## Abstract

Hyperentanglement is a promising resource in quantum information processing, especially for increasing the channel capacity of long-distance quantum communication. Here we present a general hyper-entanglement concentration protocol (hyper-ECP) for nonlocal partially hyperentangled Bell states that decay with the interrelationship between the polarization and the spatial-mode degrees of freedom of two-photon systems, which is not taken into account in other hyper-ECPs, resorting to the optical property of the quantum-dot spins inside one-side optical microcavities. We show that the success probability of our hyper-ECP is largely increased by iteration of the hyper-ECP process. Our hyper-ECP can be straightforwardly generalized to distill nonlocal maximally hyperentangled *N*-photon Greenberger-Horne-Zeilinger (GHZ) states from arbitrary partially hyperentangled GHZ-class states.

© 2014 Optical Society of America

## 1. Introduction

Entanglement is an essential resource in quantum information processing and it always acts as the quantum channel in some important long-distance quantum communication applications, such as quantum key distribution [1–4], quantum teleportation [5], quantum dense coding [6,7], quantum secret sharing [8,9], quantum secure direct communication [10–12], and so on. Hyperentanglement, defined as the entanglement in multiple degrees of freedom (DOFs) of quantum systems [13–15], is a promising resource in quantum information processing, especially for increasing the channel capacity of quantum communication [16–21] and improving the power of quantum computation [22]. Hyperentanglement has been used for assisting Bell-state analysis [23–26], quantum repeater [27], and deterministic entanglement purification [28–31] on the polarizations of photon pairs. In 2008, Barreiro *et al.* [16] beat the channel capacity limit of the photonic superdense coding with the polarization-orbital-angular-momentum hyperentanglement in linear optics. Now, there are some interesting works for increasing the channel capacity of long-distance quantum communication with hyperentanglement, such as quantum teleporation with two DOFs of photon pairs [17], entanglement swapping with photonic spatial-polarization hyperentanglement [17,18], hyperentangled Bell-state analysis [17–21], and so on.

The entangled photon systems used in quantum communication are always produced locally, and their entanglement will decrease due to the environment noise in the distribution process among the remote parties. Moreover, the photon signals can be transmitted only several hundreds of kilometers either in an optical fiber or in a free space with current technology and quantum repeaters are required to connect the neighboring nodes for long-distance quantum communication. The entanglement of photon systems will decrease in the storage process as well. Therefore, the fidelity and the security of long-distance quantum communication protocols will decrease, and entanglement purification and entanglement concentration are required to depress the noise effect on the entanglement.

Entanglement purification is used to extract nonlocally some high-fidelity entangled systems from a less-entangled ensemble in a mixed entangled state [28–40]. So far, some important entanglement purification protocols (EPPs) were proposed for photon systems entangled either in one DOF [28–39] or in two DOFs [40], including those assisted by the hyperentanglement [28–31,36,37]. Entanglement concentration is used to distill a subset of quantum systems in a maximally entangled pure state from those in a partially entangled pure state [40–49]. In 1996, Bennett *et al.* [41] proposed the first entanglement concentration protocol (ECP) with the Schmidt projection method and collective measurements. Since this pioneering work, many interesting ECPs were proposed for photon systems in a partially entangled pure state with its parameters either known [41–46] or unknown [46–49] to the remote users. In 2008, Sheng and Deng [49] presented a high-efficiency ECP for photon pairs by iteration of their ECP process. This interesting idea is very useful in the ECPs for other photon systems [43, 44]. In 2013, Ren *et al.* [46] proposed some hyperentanglement concentration protocols (hyper-ECPs) for two-photon systems in polarization-spatial hyperentangled states with the parameters either known or unknown to the remote users, resorting to linear optical elements. With the parameter-splitting method [46], the hyper-ECP for a partially hyperentangled pure state with known parameters is the optimal one as it has the maximal success probability and it requires linear optical elements only. With the Schmidt projection method [41], the success probability of the hyper-ECP for partially hyperentangled pure states with unknown parameters is relatively low resorting to linear optical elements, and it can be improved largely by using the parity-check quantum nondemolition detectors (QNDs) based on nonlinear optics [40]. In essence, all these hyper-ECPs are focused on the partially hyperentangled pure states with the independent de-coherence of the entanglement in the two DOFs. That is, the optical fibers for the two spatial modes of the photons should be exactly the same ones in the long-distance quantum communication proposals based on hyperentanglement.

In a practical quantum communication, there is difference between the two optical fibers for the transmission of the two photons from the two spatial modes of the photon systems in hyperentangled states and the decay rates of the polarization states in the two spatial modes are usually different, so the decoherence of the entanglement in both the spatial-mode and the polarization DOFs of photon systems are not independent. In this paper, we investigate a general hyper-ECP for two-photon systems in an arbitrary partially hyperentangled unknown Bell state that decays with the interrelationship between the polarization and the spatial-mode DOFs. Our hyper-ECP is achieved by the Schmidt projection method and some parity-check QNDs that are constructed with the optical property of the quantum dots (QDs) embedded in one-side optical microcavities (QD-cavity), which can preserve the photon pairs in the failed hyper-ECP processes. By iteration of the hyper-ECP process, the success probability of our hyper-ECP becomes much higher than that in the hyper-ECP with linear optics. At last, we show that our hyper-ECP is suitable for arbitrary partially hyperentangled multipartite GHZ-class states.

## 2. Hyperentanglement concentration for arbitrary partially hyperentangled Bell states

In a long-distance quantum communication, the maximally hyperentangled Bell state |*ψ*〉* _{AB}* may decay to the partially hyperentangled Bell-type state |

*ϕ*

_{0}〉

*by the independent decoherence of the entanglement in these two DOFs. If the interrelationship between the polarization and the spatial-mode DOFs is taken into account, the maximally hyperentangled Bell state |*

_{AB}*ψ*〉

*will decay to an arbitrary partially hyperentangled Bell state |*

_{AB}*ϕ*〉

*. Here*

_{AB}*A*and

*B*represent the two photons that belong to the two remote users, say Alice and Bob, respectively.

*R*and

*L*represent the left and the right circularly polarized photons, respectively, and

*i*

_{1}and

*i*

_{2}represent the two spatial modes of the photon

*i*(

*i*=

*a*,

*b*).

*α*

_{0},

*β*

_{0},

*γ*

_{0}, and

*δ*

_{0}are four parameters unknown to the two remote users, and they satisfy the relations |

*α*

_{0}|

^{2}+ |

*β*

_{0}|

^{2}= 1 and |

*γ*

_{0}|

^{2}+ |

*δ*

_{0}|

^{2}= 1.

*α*,

*β*,

*γ*, and

*δ*are another four parameters unknown to the two remote users, and they satisfy the relation |

*α*|

^{2}+ |

*β*|

^{2}+ |

*γ*|

^{2}+ |

*δ*|

^{2}= 1. In this section, we present a general hyper-ECP for two-photon systems in an arbitrary partially hyperentangled Bell state |

*ϕ*〉

*with the parity-check QNDs for the polarization and the spatial-mode DOFs, resorting to the optical property of QD-cavity systems.*

_{AB}#### 2.1. Parity-check QNDs for the polarization and spatial-mode DOFs of two-photon systems

The parity-check QND for the polarization (spatial-mode) DOF of a photon pair is constructed with a singly charged QD (e.g., a self-assembled In(Ga)As QD or a GaAs interface QD) embedded in the center of a one-side optical resonant microcavity (the bottom distributed Bragg reflectors are 100% reflective and the top distributed Bragg reflectors are partially reflective). If an excess electron is injected into a QD, the negatively charged exciton *X*^{−} (consists of two antiparallel electrons bound to one hole) [50] is created by resonantly absorbing a circularly polarized light, and there are two spin-dependent transitions in the exciton *X*^{−} according to Pauli’s exclusion principle [51], shown in Fig. 1. That is, when the excess electron spin is in the state |↑〉 (|↓〉), the negatively charged exciton in the state |↑↓⇑〉 (|↓↑⇓〉) will be created by resonantly absorbing a left (right) circularly polarized light |*L*〉 (|*R*〉). Here |⇑〉 (|⇓〉) represents the heavy-hole spin state
$|+\frac{3}{2}\u3009$ (
$|-\frac{3}{2}\u3009$).

The input-output process of the circularly polarized light with a QD-cavity system can be represented by Heisenberg equations for the cavity field operator *â* and *X*^{−} dipole operator *σ̂*_{−} in the interaction picture [52]. That is,

*ω*,

_{c}*ω*, and

*ω*

_{X−}are the frequencies of the cavity mode, the input probe photon, and the

*X*

^{−}transition, respectively.

*κ*/2 and

*κ*/2 are the decay rate and the side leakage rate of the cavity mode, respectively.

_{s}*g*is the coupling strength between

*X*

^{−}and the cavity mode, and

*γ*/2 is the decay rate of

*X*

^{−}. In the weak excitation condition (

*X*

^{−}is dominantly in the ground sate and 〈

*σ̂*〉 = −1), the reflection coefficient for this QD-cavity system can be obtained [53],

_{z}*r*

_{0}(

*ω*)| ≃ 1 for the case

*g*= 0. In the strong coupling regime (

*g*≫

*κ*,

*γ*), the reflection coefficient is |

*r*(

*ω*)| ≃ 1.

The |*L*〉 and |*R*〉 photons will get different phase shifts when they are reflected by a QD-cavity system. With the excess electron spin in the state |↑〉, the |*L*〉 photon gets a phase shift *φ* and the |*R*〉 photon gets a phase shift *φ*_{0}. Otherwise, the |*L*〉 photon gets a phase shift *φ*_{0} and the |*R*〉 photon gets a phase shift *φ*. Here *φ*_{0} = *arg*[*r*_{0}(*ω*)] and *φ* = *arg*[*r*(*ω*)], and
${\theta}_{F}^{\uparrow}=\left({\phi}_{0}-\phi \right)/2=-{\theta}_{F}^{\downarrow}$ is the Faraday rotation angle. If the frequencies are adjusted to be *ω* − *ω _{c}* ≈

*κ*/2 (

*ω*=

_{c}*ω*

_{X}_{−}), the phase shifts of the |

*L*〉 and |

*R*〉 photons are ${\phi}_{0}=-\frac{\pi}{2}$ and

*φ*= 0. If the photon interacts with the QD-cavity system twice (shown in Fig. 2), the output states of the |

*L*〉 and |

*R*〉 photons can be obtained as

Our parity-check QNDs for the polarization and the spatial-mode DOFs of a two-photon system are constructed with the optical property of QD-cavity systems, shown in Fig. 2. The polarization parity-check QND is composed of QD_{1}, CPBSs, X_{1}, X_{2}, Z_{1}, X_{3}, X_{4}, and Z_{2}. The spatial-mode parity-check QND is composed of QD_{2}, Z_{3}, and Z_{4}.

- (1)
*Polarization parity-check QND (P-QND).*Suppose that the initial state of QD_{1}is ${|+\u3009}_{1}\left({|\pm \u3009}_{1}=\left({|\uparrow \u3009}_{1}\pm {|\downarrow \u3009}_{1}\right)/\sqrt{2}\right)$ and the photon*A*is in the state |*χ*〉= (_{A}*α*_{0}|*R*〉 +*β*_{0}|*L*〉)(_{A}*γ*_{0}|*a*_{1}〉 +*δ*_{0}|*a*_{2}〉). Alice makes the photon in the |*R*〉 polarization from the spatial modes |*a*_{1}〉 and |*a*_{2}〉 interact with QD_{1}by performing X_{1}and CPBSs on these two spatial modes. After the photon*A*coming from both the spatial modes |*a*_{1}〉 and |*a*_{2}〉 passes through Z_{1}and X_{2}, respectively, the state of the system composed of QD_{1}and the photon*A*is transformed into$${|+\u3009}_{1}\otimes {|\chi \u3009}_{A}\to \left[{|\uparrow \u3009}_{1}{\left({\alpha}_{0}|R\u3009+{\beta}_{0}|L\u3009\right)}_{A}+{|\downarrow \u3009}_{1}{\left(-{\alpha}_{0}|R\u3009+{\beta}_{0}|L\u3009\right)}_{A}\right]\left({\gamma}_{0}|{a}_{1}\u3009+{\delta}_{0}|{a}_{2}\u3009\right).$$If Alice has a photon pair*AC*in the hyperentangled Bell state |*ϕ*^{±}〉= (_{AC}*α*_{0}|*RR*〉 ±*β*_{0}|*LL*〉)⊗ (_{AC}*γ*_{0}|*a*_{1}*c*_{1}〉 +*δ*_{0}|*a*_{2}*c*_{2}〉) $\left({|{\varphi}_{1}^{\pm}\u3009}_{AC}={\left({\alpha}_{0}|RL\u3009\pm {\beta}_{0}|LR\u3009\right)}_{AC}\otimes \left({\gamma}_{0}|{a}_{1}{c}_{1}\u3009+{\delta}_{0}|{a}_{2}{c}_{2}\u3009\right)\right)$, after they interact with QD_{1}, CPBSs, X_{1}, X_{2}, Z_{1}, X_{3}, X_{4}, and Z_{2}in sequence as shown in Fig. 2, the state of the system composed of QD_{1}and the photon pair*AC*is transformed into$${|+\u3009}_{1}\otimes {|{\varphi}^{\pm}\u3009}_{AC}\to {|+\u3009}_{1}\otimes {|{\varphi}^{\pm}\u3009}_{AC},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{|+\u3009}_{1}\otimes {|{\varphi}_{1}^{\pm}\u3009}_{AC}\to {|-\u3009}_{1}\otimes {|{\varphi}_{1}^{\pm}\u3009}_{AC}.$$By measuring the excess electron spin in the orthogonal basis {|+〉_{1}, |−〉_{1}} [18], one can distinguish the even-parity polarization mode (|*ϕ*^{±}〉) of the photon pair_{AC}*AC*from the odd-parity polarization mode ( ${|{\varphi}_{1}^{\pm}\u3009}_{AC}$) without disturbing its state in the spatial-mode DOF. That is, if the excess electron spin is in the state |+〉_{1}, the polarization DOF of the hyperentangled state is in the even-parity mode. Otherwise, the polarization DOF of the hyperentangled state is in the odd-parity mode. If the spatial-mode DOF of the two-photon system*AC*is in any other state, the outcome of our polarization parity-check QND is exactly the same as that in Eq. (6). - (2)
*Spatial-mode parity-check QND (S-QND).*Suppose that the initial state of QD_{2}is |+〉_{2}and the photon*B*is in the state |*χ*〉= (_{B}*α*_{0}|*R*〉 +*β*_{0}|*L*〉)(_{B}*γ*_{0}|*b*_{1}〉 +*δ*_{0}|*b*_{2}〉). Bob lets the photon from the spatial mode |*b*_{2}〉 interact with QD_{2}and then pass through Z_{3}. The state of the system composed of QD_{2}and the photon*B*is transformed into$${|+\u3009}_{2}\otimes {|\chi \u3009}_{B}\to \left[{|\uparrow \u3009}_{2}\left({\gamma}_{0}|{b}_{1}\u3009+{\delta}_{0}|{b}_{2}\u3009\right)+{|\downarrow \u3009}_{2}\left({\gamma}_{0}|{b}_{1}\u3009-{\delta}_{0}|{b}_{2}\u3009\right)\right]{\left({\alpha}_{0}|R\u3009+{\beta}_{0}|L\u3009\right)}_{B}.$$If Bob has two photons*BD*in the hyperentangled Bell state |*ϕ*^{±}〉= (_{BD}*α*_{0}|*RR*〉 +*β*_{0}|*LL*〉)⊗ (_{BD}*γ*_{0}|*b*_{1}*d*_{1}〉 ±*δ*_{0}|*b*_{2}*d*_{2}〉) $\left({|{\varphi}_{1}^{\pm}\u3009}_{BD}={\left({\alpha}_{0}|RR\u3009+{\beta}_{0}|LL\u3009\right)}_{BD}\otimes \left({\gamma}_{0}|{b}_{1}{d}_{2}\u3009\pm {\delta}_{0}|{b}_{2}{d}_{1}\u3009\right)\right)$, after they interact with QD_{2}, and pass though Z_{3}and Z_{4}in sequence as shown in Fig. 2, the state of the system composed of QD_{2}and the photon pair*BD*is transformed into$${|+\u3009}_{2}\otimes {|{\varphi}^{\pm}\u3009}_{BD}\to {|+\u3009}_{2}\otimes {|{\varphi}^{\pm}\u3009}_{BD},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{|+\u3009}_{2}\otimes {|{\varphi}_{1}^{\pm}\u3009}_{BD}\to {|-\u3009}_{2}\otimes {|{\varphi}_{1}^{\pm}\u3009}_{BD}.$$By measuring the excess electron spin in the orthogonal basis {|+〉_{2}, |−〉_{2}} [18], one can distinguish the even-parity spatial mode (|*ϕ*^{±}〉) from the odd-parity spatial mode ( ${|{\varphi}_{1}^{\pm}\u3009}_{BD}$) without disturbing the state of the polarization DOF of the photon pair_{BD}*BD*. That is, if the excess electron spin is in the state |+〉_{2}, the spatial-mode DOF of the hyperentangled state is in the even-parity mode. Otherwise, the spatial-mode DOF of the hyperentangled state is in the odd-parity mode. If the polarization DOF of the two-photon system*BD*is in any other state, the outcome of our spatial-mode parity-check QND is exactly the same as that in Eq. (8).

#### 2.2. Hyperentanglement concentration for arbitrary partially hyperentangled Bell states

In this section, we investigate the general hyperentanglement concentration for two-photon systems in an arbitrary partially hyperentangled Bell state |*ϕ*〉* _{AB}* that decays with the interrelationship between the polarization and the spatial-mode DOFs.

### I. The first round of the hyperentanglement concentration process

As there are three independent unknown parameters in the arbitrary partially hyperentangled Bell state, four copies of the two-photon systems in the state |*ϕ*〉* _{AB}* are required to implement the hyper-ECP with the Schmidt projection method. That is,

*AB*,

*CD*,

*A′B′*, and

*C′D′*represent the four copies of the two-photon pairs. The four photons

*A*,

*C*,

*A′*, and

*C′*belong to Alice, and the four photons

*B*,

*D*,

*B′*, and

*D′*belong to Bob. Alice and Bob divide their four photons into two groups. That is, Alice has two photons

*A*and

*C*in one group, and the other two photons

*A′*and

*C′*in the other group. Bob has two photons

*B*and

*D*in one group, and the other two photons

*B′*and

*D′*in the other group.

- (1)
*The first step.*The principle of our hyper-ECP for the photon pairs*AB*and*CD*is shown in Fig. 2, and the same operations are performed on the photon pairs*A′B′*and*C′D′*. The initial state of the four-photon system*ABCD*is described as |Φ〉= |_{ABCD}*ϕ*〉⊗ |_{AB}*ϕ*〉. Alice and Bob perform the P-QND and S-QND on the photon pairs_{CD}*AC*and*BD*, respectively. The outcomes can be divided into four groups, and they are discussed in detail as follows.- (1.1) If the outcomes of the P-QND and S-QND are in an even-parity polarization mode and an odd-parity spatial mode, respectively, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{1}\u3009}_{ABCD}^{1}$ with the probability of
*p*(1)_{1}= 2(|*αγ*|^{2}+ |*βδ*|^{2}). Here$${|{\mathrm{\Phi}}_{1}\u3009}_{ABCD}^{1}=\frac{1}{\sqrt{p{(1)}_{1}}}{\left(\alpha \gamma |RRRR\u3009+\beta \delta |LLLL\u3009\right)}_{ABCD}\otimes \left(|{a}_{1}{b}_{1}{c}_{2}{d}_{2}\u3009+|{a}_{2}{b}_{2}{c}_{1}{d}_{1}\u3009\right).$$Alice and Bob perform Hadamard operations on the polarization and the spatial-mode DOFs of the photons*C*and*D*, respectively, and then the state ${|{\mathrm{\Phi}}_{1}\u3009}_{ABCD}^{1}$ is transformed into the state ${|{\mathrm{\Phi}}_{1}^{\prime}\u3009}_{ABCD}^{1}$. Here$$\begin{array}{ll}{|{\mathrm{\Phi}}_{1}^{\prime}\u3009}_{ABCD}^{1}=\hfill & \frac{1}{4\sqrt{p{(1)}_{1}}}[\left(\alpha \gamma |RR\u3009+\beta \delta |LL\u3009\right)\left(|RR\u3009+|LL\u3009\right)+\left(\alpha \gamma |RR\u3009-\beta \delta |LL\u3009\right)\hfill \\ \hfill & {\otimes \left(|RL\u3009+|LR\u3009\right)]}_{ABCD}\otimes [\left(|{a}_{1}{b}_{1}\u3009+|{a}_{2}{b}_{2}\u3009\right)\left(|{c}_{1}{d}_{1}\u3009+|{c}_{2}{d}_{2}\u3009\right)\hfill \\ \hfill & +\left(-|{a}_{1}{b}_{1}\u3009+|{a}_{2}{b}_{2}\u3009\right)\left(|{c}_{1}{d}_{2}\u3009+|{c}_{2}{d}_{1}\u3009\right)].\hfill \end{array}$$At last, the two photons*C*and*D*are detected by the single-photon detectors shown in Fig. 2. If the outcome of the two clicked detectors is in an even-parity polarization mode and an even-parity spatial mode, the two-photon system*AB*is projected into the state ${|{\varphi}_{1}\u3009}_{AB}^{1}=\frac{1}{\sqrt{p{(1)}_{1}}}{\left(\alpha \gamma |RR\u3009+\beta \delta |LL\u3009\right)}_{AB}\otimes \left(|{a}_{1}{b}_{1}\u3009+|{a}_{2}{b}_{2}\u3009\right)$. This is a partially hyperentangled Bell-type state with the spatial-mode DOF in a maximally entangled Bell state. If the outcome of the two clicked detectors is in an odd-parity polarization mode (an odd-parity spatial mode), a phase-flip operation ${\sigma}_{z}^{p}=|R\u3009\u3008R|-|L\u3009\u3008L|\left({\sigma}_{z}^{s}=-|{b}_{1}\u3009\u3008{b}_{1}|+|{b}_{2}\u3009\u3008{b}_{2}|\right)$ on the photon*B*is required to obtain the partially hyperentangled Bell-type state ${|{\varphi}_{1}\u3009}_{AB}^{1}$. - (1.2) If the outcomes of the P-QND and S-QND are in an odd-parity polarization mode and an even-parity spatial mode, respectively, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{2}\u3009}_{ABCD}^{1}$ with the probability of
*p*(1)_{2}= 2(|*αβ*|^{2}+ |*γδ*|^{2}). Here$${|{\mathrm{\Phi}}_{2}\u3009}_{ABCD}^{1}=\frac{1}{\sqrt{p{(1)}_{2}}}{\left(|RRLL\u3009+|LLRR\u3009\right)}_{ABCD}\otimes \left(\alpha \beta |{a}_{1}{b}_{1}{c}_{1}{d}_{1}\u3009+\gamma \delta |{a}_{2}{b}_{2}{c}_{2}{d}_{2}\u3009\right).$$After performing Hadamard operations and detections on the photons*C*and*D*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the state of the two-photon system*AB*is transformed into ${|{\varphi}_{2}\u3009}_{AB}^{1}=\frac{1}{\sqrt{p{(1)}_{2}}}{\left(|RR\u3009+|LL\u3009\right)}_{AB}\otimes \left(\alpha \beta |{a}_{1}{b}_{1}\u3009+\gamma \delta |{a}_{2}{b}_{2}\u3009\right)$, which is a partially hyperentangled Bell-type state with the polarization DOF in a maximally entangled Bell state. - (1.3) If the outcomes of the P-QND and S-QND are in an odd-parity polarization mode and an odd-parity spatial mode, respectively, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{3}\u3009}_{ABCD}^{1}$with the probability of
*p*(1)_{3}= 2(|*αδ*|^{2}+ |*βγ*|^{2}). Here$$\begin{array}{ll}{|{\mathrm{\Phi}}_{3}\u3009}_{ABCD}^{1}=\hfill & \frac{1}{\sqrt{p{(1)}_{3}}}[{\left(\alpha \delta |RRLL\u3009+\beta \gamma |LLRR\u3009\right)}_{ABCD}|{a}_{1}{b}_{1}{c}_{2}{d}_{2}\u3009\hfill \\ \hfill & +{\left(\alpha \delta |LLRR\u3009+\beta \gamma |RRLL\u3009\right)}_{ABCD}|{a}_{2}{b}_{2}{c}_{1}{d}_{1}\u3009\left)\right].\hfill \end{array}$$Alice and Bob perform the polarization bit-flip operations ${\sigma}_{x}^{p}=|R\u3009\u3008L|+|L\u3009\u3008R|$ on the spatial modes |*a*_{2}〉, |*b*_{2}〉, and |*c*_{1}〉, |*d*_{1}〉, respectively, and then the state ${|{\mathrm{\Phi}}_{3}\u3009}_{ABCD}^{1}$ is transformed into ${|{\mathrm{\Phi}}_{3}^{\prime}\u3009}_{ABCD}^{1}$. Here$${|{\mathrm{\Phi}}_{3}^{\prime}\u3009}_{ABCD}^{1}=\frac{1}{\sqrt{p{(1)}_{3}}}[{\left(\alpha \delta |RRLL\u3009+\beta \gamma |LLRR\u3009\right)}_{ABCD}\otimes \left(|{a}_{1}{b}_{1}{c}_{2}{d}_{2}\u3009+|{a}_{2}{b}_{2}{c}_{1}{d}_{1}\u3009\right).$$After performing Hadamard operations and detections on the photons*C*and*D*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the state of the two-photon system*AB*is transformed into ${|{\varphi}_{3}\u3009}_{AB}^{1}=\frac{1}{\sqrt{p{(1)}_{3}}}{\left(\alpha \delta |RR\u3009+\beta \gamma |LL\u3009\right)}_{AB}\otimes \left(|{a}_{1}{b}_{1}\u3009+|{a}_{2}{b}_{2}\u3009\right)$, which is similar to ${|{\varphi}_{1}\u3009}_{AB}^{1}$ by replacing*αγ*and*βδ*with*αδ*and*βγ*, respectively. - (1.4) If the outcomes of the P-QND and S-QND are in an even-parity polarization mode and an even-parity spatial mode, respectively, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{4}\u3009}_{ABCD}^{1}$ with the probability of
*p*(1)_{4}= |*α*^{2}|^{2}+ |*β*^{2}|^{2}+ |*γ*^{2}|^{2}+ |*δ*^{2}|^{2}. Here$$\begin{array}{ll}{|{\mathrm{\Phi}}_{4}\u3009}_{ABCD}^{1}=\hfill & \frac{1}{\sqrt{p{(1)}_{4}}}({\alpha}^{2}|RRRR\u3009|{a}_{1}{b}_{1}{c}_{1}{d}_{1}\u3009+{\beta}^{2}|LLLL\u3009|{a}_{1}{b}_{1}{c}_{1}{d}_{1}\u3009\hfill \\ \hfill & {+{\gamma}^{2}|RRRR\u3009|{a}_{2}{b}_{2}{c}_{2}{d}_{2}\u3009+{\delta}^{2}|LLLL\u3009|{a}_{2}{b}_{2}{c}_{2}{d}_{2}\u3009)}_{ABCD}.\hfill \end{array}$$After performing Hadamard operations and detections on the photons*C*and*D*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the state of the two-photon system*AB*is transformed into ${|{\varphi}_{4}\u3009}_{AB}^{1}=\frac{1}{\sqrt{p{(1)}_{4}}}{\left({\alpha}^{2}|RR\u3009|{a}_{1}{b}_{1}\u3009+{\beta}^{2}|LL\u3009{a}_{1}{b}_{1}+{\gamma}^{2}|RR\u3009|{a}_{2}{b}_{2}\u3009+{\delta}^{2}|LL\u3009|{a}_{2}{b}_{2}\u3009\right)}_{AB}$, which is an arbitrary partially hyperentangled Bell state with less entanglement. In this condition, another round of the hyper-ECP process is required.Alice and Bob perform the same operations on the photon pairs

*A′B′*and*C′D′*as those on the photon pairs*AB*and*CD*, and the same results can be obtained. That is, the four cases in Eqs. (10), (12), (14), and (15) are obtained with the probabilities*p*(1)_{1},*p*(1)_{2},*p*(1)_{3}, and*p*(1)_{4}, respectively, by replacing the four photons*ABCD*with*A′B′C′D′*.

- (2)
*The second step.*In this step, we are going to distill a maximally hyperentangled Bell state from the partially hyperentangled Bell-type states obtained in the first three cases of the first step. The principle of this step is the same as the first one (shown in Fig. 2) by replacing the photons*ABCD*with*ABA′B′*. That is, Alice and Bob perform the P-QND and S-QND on the photon pairs*AA′*and*BB′*, respectively.- (2.1) For the case in (1.1), the state of the four-photon system
*ABA′B′*is ${|{\mathrm{\Phi}}_{1}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}={|{\varphi}_{1}\u3009}_{AB}^{1}\otimes {|{\varphi}_{1}\u3009}_{{A}^{\prime}{B}^{\prime}}^{1}$. Alice and Bob pick up the case when the outcome of the P-QND is in an odd-parity polarization mode, and a maximally hyperentangled Bell state can be obtained whether the outcome of the S-QND is in an odd-parity spatial mode or in an even-parity spatial mode. That is, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{1}^{1}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ or ${|{\mathrm{\Phi}}_{1}^{2}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ with the same probability*P*(1)_{1}= 4|*αβγδ*|^{2}/*p*(1)_{1}. Here,$$\begin{array}{l}{|{\mathrm{\Phi}}_{1}^{1}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{2}{\left(|RRLL\u3009+|LLRR\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left(|{a}_{1}{b}_{1}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009+|{a}_{2}{b}_{2}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009\right),\\ {|{\mathrm{\Phi}}_{1}^{2}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{2}{\left(|RRLL\u3009+|LLRR\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left(|{a}_{1}{b}_{1}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009+|{a}_{2}{b}_{2}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009\right).\end{array}$$After performing Hadamard operations and detections on the photons*A′*and*B′*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the two-photon system*AB*is projected into the maximally hyperentangled Bell state |*ψ*〉._{AB}If the outcome of the P-QND is in an even-parity polarization mode, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{1}^{3}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ or ${|{\mathrm{\Phi}}_{1}^{4}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ with the same probability ${P}^{\prime}{(1)}_{1}={y}_{1}^{2}/p{(1)}_{1}$. Here

$$\begin{array}{l}{|{\mathrm{\Phi}}_{1}^{3}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{{y}_{1}}{\left({\alpha}^{2}{\gamma}^{2}|RRRR\u3009+{\beta}^{2}{\delta}^{2}|LLLL\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left(|{a}_{1}{b}_{1}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009+|{a}_{2}{b}_{2}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009\right),\\ {|{\mathrm{\Phi}}_{1}^{4}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{{y}_{1}}{\left({\alpha}^{2}{\gamma}^{2}|RRRR\u3009+{\beta}^{2}{\delta}^{2}|LLLL\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left(|{a}_{1}{b}_{1}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009+|{a}_{2}{b}_{2}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009\right),\end{array}$$where ${y}_{1}=\sqrt{2\left({\left|{\alpha}^{2}{\gamma}^{2}\right|}^{2}+{\left|{\beta}^{2}{\delta}^{2}\right|}^{2}\right)}$. After performing Hadamard operations and detections on the photons*A′*and*B′*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the two-photon system*AB*is projected into the partially hyperentangled Bell-type state ${|{\mathrm{\Phi}}_{1}^{3}\u3009}_{AB}^{1}=\frac{1}{{y}_{1}}{\left({\alpha}^{2}{\gamma}^{2}|RR\u3009+{\beta}^{2}{\delta}^{2}|LL\u3009\right)}_{AB}\left(|{a}_{1}{b}_{1}\u3009+|{a}_{2}{b}_{2}\u3009\right)$. In this condition, another round of the hyper-ECP process (only the second step) is required. - (2.2) For the case in (1.2), the state of the four-photon system
*ABA′B′*is ${|{\mathrm{\Phi}}_{2}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}={|{\varphi}_{2}\u3009}_{AB}^{1}\otimes {|{\varphi}_{2}\u3009}_{{A}^{\prime}{B}^{\prime}}^{1}$. Alice and Bob pick up the case when the outcome of the S-QND is in an odd-parity spatial mode, and a maximally hyperentangled Bell state can be obtained whether the outcome of the P-QND is in an odd-parity polarization mode or in an even-parity polarization mode. That is, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{2}^{1}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ or ${|{\mathrm{\Phi}}_{2}^{2}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ with the same probability*P*(1)_{2}= 4|*αβγδ*|^{2}/*p*(1)_{2}. Here$$\begin{array}{l}{|{\mathrm{\Phi}}_{2}^{1}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{2}{\left(|RRRR\u3009+|LLLL\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left(|{a}_{1}{b}_{1}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009+|{a}_{2}{b}_{2}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009\right),\\ {|{\mathrm{\Phi}}_{2}^{2}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{2}{\left(|RRLL\u3009+|LLRR\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left(|{a}_{1}{b}_{1}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009+|{a}_{2}{b}_{2}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009\right).\end{array}$$After performing Hadamard operations and detections on the photons*A′*and*B′*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the two-photon system*AB*is projected into the maximally hyperentangled Bell state |*ψ*〉._{AB}If the outcome of the S-QND is in an even-parity spatial mode, the four-photon system is projected into the state ${|{\mathrm{\Phi}}_{2}^{3}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ or ${|{\mathrm{\Phi}}_{2}^{4}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ with the same probability ${P}^{\prime}{(1)}_{2}={y}_{2}^{2}/p{(1)}_{2}$. Here

$$\begin{array}{l}{|{\mathrm{\Phi}}_{2}^{3}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{{y}_{2}}{\left(|RRRR\u3009+|LLLL\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left({\alpha}^{2}{\beta}^{2}|{a}_{1}{b}_{1}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009+{\gamma}^{2}{\delta}^{2}|{a}_{2}{b}_{2}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009\right),\\ {|{\mathrm{\Phi}}_{2}^{4}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}=\frac{1}{{y}_{2}}{\left(|RRLL\u3009+|LLRR\u3009\right)}_{AB{A}^{\prime}{B}^{\prime}}\otimes \left({\alpha}^{2}{\beta}^{2}|{a}_{1}{b}_{1}{a}_{1}^{\prime}{b}_{1}^{\prime}\u3009+{\gamma}^{2}{\delta}^{2}|{a}_{2}{b}_{2}{a}_{2}^{\prime}{b}_{2}^{\prime}\u3009\right),\end{array}$$where ${y}_{2}=\sqrt{2\left({\left|{\alpha}^{2}{\beta}^{2}\right|}^{2}+{\left|{\gamma}^{2}{\delta}^{2}\right|}^{2}\right)}$. After performing Hadamard operations and detections on the photons*A′*and*B′*and performing the conditional phase-flip operation ${\sigma}_{z}^{p}$ ( ${\sigma}_{z}^{s}$) on the photon*B*, the two-photon system*AB*is projected into the partially hyperentangled Bell-type state ${|{\mathrm{\Phi}}_{2}^{3}\u3009}_{AB}^{1}=\frac{1}{{y}_{2}}{\left(|RR\u3009+|LL\u3009\right)}_{AB}\left({\alpha}^{2}{\beta}^{2}|{a}_{1}{b}_{1}\u3009+{\gamma}^{2}{\delta}^{2}|{a}_{2}{b}_{2}\u3009\right)$. In this condition, another round of the hyper-ECP process (only the second step) is required. - (2.3) For the case in (1.3), the same operations are performed on the state ${|{\mathrm{\Phi}}_{3}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}={|{\varphi}_{3}\u3009}_{AB}^{1}\otimes {|{\varphi}_{3}\u3009}_{{A}^{\prime}{B}^{\prime}}^{1}$ as those on the state ${|{\mathrm{\Phi}}_{1}\u3009}_{AB{A}^{\prime}{B}^{\prime}}^{1}$ in case (2.1). The maximally hyperentangled Bell state |
*ψ*〉can be obtained with the probability of 2_{AB}*P*(1)_{3}= 8|*αβγδ*|^{2}/*p*(1)_{3}. And the state ${|{\mathrm{\Phi}}_{3}^{3}\u3009}_{AB}^{1}=\frac{1}{{y}_{3}}\left({\alpha}^{2}{\delta}^{2}|RR\u3009+{\beta}^{2}{\gamma}^{2}{|LL\u3009}_{AB}\otimes |{a}_{1}{b}_{1}\u3009+|{a}_{2}{b}_{2}\u3009\right)$ can be obtained with the probability of $2{P}^{\prime}{(1)}_{3}=2{y}_{3}^{2}/p{(1)}_{3}$, which requires another round of the hyper-ECP process (only the second step). Here ${y}_{3}=\sqrt{2\left({\left|{\alpha}^{2}{\delta}^{2}\right|}^{2}+{\left|{\beta}^{2}{\gamma}^{2}\right|}^{2}\right)}$.

### II. The iteration of the hyper-ECP process

For the arbitrary partially hyperentangled Bell state
${|{\varphi}_{4}\u3009}_{AB}^{1}$ preserved in the case (1.4), four copies of the two-photon systems in this state are required to complete the two steps in the second round of the hyper-ECP process. While for the partially hyperentangled Bell-type states
${|{\mathrm{\Phi}}_{1}^{3}\u3009}_{AB}^{1}$,
${|{\mathrm{\Phi}}_{2}^{3}\u3009}_{AB}^{1}$, and
${|{\mathrm{\Phi}}_{3}^{3}\u3009}_{AB}^{1}$ preserved in the second step, two copies of the photon systems in each state are required to complete the second step in the second round of the hyper-ECP process. Therefore, the partially hyperentangled Bell states preserved in the *n*-th round of the hyper-ECP process can be used to distill the maximally hyperentangled Bell state in the (*n*+1)-th round. The success probability of the hyperentanglement concentration process in each round can be described as follows:

*n*-th round iteration of the hyper-ECP process, the total success probability is

In Fig. 3, the success probability *P* of our hyper-ECP for the case |*β*| = |*δ*| is shown with the parameters |*α*| and |*γ*|. In the hyper-ECP for the two-photon systems in an arbitrary partially hyperentangled Bell state with linear optics, only one of the three cases (1.1), (1.2), and (1.3) can be preserved, and only one of the two spatial-mode (polarization) parity cases is preserved in the second step. These six cases can all be preserved in our hyper-ECP with nonlinear optical elements, so the success probability *P*_{1} of the first round of the hyper-ECP process is almost five times larger than that in the hyper-ECP with linear optical elements. With the iteration of our hyper-ECP process, the success probability *P* will be increased largely.

## 3. Discussion and summary

It is obvious that our hyper-ECP can be generalized to distill some multi-photon systems in a maximally hyperentangled *N*-photon GHZ state from those in an arbitrary partially hyper-entangled GHZ-class state that decays with the interrelationship between the polarization and the spatial-mode DOFs. Suppose an arbitrary partially hyperentangled *N*-photon GHZ state is described as

*α*,

*β*,

*γ*, and

*δ*are the four parameters unknown to the

*N*remote users in quantum communication, and they satisfy the relation |

*α*|

^{2}+ |

*β*|

^{2}+ |

*γ*|

^{2}+ |

*δ*|

^{2}= 1. The subscripts

*A*,

*B*, ···, and

*Z*represent the photons that are kept by the remote users, called Alice, Bob,

*···*, and Zach, respectively. As there are also three independent parameters in the state |

*ϕ*〉, which is similar to the state |

_{N}*ϕ*〉

*, our hyper-ECP can be straightforwardly used to distill maximally hyperentangled*

_{AB}*N*-photon GHZ states with any two remote users of the

*N*parities performing the S-QND and P-QND on their photon pairs, respectively. The success probability of our hyper-ECP for

*N*-photon systems in an arbitrary partially hyperentangled GHZ state is the same as the one for two-photon systems in an arbitrary partially hyperentangled Bell state.

Hyperentanglement is a promising resource in quantum information processing, and several hyper-ECPs were proposed for partially hyperentangled states with the independent decoher-ence of the entanglement in the polarization and the spatial-mode (angular-momentum) DOFs of photon pairs [40, 45, 46]. The present hyper-ECP is proposed to concentrate the photon systems in an arbitrary partially hyperentangled state with unknown parameters, resorting to the P-QNDs and S-QNDs that are constructed with the optical property of QD-cavity systems.

In experiment, the input-output rule of a QD-cavity system is not perfect, which could reduce the fidelities and efficiencies of the P-QND and S-QND. The heavy hole mixing can decrease the fidelity by a few percent [54], but it can be reduced by engineering the shape, size, and type of QDs. The fine structure splitting is immune in this scheme with charged excitons [55, 56]. The optical dephasing can slightly reduce the fidelity due to its several hundred picosecond coherence time [57–61]. The spin dephasing of *X*^{−} comes mainly from the hole-spin dephasing and it can be safely neglected with the hole spin coherence time three orders of magnitude longer than the cavity photon life time (around tens of picoseconds in the strong coupling regime) [62–64]. The electron spin coherence time, which is reduced by the hyperfine interaction, could be extended to *μ*s using spin echo techniques (using single-photon pluses to play the role of the *π* pulse) [65]. The weak excitation condition in the QD-cavity system requires the intracavity photon number to be less than the critical photon number (*n*_{0} = *γ*^{2}/2*g*^{2}) [66,67]. That is, the time interval of the input photons is required to be Δ*t* = *τ/n*_{0}, where *τ* is the cavity photon life time. In the strong coupling regime for the cavity Q-factor of 10^{4} − 10^{5}, the critical photon number is *n*_{0} ∼ 10^{−3}, and the time interval of the input photons is Δ*t* ∼ *n*s [65], which is shorter than the electron spin coherence time. The speed of the photon-spin interaction is determined by the cavity photon life time (*p*s) [59]. And the optical spin manipulation method at single-photon levels can be used in our scheme, instead of the other optical spin manipulation [65].

The fidelity and efficiency of our scheme are mainly reduced by the coupling strength *g* and the cavity side leakage *κ _{s}*. In the case

*κ*< 1.3

_{s}*κ*, a phase shift of $\pm \frac{\pi}{2}$ can be achieved [65], and it is not very sensitive to the cavity side leakage when

*κ*<

_{s}*κ*[68]. In 2011, Young

*et al.*[69] confirmed a QD-induced phase shift of 0.2 rad in

*d*= 2.5

*μ*m pillar microcavity (

*g*> (

*κ*+

*κ*+

_{s}*γ*)/4). The fidelities and efficiencies of the P-QND (

*F*,

_{p}*η*) and S-QND (

_{p}*F*,

_{s}*η*) can be obtained (for the even parity states |

_{s}*ϕ*

^{±}〉

*and |*

_{AC}*ϕ*

^{±}〉

*),*

_{BD}*ψ*〉 and |

_{ideal}*ψ*〉 are the final states in the ideal condition and in the experimental condition, respectively. The coupling strength

_{f}*g*≅ 0.5(

*κ*+

*κ*) [70] was reported in

_{s}*d*= 1.5

*μ*m micropillar microcavities (

*Q*∼ 8800), and the coupling strength can be enhanced to

*g*≅ 2.4(

*κ*+

*κ*) (

_{s}*Q*∼ 40000) [71] by improving the sample designs, growth, and fabrication [72]. Here the quality factor is dominated by the side leakage and the cavity loss rate (

*κ*), and

_{s}/κ*κ*can be reduced by thinning down the top mirrors, which may decrease the quality factor, increase

_{s}/κ*κ*, and keep

*κ*nearly unchanged [65]. Using this method, the quality factor

_{s}*Q*∼ 17000 (

*g*≅

*κ*+

*κ*) was achieved with

_{s}*κ*∼ 0.7 [65]. In the condition

_{s}/κ*g*≅ 2.4(

*κ*+

*κ*),

_{s}*κ*∼ 0, and

_{s}/κ*γ*∼ 0.1

*κ*, the fidelities and efficiencies of the two QNDs are

*F*=

_{p}*F*= 100% and

_{s}*η*=

_{p}*η*= 98.8%, respectively. In the case

_{s}*g*≅ 1.3(

*κ*+

*κ*) and

_{s}*κ*∼ 0.3, the fidelities and efficiencies of the two QNDs are

_{s}/κ*F*=

_{p}*F*= 87% and

_{s}*η*=

_{p}*η*= 67%, respectively. While for the case

_{s}*g*≅

*κ*+

*κ*and

_{s}*κ*∼ 0.7, the fidelities and efficiencies of the two QNDs are

_{s}/κ*F*=

_{p}*F*= 72.5% and

_{s}*η*=

_{p}*η*= 58%, respectively. The strong coupling and low cavity side leakage are required in this scheme. Our scheme is implemented with a QD-induced phase shift of $\pm \frac{\pi}{2}$, which requires the photons to interact with the QD-cavity system for twice to achieve a phase shift of ±

_{s}*π*. If the QD-induced phase shift of ±

*π*can be achieved, the photons only have to interact with the QD-cavity system for once, which will increase the fidelity and efficiency of our scheme largely.

The P-QND introduced here is more convenient to complete the interaction between the QD-cavity system and the |*R*〉 polarization of the two spatial modes |*k*_{1}〉 and |*k*_{2}〉 (*k* = *a*, *c*), compared with the P-QND constructed in [40]. Moreover, our hyper-ECP presented here is used to concentrate the two-photon systems in an arbitrary partially hyperentangled Bell state with the interrelationship between the decoherence of the entanglement in the two DOFs, which is not taken into account in the other hyper-ECPs [40,45,46]. With nonlinear optical elements, the success probability of the first round of the hyper-ECP process is almost five times larger than that in the hyper-ECP with linear optical elements [46], which is caused by preserving the states that are discarded in the latter. With the iteration of the hyper-ECP process, the success probability *P* can be increased largely.

In summary, we have proposed a general hyper-ECP for improving the entanglement of the two-photon systems in an arbitrary partially hyperentangled Bell state that decays with the interrelationship between the polarization and the spatial-mode DOFs, resorting to the P-QNDs and S-QNDs that are constructed by the optical property of QD-cavity systems. This hyper-ECP can be straightforwardly generalized for arbitrary partially hyperentangled *N*-photon GHZ-class states, and it is useful in the long-distance quantum communication proposals based on hyperentanglement, especially for the case with the interrelationship between the two DOFs of multi-photon systems.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11175094 and 91221205, and the National Basic Research Program of China under Grants No. 2009CB929402 and No. 2011CB9216002. GLL is a member of the Center of Atomic and Molecular Nanosciences, Tsinghua University.

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