# One-step error correction for multipartite polarization entanglement^{1}^{1}1Published in Phys. Rev. A 83, 062316 (2011)

###### Abstract

We present two economical one-step error-correction protocols for multipartite polarization-entangled systems in a Greenberger-Horne-Zeilinger state. One uses spatial entanglement to correct errors in the polarization entanglement of an N-photon system, resorting to linear optical elements. The other uses frequency entanglement to correct errors in the polarization entanglement of an N-photon system. The parties in quantum communication can obtain a maximally entangled state from each N-photon system transmitted with one step in these two protocols, and both of their success probabilities are 100%, in principle. That is, they both work in a deterministic way, and they do not largely consume the less-entangled photon systems, which is far different from conventional multipartite entanglement purification schemes. These features may make these two protocols more useful for practical applications in long-distance quantum communication.

###### pacs:

03.67.Pp, 03.65.Ud, 03.67.Hk## I introduction

Entanglement is an important quantum resource for quantum information processing book (). The powerful speedup of quantum computation resorts to multipartite entanglement. Long-distance quantum communication should resort to entanglement for setting up the quantum channel between remote locations, including quantum teleportation teleportation (), quantum key distribution (QKD) Ekert91 (); BBM92 (); CORE (); LongLiu (); rmp (), quantum dense coding densecoding (); densecoding2 (); densecoding3 (), quantum secret sharing QSS (); QSS2 (); QSS3 (), and so on. In a practical transmission, an entangled quantum system will inevitably interact with its environment, which will degrade the entanglement of the system. In general, the interaction will make an entangled system be in a mixed state. The decoherence of entanglement in quantum system will affect quantum communication largely. For example, it will decrease the security of a QKD protocol if a maximally entangled state transmitted over a noisy channel becomes a mixed entangled state as a vicious eavesdropper can exploit the decoherence to hide her illegal action. The non-maximally entangled quantum channel will decrease the fidelity of quantum teleportation and quantum dense coding.

Entanglement purification Bennett1996 (); Deutsch (); Pan1 (); Simon (); shengpra2008 (); wangcpra (); shengpratwostep (); shengpraonestep (); lixhepp () is an interesting tool for the parties in quantum communication to improve the fidelity of entangled quantum systems after they are transmitted over a practical channel. Its task is to obtain a subset of high-fidelity entangled quantum systems from a set of quantum systems in a mixed entangled state. In 1996, Bennett et al. Bennett1996 () proposed an original entanglement purification protocol (EPP) to purify a Werner state werner (), resorting to quantum controlled-not (CNOT) gates and bidirectional unitary operations. Subsequently, Deutsh et al. Deutsch () improved this EPP with two additional specific unitary operations. In 2001, Pan et al. Pan1 () presented an EPP with linear optical elements and an ideal entanglement source by sacrificing a half of the efficiency in the EPP by Bennett et al. However, this protocol decreases largely the difficulty for the implementation of EPP with current technology as it is impossible to construct a perfect CNOT gate based on linear optics. In 2002, Simon and Pan Simon () proposed an EPP with a currently available parametric down-conversion (PDC) source, not an ideal single-pair entanglement source. This protocol exploits the spatial entanglement of a photon pair to purify its polarization entanglement. In 2003, Pan et al. Pan2 () demonstrated this EPP in experiment. In 2008, an efficient EPP shengpra2008 () based on a PDC source was proposed with cross-Kerr nonlinearity. It has the same efficiency as the EPP by Bennett et al. with perfect CNOT gates. Moreover, it can be repeated to get a high-fidelity entangled photon pairs from a practical entanglement source. However, the cross-Kerr nonlinearity is too small in nature, which increases the difficulty for the implementation of this EPP at present. In 2008, Xiao et al. wangcpra () proposed an EPP with frequency entanglement.

Recently, the concept of deterministic entanglement purification was proposed shengpratwostep () for two-photon entangled systems, which is far different from the conventional entanglement purification (CEPP) Bennett1996 (); Deutsch (); Pan1 (); Simon (); shengpra2008 (); wangcpra () as the former works in a deterministic way, while the latter works in a probabilistic way. In 2010, we introduced a two-step deterministic entanglement purification protocol (DEPP) shengpratwostep () for entangled photon pairs, resorting to hyerentanglement. In this two-step DEPP, the spatial entanglement and the frequency entanglement of a quantum system are used to correct the bit-flip error and the phase-flip error in this quantum system, respectively. With two error-correction processes, the two parties in quantum communication, say Alice and Bob, can obtain a maximally entangled state from each entangled system in theory. Certainly, it is difficult to implement this two-step DEPP in experiment at present as it requires the hyperentanglement of photon systems and a nonlinear optical interaction. Subsequently, we proposed an one-step DEPP shengpraonestep (), resorting to the spatial entanglement of a practical PDC source and linear optical elements. With this one-step DEPP, Alice and Bob can in principle obtain a maximally entangled photon pair from each photon system with only one step. Simultaneously, Li lixhepp () presented independently an interesting DEPP using spatial entanglement, called it also an one-step DEPP as it has the same role as the one-step DEPP in Ref. shengpraonestep () although there are, in essence, some differences in explaining the principle of the detereministic entanglement purification. Although the physics in the one-step DEPP in Ref. shengpraonestep () is more clear than that in Ref. lixhepp (), the one-step DEPP by Li is more economical than that in Ref. shengpraonestep () as the former needs two polarizing beam splitters (PBSs), while the latter needs eight PBSs.

In essence, each of all existing EPPs Bennett1996 (); Deutsch (); Pan1 (); Simon (); shengpra2008 (); wangcpra (); shengpratwostep (); shengpraonestep (); lixhepp () can be considered as a quantum error-correction protocol. In CEPPs, the process for error correction is divided into two steps which are repeated some times for improving the fidelity of entanglement largely. One step is used to correct the bit-flip error and the other is used correct the phase-flip error. That is, all the CEPPs Bennett1996 (); Deutsch (); Pan1 (); Simon (); shengpra2008 (); wangcpra () work in a probabilistic way for the correction of the errors in only one degree of freedom of photons. In DEPPs shengpratwostep (); shengpraonestep (); lixhepp (), the error correction is completed with one step. That is, they work in a deterministic way, but they should resort to at least another degree of freedom of photons.

By far, there have been several interesting EPPs Bennett1996 (); Deutsch (); Pan1 (); Simon (); shengpra2008 (); shengpratwostep (); shengpraonestep (); lixhepp (); wangcpra () focusing on the bipartite entangled quantum systems, while the number of multipartite entanglement purification protocols (MEPP) Murao (); shengepjd (); shengpla () is very small as the structure of a multipartite quantum system is more complicated than that of a bipartite quantum system. Moreover, it is difficult to optimize a MEPP as its efficiency depends on a great number of parameters coming from the channel noise. In 1998, Murao et al. Murao () proposed the first multipartite entanglement purification protocol (MEPP) to purify multipartite quantum system in a Greenberger-Horne-Zeilinger (GHZ) with CNOT gates, following some ideas in the EPP by Bennett et al Bennett1996 (). In 2009, a MEPP based on cross-Kerr nonlinearities was proposed shengepjd (). In this protocol, the cross-Kerr nonlinearity is used to construct a nondestructive quantum nondemolition detector (QND) QND1 () which has the functions of both a parity-check detector and a single-photon detector. With QNDs, the parties can obtain some high-fidelity GHZ states from an ensemble in a mixed entangled state by performing this MEPP repeatedly. In 2009, a multipartite electronic entanglement purification was proposed with charge detection shengpla (). All these three MEPPs work in a conventional way. That is, the parties can only obtain a subset of high-fidelity entangled states from an ensemble in a mixed less-entangled state by performing the MEPPs repeatedly and sacrificing a great deal of quantum resource.

In this paper, we will present two economical one-step error-correction protocols for multipartite polarization-entangled systems in a GHZ state. The first one-step multipartite polarization entanglement error-correction protocol (MPEECP) is based on spatial entanglement and simple linear optical elements. The parties in quantum communication can obtain a maximally entangled state from each system polluted by the channel noise on the polarization degree of freedom. Also, the number of the linear optical elements in this protocol is reduced to be a minimal one. Moreover, this protocol works in a deterministic way, which is far different from the polarization entanglement purification protocol for two-photon system by using the spatial entanglement Simon () as the latter is used to improve the fidelity of an ensemble in a mixed entangled state by repeating the protocol again and again. The second MPEECP uses the frequency entanglement to correct the errors in the polarization entanglement of an -photon system and reduces the number of the channels for the transmission of the system. The success probabilities of these two economical one-step MPEECPs are in principle 100%. Compared with the conventional multipartite entanglement purification protocols Murao (); shengepjd (); shengpla (), these two economical one-step MPEECPs reduces the quantum resource consumed largely. These advantages maybe make these two MPEECPs more useful in practical applications in long-distance quantum communication in future.

## Ii Economic one-step MPEECP with spatial entanglement

As shown by Simon and Pan Simon (), an entangled state of a photon pair in two degrees of freedom can be written as

(1) |

where and represent the horizonal and the vertical polarizations of photons, respectively. The subscripts and represent the two photons sent to the two parties, say Alice and Bob, respectively. () and () are the two spatial modes for the photon sent to Alice (Bob). As the spatial entanglement is far more stable than the polarization entanglement over an optical-fiber channel, Simon and Pan Simon () exploited the spatial entanglement of a photon pair coming from a PDC source to purify the polarization of a photon pair. By controlling the phase stability, the EPP by Simon and Pan can improve the fidelity of an ensemble in a mixed entangled state, as shown in the experiment by Pan et al. Pan2 ().

The GHZ state of a multipartite entangled system composed of two-level particles can be described as

(2) |

Here the subscripts , , , and represent the photons sent to the parties Alice, Bob, , and Charlie, respectively. Certainly, there are other GHZ state for an -qubit system and can be written as and . Here , , , and . and . An entangled state of an -photon system in the polarization and the spatial-mode degrees of freedom can be written as

(3) | |||||

Here represents the spatial entanglement of an -photon system.

The principle of the present economic one-step MPEECP using spatial entanglement is shown in Fig. 1. Suppose that the original state of an -photon system is . After the photons suffer from the noise in the channels, the state of the system becomes

(4) |

where and represent the states of the multi-photon systems in the polarization and the spatial degrees of freedom, respectively, i.e.,

(5) |

(6) |

(7) |

Here () is the probability that the -photon system is in the state () after it is transmitted over a noisy channel. That is, the mixed state shown in Eq.(4) can be viewed as a probabilistic mixture of pure states. The system is in the state or the state with the probabilities and , respectively.

Let us first discuss the case that the -photon system is in the state . After the setup shown in Fig.1, the evolution of the state can be described as

(8) | |||||

That is, the parties in quantum communication can determine the state of their -photon system by postselection. If the photons emit from the outports or of the PBSs, the system is in the state . The parties can obtain the state with some unitary single-photon operations. For example, if the photons emit from the outports , the system is in the state and the parties can obtain the state with only a bit-flip operation on the first qubit .

When the -photon system is in the state , one can obtain the similar result. After the setup shown in Fig.1, the evolution of the state can be described as

(9) | |||||

The parties in quantum communication can also determine the state of their -photon system by postselection on the spatial modes of the photons. If the photons emit from the outports or of the PBSs, the system is in the state . That is, the phase-flip error does not affect the result and the parties can obtain the state with some unitary single-photon operations.

## Iii Economic one-step MPEECP with frequency entanglement

Experimental results showed that the frequency of a photon is also far more stable than its polarization as previous experiments showed that the polarization entanglement is quite unsuitable for transmission over distances of more than a few kilometers in an optical fiber rmp (). For example, Naik et al. experiment1 () demonstrated the Ekert protocol Ekert91 () by only a few meters . Also, they observed the quantum bit error rate (QBER) increase to 33% in the experiment implementation of the six-state protocol sixstate1 (); sixstate2 (). For frequency coding experiment2 (); experiment5 (); experiment3 (); experiment4 (); frequencystable1 (); frequencystable2 (), for example, the Besancon group performed a key distribution over a 20-km single-mode optical fiber spool. They recorded a QBER contribution of approximately 4%, and estimated that 2% could be attributed to the transmission of the central frequency by the Fabry-Perot cavity experiment5 (). That is, the parties in quantum communication can also exploit the frequency entanglement of a multipartite entangled system to correct the errors in its polarization entanglement, similar to the case with a spatial entanglement. In this way, the parties can obtain the maximally entangled state with less channels than that with spatial entanglement.

An entangled state of an -photon system in both the polarization and the frequency degrees of freedom can be written as

(10) |

where and are the two frequency modes for photons. represents the frequency entanglement of an -photon system.

The principle of the present economic one-step MPEECP using frequency entanglement is shown in Fig. 2. Suppose that the original state of an -photon system is . After the photons suffer from the noise in the channels, the state of the system becomes

(11) |

where

(12) |

represents the density matrix in the polarization degree of freedom shown in Eq. (6). The mixed state shown in Eq.(11) means that the system is in the state or the state with the probabilities and , respectively.

Similar to the case with the spatial entanglement of an -photon system, if the system is in the state , after the setup shown in Fig.2, the evolution of the state can be described as

(13) | |||||

That is, the parties in quantum communication can determine the state of their -photon system by postselection on the spatial modes of the photons. If the photons emit from the outports or of the PBSs, the system is in the state . The parties can obtain the state with some unitary single-photon operations.

When the -photon system is in the state , one can obtain the similar result. After the setup shown in Fig.2, the evolution of the state can be described as

(14) | |||||

That is, the phase-flip error does not affect the result yet, as the same as the case with spatial entanglement.

## Iv discussion and summary

It is interesting to compare these two economic one-step MPEECPs with the conventional multipartite entanglement purification protocols (CMEPPs) in Refs.Murao (); shengepjd (). In Ref.Murao (), Murao et al. divided their CMEPP into two steps. One is used to purify the bit-flip errors and the other is used to purify the phase-flip errors, by resorting to CNOT gates. The parties in quantum communication can in principle improve the fidelity of an ensemble in a multipartite mixed entangled state by repeating these two steps again and again. The CMEPP in Ref.shengepjd () is similar to that by Murao et al. but the CNOT gates are replaced with some parity-check detectors based on nonlinear optics. That is, the two existing CMEPPs Murao (); shengepjd () for -photon systems can be used to improve the fidelity of an ensemble in a mixed entangled state and they works in a probabilistic way. In these two economic one-step MPEECPs, the parties can in principle obtain a maximally entangled state from each system transmitted and they works in a completely deterministic way. On the other hand, the system after it is transmitted over a noisy channel in the CMEPPs Murao (); shengepjd () should be in an entangled state in the polarization degree of freedom. In these two MPEECPs, they do not require the system to be entangled in the polarization degree of freedom but they require another degree of freedom of photons to keep the entanglement.

In essence, these two MPEECPs exploit the entanglement transformation between two degrees of freedom, not that between two systems, to correct the errors in the polarization entanglement, which is different from the CMEPPs Murao (); shengepjd (). In our two MPEECPs, the polarization degree of freedom of an -photon system is polluted by the noisy channels. The noise may make the system be in a completely mixed state in the polarization degree of freedom. In our first MPEECP, the spatial entanglement is kept as spatial entanglement is far more stable than the polarization entanglement over an optical-fiber channel. So does the frequency entanglement in our second MPEECP. After the transmission over noisy channels, the parties transform the spatial entanglement into the polarization entanglement completely in our first MPEECP, by using some linear optical elements to remove the noise effect on the polarization first, which is in principle different from the EPP using spatial entanglement for two-photon systems in Ref. Simon (). In our second MPEECP, the parties first exploit WDMs to transform the frequency entanglement of an -photon system into its spatial entanglement. With FSs, the parties erase the distinguishability for the frequencies of their photons. In this way, the frequency entanglement is completely transformed into the spatial entanglement. The other precesses are the same as the case in our first MPEECP. Our second MPEECP only exploits a fiber channel for each party, not two channels in our first MPEECP. The parties can obtain this economization on the number of channels but they should resort to some nonlinear elements to complete this MPEECP. This is a tradoff between our two MPEECPs.

In summary, we have presented two economic one-step MPEECPs for an -photon system in a GHZ state. In our first MPEECP, the parties exploit the spatial entanglement of an -photon system to correct the errors in its polarization entanglement with some linear optical elements. Moreover, this protocol works in a deterministic way, which is far different from the polarization entanglement purification protocol for two-photon systems by using the spatial entanglement Simon () as the latter is used to improve the fidelity of an ensemble in a mixed entangled state by repeating the protocol again and again. In our second MPEECP, the parties exploit the frequency entanglement of an -photon system to correct the errors in its polarization entanglement, resorting to some nonlinear optical elements but reducing a fiber channel for each party. As both the spatial entanglement and the frequency entanglement of an -photon system are far more stable than the polarization entanglement over an optical-fiber channel, these two economic one-step MPEECPs will reduce a greet deal of quantum resource consumed as they both work in a deterministic way, not a probabilistic way. This advantage maybe make these two MPEECPs more useful in practical applications in long-distance quantum communication in future.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 10974020, A Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China under Grant No. 200723, Beijing Natural Science Foundation under Grant No. 1082008, and the Fundamental Research Funds for the Central Universities.

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