4}) respectively to swap them. A swap gate is an important twoqubit logic gate. In terms of the basis of {00〉, 01〉, 10〉, 11〉}, the swap gate can be represented as the following matrix:
In practice, the swap gate transformation can be yielded by the HongOuMandel interference^{41} in the MachZehnder interferometer^{39,42}, illustrated in Fig. 3. Two beam splitters constitute a MachZehnder interferometer. Additionally, the phase shifter PS π denotes the phase shift π executed on the photon passing through the line it is inserted.
After two swap gate operations, the state denoted in Eq. (12) is changed to
Then, a local unitary operation σ_{y} should be performed on photon 3 and 4, respectively, which can be realized by the combination of a HWP45° and a HWP. So the above state can be denoted as
Due to the presence of BS_{3} and BS_{4}, the photons (1, 2) leave the paths (S_{11}, S_{12}) and the paths (S_{21}, S_{22}) to the paths and the paths according to the following rules, , , , . Correspondingly, at the output ports, the state of four photons expressed as Eq. (15) is transformed to
From the above equation, we can see that by detecting the outputs of the four photons, the logical qubit 0〉_{L} can be obtained with the total prabability of 25% at the output ports of or . As for the logical qubit 1〉_{L}, it can be obtained with the total prabability of 75% at the output ports of or .
As for another scenario, if we obtain the spatial entangled state denoted as Eq. (9), with a similar process, we can also obtain the fourphoton polarization entangled decoherencefree states. It is worth noting that the state denoted as Eq. (9) is the same as Eq. (8) when a swap gate is inserted into the path S_{21} and S_{22}, so far, the preparation of fourphoton polarization entangled decoherencefree states if fullfilled.
Discussion and Conclusion
We now give a brief discussion about the experimental feasibility of protocol with the current experimental technology. First of all, in the input ports, singlephoton resources are used. The complete technology of these single photons is yet to be established^{43,44,45,46}. Currently singlephoton sources in signal modes can be achieved from the collinear type II spontaneous parametric down conversion^{47}. As down conversion experiments are intrinsically probabilistic due to the statistical creation property of the photon pairs, the scheme will be in a sense probabilistic too in view of the usage of singlephoton sources. Thus, more efficient ones are demanded for our setup. Second, in the present scheme, two CNOT gates are performed, which can be realized following the refs 26,^{38,39,40}. However, these methods are at the best, nearly deterministic, so our scheme could be nearly deterministic. Third, in our protocol, we exploit the crossKerr nonlinearities medium in the spatial entanglement process and performing the polarization entanglement gate. It should be noted that in actual experiments, many factors will affect the perfect performance of crossKerr nonlinearities, such as dispersion, selfphase modulation, molecular vibrations in Kerr media, etc. Shapiro et al.^{48} analyzed the crossKerr nonlinear interaction and showed that singlemode crossKerr nonlinearities is not available for quantum information processing. Recently, GeaBanacloche^{49} pointed out that the large phase shifts via the giant Kerr effect with singlephoton wave packets is impossible at present. A proper physics systems providing larger strength of 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}. Finally, 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. (4) 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.^{51} investigated quantum measurement strategies capable of discriminating two coherent states using a homodyne detector and a photon number resolving (PNR) detector. 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 a theoretical protocol for preparing fourphoton polarization entangled decoherencefree states with the assistance of the crossKerr nonlinearity. In our protocol, combined with some swap gates and simple linear optical elements, a twophoton spatial entanglement gate, two CNOT gates, a fourphoton polarization entanglement gate are applied. We hope our work will afford facilities for other practical implementations of quantum information processing based on optics.
Additional Information
How to cite this article: Wang, M. et al. Generation of fourphoton polarization entangled decoherencefree states with crossKerr nonlinearity. Sci. Rep. 6, 38233; doi: 10.1038/srep38233 (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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