1. Introduction

Quantum entanglement is a special property of quantum system which can serve as a significant resource in quantum communication, such as quantum key distribution [1,2], quantum dense coding [3,4], quantum secret sharing [5], quantum secure direct communication [6–8], and so on. The high-quality entanglement between distant quantum nodes constitutes an efficient basis for quantum communication. Photon system, possessing various degrees of freedom (DOF), is regarded as an essential quantum system for realizing quantum communication due to its manipulability. Hyperentanglement, defined as the simultaneous entanglement in multiple DOFs of photon systems [9–11], has attracted considerable attention recently, which can largely increase the capacity and improve security of quantum communication [12], and speed up the quantum parallel computation assisted by hyperparallel quantum gates [13–15].

The generation of quantum entanglement is local and the distribution of quantum entanglement is a prerequisite for realizing the effective quantum communication process. However, the random influence originated from quantum channels during quantum state transmission, referred to as channel noise, will inevitably decrease the photonic entanglement or even turn the photon systems into mixed states. This will significantly decrease the security and the transmission efficiency of quantum communication. In order to build an extensive quantum network, the quantum repeater is proposed to suppress the decoherence caused by environmental noise [16]. Entanglement purification is one of the core constituents of quantum repeaters in long-distance quantum communication, and it can distil some high-fidelity entangled quantum systems from the mixed entangled ones [17,18]. Since the first entanglement purification protocol (EPP) was proposed with the help of quantum controlled not (CNOT) gates and bilateral rotations [19], some practical EPPs have been presented with different ways, mainly including conventional EPPs [20–24] and the deterministic EPPs [25,26]. These protocols are very useful for obtaining entanglement in a single DOF.

When high-capacity quantum communication with the help of hyperentanglement is involved, the quantum communicating parties need to distil the high-fidelity hyperentanglement from the mixed hyperentangled ones, contaminated by the channel noise in the entanglement distribution process. In 2013, Ren et al. proposed an initial hyperentanglement purification protocol (hyper-EPP) with parity-check quantum nondemolition detectors (QNDs) for two-photon systems [27]. In 2015, Wang et al. proposed an efficient hyper-EPP for Bell states from photon losses and decoherence in a heralded way, assisted by the linear optics and four-qubit entangled Greenberger-Horne-Zeilinger (GHZ) state [28], in which the GHZ state was regarded as auxiliary resource. Subsequently, Du et al. presented a refined hyper-EPP for two-photon systems with cavity-assisted interaction [29].

Up to now, there have been several interesting EPPs [19–26] and hyper-EPPs [27–29] focusing on entangled and hyperentangled two-photon systems, respectively. However, there are few EPPs for multiple photon systems [30–34] in single DOF. For example, a multipartite entanglement purification protocol (MEPP) to purify the GHZ state with CNOT gates [30] and another MEPP with the QNDs relying on cross-Kerr nonlinearities [31] were proposed. In two MEPPs [30,31], the original fidelities are required to be larger than 1/2, and meanwhile plenty of entangled quantum resources are discarded. In 2011, Deng presented a MEPP with the entanglement link [32], which had a higher efficiency than the conventional MEPPs [30,31]. Therefore, it becomes very important to study an efficient multiphoton hyper-EPP with lower original fidelity in each DOF by fully utilizing quantum resources, for nonlocal mixed hyperentangled GHZ states in both the spatial-mode and polarization DOFs.

Based on the previous discussion, we propose an efficient spatial-polarization hyper-EPP for three-photon systems with bit-flip errors in two DOFs. Our hyper-EPP contains two steps. The first step resorts to fidelity-robust spatial-polarization parity-check gates, which will retain the effective three-photon system with higher fidelities in either DOF. The second one makes full use of the cross combinations with swap gates and hyperentanglement link (HL). By utilizing quantum swap gates, one transforms the higher-fidelity hyperentanglement of different three-photon systems into one three-photon system, meanwhile HL is used to reproduce some three-photon hyperentangled systems from hyperentangled two-photon subsystems. Moreover, the quantum circuits make this hyper-EPP works faithfully, as the errors coming from practical scattering, are transferred into a detectable failure rather than infidelity with the modified NV union. Furthermore, our hyper-EPP is suitable to purify the multiphoton systems entangled in one DOF (spatial-mode, polarization, orbital angular momentum or time-bin) by utilizing our fidelity-robust parity-check gates, and meanwhile it can be directly extended to purify the photon systems hyperentangled in multiple DOFs.

2. Interaction rules for a circularly polarized photon with a NV unit

Nitrogen-vacancy (NV) center is constituted by a substitutional nitrogen atom and an adjacent vacancy in the diamond lattice, in which there are six electrons from the nitrogen and three carbons surrounding the vacancy. Due to its optical properties and superb spin coherence, the NV center becomes one of the most promising platforms to complete quantum communication at room temperature. The ground state of the NV center is an electron spin triplet with the splitting at $2.87$ GHz between the magnetic sublevels $|0\rangle$ ($|m_s=0\rangle$) and $|\pm 1\rangle$ ($|m_s=\pm 1\rangle$). Concurrently, six electronic excited states are existed according to the spin-orbit and spin-spin interactions [35]. Optical transitions between the ground and the excited states are spin conservation, while the electronic orbital angular momentum is changed depending on the photon polarization. The excited state $|A_2\rangle$ is robust due to the stable symmetric properties, decay with an equal probability to the ground states $|-1\rangle$ and $|+1\rangle$ through the polarization radiations $\hat {\sigma }_+=+1$ and $\hat {\sigma }_-=-1$ along the NV axis (z axis) [36], respectively, shown in Fig. 1(a).

 

Fig. 1. (a) Schematic diagrams of an NV-center-cavity system, and the optical transitions of the NV center with the circularly polarized photons. $R^{\uparrow } (R^{\downarrow })$ and $L^{\uparrow } (L^{\downarrow })$ represent the right- and left-circularly polarized photons propagating along (against) the quantization axis $z$, respectively. (b) Schematic diagram of a modified NV union. SW is an optical switch, which makes the photons entering into and going out of the circuit unit in sequence. M is a mirror and D is a single-photon detector. BS is a $50:50$ beam splitter, which performs the Hadamard operation, that is, $|i_{1}\rangle \rightarrow (|j_{1}\rangle +|j_{2}\rangle )/\sqrt {2}$, or $|i_{2}\rangle \rightarrow (|j_{1}\rangle -|j_{2}\rangle )/\sqrt {2}$, in the spatial-mode DOF of one photon. H represents a quarter-wave plate, which performs the Hadamard operation, that is, $|R\rangle \rightarrow (|R\rangle +|L\rangle )/\sqrt {2}$, or $|L\rangle \rightarrow (|R\rangle -|L\rangle )/\sqrt {2}$, in the polarization DOF of one photon.

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In the double-sided cavity-NV-center system, two waveguides are simultaneously coupled to the cavity mode with the coupled constants $\mu _1$ and $\mu _2$, respectively. Suppose the coupled rates of the two waveguides can achieve nearly the same in both directions, that is, $\mu _1\cong \mu _2=\mu$. The cavity-NV-center system, shown in Fig. 1(a), can be calculated by the Heisenberg equations of motion for the cavity field operator $\hat {a}$ and the diploe operator $\hat {\sigma }_-$ [13,37–39],

We can convert the above NV-cavity system into a modified NV unit by assisting some linear optical elements shown in Fig. 1(b), which work efficiently and realize unitary transformations. The electron spin of the NV is initially prepared in the quantum state $|\varphi ^{+}\rangle =\frac {1}{\sqrt {2}}(|+1\rangle +|-1\rangle )$ and a right-circularly polarized photon $|R\rangle$ enters into the upper mode $i_{1}$. After the photon is performed on Hadamard operations in both the spatial-mode DOF with BS and the polarization DOF with H, described in the caption of Fig. 1, and is reflected by each of the mirrors (Ms), then it will interact with the NV-cavity system. Generally, the polarization state of the photon will be changed once reflected by mirrors and BS [40], yet one can assist a half-wave plate performing a polarization bit-flip operation after each reflection [41] to make it unchanged.

Sequentially, the photon with different polarizations will be reflected by either M, passes through H, and then converges on the BS again. For the photon in the upper mode $i_{1}$, it will traverse an optical switch (SW), which can make the photon to reach an appropriate mode $i'_{1}$, and click the single-photon detector (D). Therefore, the states of the hybrid system consisting of the photon and the NV-cavity system before the click of the D can be denoted as

It is self-evident that our modified NV union is robust, shown in Fig. 1(b), which makes the right-circularly polarized photon $|R\rangle$ possess two output modes no matter what the NV state is. If the ejected photon is in the upper mode $i'_{1}$, it will trigger the D and the electron spin of the NV will remain unchanged for recycling in the next round; while the ejected photon in the lower mode $i'_{2}$ transforms into the left-circularly polarized photon $|L\rangle$ accompanied by a phase-flip operation on the electron spin, shown in Eq. (4). Based on the modified NV union, we propose three fidelity-robust quantum circuits below and they could be used in quantum computing and quantum networks in a heralded way.

3. Establishing fidelity-robust S-P-PCG and swap gates

3.1 Establishing fidelity-robust spatial-polarization parity-check gate (S-P-PCG)

A hyperentangled Bell state of two-photon system in the spatial-mode and polarization DOFs can be written as $|\Psi \rangle =|\phi \rangle ^{S}\otimes |\phi \rangle ^{P}$, where $|\phi \rangle ^{S}$ is one of the four Bell states in the spatial-mode DOF ${(}|\phi _{1}^{\pm }\rangle ^{S} =1/\sqrt {2}~(|a_{1}\bar {a}_{1}\rangle \pm |a_{2}\bar {a}_{2}\rangle )$, $|\phi _{2}^{\pm }\rangle ^{S} =1/\sqrt {2}~(|a_{1}\bar {a}_{2}\rangle \pm |a_{2}\bar {a}_{1}\rangle ))$, and $|\phi \rangle ^{P}$ is one of the four Bell states in the polarization DOF $|\phi _{1}^{\pm }\rangle ^{P} =1/\sqrt {2}~(|RR\rangle \pm |LL\rangle)$, $|\phi _{2}^{\pm }\rangle ^{P} =1/\sqrt {2}~(|RL\rangle \pm |LR\rangle ))$.

To establish the fidelity-robust spatial-polarization parity-check gate (S-P-PCG) of the two photons $A_{1}A_{2}$, one initializes the electron spins to the state $|\varphi ^{+}_{k}\rangle =\frac {1}{\sqrt {2}}(|+1\rangle +|-1\rangle )(k=1,2)$ in each modified NV union, and then inputs photons $A_{1}$ in spatial mode $a_{1}$ and $a_{2} (l = a)$ and $A_{2}$ in spatial mode $\bar {a}_{1}$ and $\bar {a}_{2} (l = \bar {a})$ into the quantum circuits on the left of Fig. 2, corresponding to the spatial parity-check gate (S-PCG). The left-circularly polarized component $|L\rangle$ in spatial mode $l_{2} (l = a, \bar {a})$ will first pass through a half-wave plate X performing a bit-flip operation on the polarization DOF of the photon $\sigma _{X}^{P}= |R\rangle \langle L| + |L\rangle \langle R|$, and then be emitted by the modified NV union with the reflection coefficient $P_{1}$ and the transmission coefficient $P_{2}$, detailed in Eq. (4), while the right-circularly polarized component $|R\rangle$ in spatial mode $l_{2}$ will first be emitted by the modified NV union and then pass through X. If two photons traverse the modified NV$_{1}$ union with the detector (D) unresponsive, the two components will converge on another CPBS again and then export to the spatial mode $l_{2}$. However, for the spatial mode $l_{1}$, both the left-and right-circularly polarized wave packets will directly pass through a partially transmitted mirror (T) with the transmission coefficient $P_{2}$. The S-PCG is finished.

 

Fig. 2. Schematic diagram of the fidelity-robust spatial-polarization parity-check gate (S-P-PCG) for a two-photon system. CPBS is a circularly polarizing beam splitter, which reflects the left-circular-polarization photon $\vert L\rangle$ and transmits the right-circular-polarization photon $\vert R\rangle$, respectively. X is a half-wave plate, which performs a bit-flip operation on the polarization DOF of the photon. T is a partially transmitting mirror.

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Subsequently, schematic diagram of the polarization parity-check gate (P-PCG) of the two photons $A_{1}A_{2}$ is shown on the right of the Fig. 2. The left-circularly polarized component $|L\rangle$ in two spatial modes $l_{1}$ and $l_{2}(l = a, \bar {a})$ will pass through X and then be ejected by the modified NV$_{2}$ union with D unresponsive; while the right-circularly polarized photon $|R\rangle$ in both spatial modes will directly pass through T. Then the two components of each photon will converge on another CPBS again and then export to the spatial modes $l_{1}$ and $l_{2}$, realizing the parity check in the polarization DOF. If the photon $A_{1}$($A_{2}$) is reflected by the two modified NV unions, it will not trigger either of the two Ds, which marks the success of the S-P-PCG. Therefore, the evolutions of the whole system consisting of photons $A_{1}A_{2}$ and the electron spins are described as follows

The failure of the S-P-PCG for photons in two DOFs can be heralded by the responses of the single-photon detectors in the two modified NV unions. However, once the S-P-PCG succeeds, their fidelities approach unity, as original infidelity term is transformed into the failure of detection. The efficiency of the S-PCG (or P-PCG) is $\eta _{p} = |P_{2}|^{4}$, in principle, ignoring nonradiative decay of excited state of the NV center, photon loss in propagation and detector failure, mode mismatch and crosstalk [42,43].

3.2 Establishing fidelity-robust swap gates

3.2.1 Establishing fidelity-robust S-S-swap gate

Suppose two photons $A$ and $B$ encoded in the spatial-mode and polarization DOFs are

 

Fig. 3. (a) Schematic diagram of the fidelity-robust spatial-spatial-swap (S-S swap ) gate. (b) Schematic diagram of the fidelity-robust polarization-polarization-swap (P-P-swap) gate. The red quantum circuit represent that the photons will enter again for the second round.

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Subsequently, each component of the two photons will passes through the same optical path as it does in the first procedure called the second round. After Hadamard operations performed on the photons $AB$ and the electron spin again, the above operations (S-PCG $\rightarrow$ $H^{S}_{AB}$, $H_{e}\rightarrow$ S-PCG $\rightarrow$ $H^{S}_{AB}$, $H_{e}$) transform the state $|\phi \rangle _{A}\otimes |\phi \rangle _{B}\otimes |\varphi ^{+}\rangle$ of the whole system into

3.2.2 Establishing fidelity-robust P-P-swap gate

To implement the fidelity-robust polarization-polarization-swap (P-P-swap) gate in polarization states of two photons, shown in Fig. 3(b), DL is a time-delay device [44], which makes the two wave packets coming from different paths successively reach the SWs. The photons $A$ and $B$ pass through the same optical path. Similar to the above discussion of S-S-swap gate, one first performs the operations (P-PCG $\rightarrow$ $H^{P}_{AB}$, $H_{e}\rightarrow$ P-PCG $\rightarrow$ $H^{P}_{AB}$, $H_{e}$) and then measures the electron spin of the NV in the basis $\{|+1\rangle ,|-1\rangle \}$, adding to the single-qubit feedback according to the outcomes of the electron measurement. That is, the polarization states of the photons $A$ and $B$ swap each other as follows

4. Efficient spatial-polarization hyper-EPP for three-photon systems

In the practical transmission of photons in hyperentangled GHZ states for high-capacity quantum communication, both the bit-flip and phase-flip errors will occur randomly. Usually, the phase-flip error can be transformed into the bit-flip error with bilateral local operations. Therefore, we only discuss the purification for bit-flip errors of three-photon hyperentangled mixed states below.

In this section, we generalize our hyper-EPP for mixed hyperentangled GHZ states of three-photon systems. There are eight spatial-mode GHZ states and eight polarization GHZ states, and they are described as follows

4.1 The first step of our hyper-EPP with S-P-PCG

The process of the first step of our hyper-EPP is shown in Fig. 4. In Alice’s side, it can be rebuilt in two halves, which consist of performing the S-P-PCG on the photons $A_{1}A_{2}$ in the spatial-mode and polarization DOFs shown in Fig. 4(a) at first, and then detecting the photon ${A_{2}}$ with the single-photon measurement device (SPMD) shown in Fig. 4(b). So do Bob and Charlie. After Alice, Bob, and Charlie perform the S-P-PCGs on the photons $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ in two DOFs, respectively, the outcomes of the parity are divided into four instances.

 

Fig. 4. (a) Schematic diagram of the first step of our hyper-EPP with S-P-PCGs. (b) Schematic diagram of a single-photon measurement device (SPMD).

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In instance (1), the states of the photons $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ are all of the same parity (i.e., even or odd) in two DOFs, which corresponds to the identity combinations $|\psi ^{+}_{l}\rangle ^{S}_{A_{1}B_{1}C_{1}}\otimes |\psi ^{+}_{l}\rangle ^{S}_{A_{2}B_{2}C_{2}}(l=0, 1, 2, 3)$ in the spatial-mode DOF and the identity combinations $|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}}\otimes |\psi ^{+}_{m}\rangle ^{P}_{A_{2}B_{2}C_{2}}(m=0, 1, 2, 3)$ in the polarization DOF. In detail, if the parities of the photons $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ are even in two DOFs, the states of the six-photon system $A_{1}B_{1}C_{1}A_{2}B_{2}C_{2}$ is in a new mixed state composed of the eight states

Subsequently, Alice, Bob, and Charlie detect the photons $A_{2}$, $B_{2}$, and $C_{2}$ with the SPMDs, shown in Fig. 4(b), respectively. The outcomes will divide the instances into two groups. In the first group, the numbers of the outcomes $|k_{1}\rangle (k=\bar {a}, \bar {b}, \bar {c})$ $ (i.e., |\bar {a}_{1}\bar {b}_{1}\bar {c}_{1}\rangle$, $|\bar {a}_{1}\bar {b}_{2}\bar {c}_{2}\rangle$, $|\bar {a}_{2}\bar {b}_{1}\bar {c}_{2}\rangle$, or $|\bar {a}_{2}\bar {b}_{2}\bar {c}_{1}\rangle )$ in the spatial-mode DOF and the outcomes $|R\rangle$ $(i.e., |RRR\rangle$, $|RLL\rangle$, $|LRL\rangle$, or $|LLR\rangle )$ in the polarization DOF are odd. In this time, Alice, Bob, and Charlie obtain the state $|\psi _{l}^{+}\rangle ^{S}_{A_{1}B_{1}C_{1}} (l=0, 1, 2, 3)$ and $|\psi _{l}^{+}\rangle ^{P}_{A_{1}B_{1}C_{1}} (m=0, 1, 2, 3)$ with the probabilities $\frac {1}{2}g_{l}^{2}$ and $\frac {1}{2}f_{m}^{2}$, respectively, the first step of the hyper-EPP is finished directly. In the second group, the number of the outcomes $|k_{1}\rangle (k=\bar {a}, \bar {b}, \bar {c})$ $(i.e., |\bar {a}_{2}\bar {b}_{2}\bar {c}_{2}\rangle$, $|\bar {a}_{2}\bar {b}_{1}\bar {c}_{1}\rangle$, $|\bar {a}_{1}\bar {b}_{2}\bar {c}_{1}\rangle$, or $|\bar {a}_{1}\bar {b}_{1}\bar {c}_{2}\rangle )$ in the spatial-mode DOF and $|R\rangle$ $(i.e., |LLL\rangle$, $|LRR\rangle$, $|RLR\rangle$, or $|RRL\rangle )$ in the polarization DOF are even, and the three parties obtain the states $|\psi _{l}^{-}\rangle ^{S}_{A_{1}B_{1}C_{1}} (l=0, 1, 2, 3)$ and $|\psi _{m}^{-}\rangle ^{P}_{A_{1}B_{1}C_{1}} (m=0, 1, 2, 3)$ with the probabilities $\frac {1}{2}g_{l}^{2}$ and $\frac {1}{2}f_{m}^{2}$, respectively. Alice, Bob, and Charlie can transform the states $|\psi _{l}^{-}\rangle ^{S}_{A_{1}B_{1}C_{1}}$ $( |\psi _{m}^{-}\rangle ^{P})_{A_{1}B_{1}C_{1}}$ into the states $|\psi _{l}^{+}\rangle ^{S}_{A_{1}B_{1}C_{1}}$ $( |\psi _{m}^{+}\rangle ^{P}_{A_{1}B_{1}C_{1}})$ with the spatial-mode (polarization) phase-flip operation $\sigma _{Z}^{S}=|\bar {a}_{1}\rangle \langle \bar {a}_{1} |-|\bar {a}_{2}\rangle \langle \bar {a}_{2}|$ $(\sigma _{Z}^{P}=|R\rangle \langle R|-|L\rangle \langle L|)$ on the first photon $A_{ 1}$, then the first step of the hyper-EPP is finished. In this case, Alice, Bob, and Charlie will reserve the three-photon system $A_{1}B_{1}C_{1}$, which will contribute to the final purified states of the hyper-EPP. That is, the parties only distill some high-fidelity three-photon hyperentangled systems from the identical combinations by keeping the instances, in which all the three parties obtain the same parity in two DOFs, and measure the photons $A_{2}B_{2}C_{2}$, similar to all existing EPPs for GHZ state only in the polarization DOF [30–32].

In instance (2), the states of two photons $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ are all of the different parities in two DOFs, which correspond to the cross combinations $|\psi ^{+}_{l}\rangle ^{S}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{q}\rangle ^{S}_{A_{2}B_{2}C_{2}}(l \neq q \in \{0, 1, 2, 3\})$ with the probability of $g_{l}g_{q}$ in the spatial-mode DOF and the cross combinations $|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{n}\rangle ^{P}_{A_{2}B_{2}C_{2}}(m\neq n \in \{0, 1, 2, 3\})$ with the probability of $f_{n}f_{m}$ in the polarization DOF. Alice, Bob, and Charlie discard the instance in the conventional three-photon EPPs [30,31], as the probabilities of the spatial-mode (polarization) states $|\psi ^{+}_{l}\rangle ^{S}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{q}\rangle ^{S}_{A_{2}B_{2}C_{2}}$ $ (|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{n}\rangle ^{P}_{A_{2}B_{2}C_{2}} )$ and $|\psi ^{+}_{q}\rangle ^{S}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{l}\rangle ^{S}_{A_{2}B_{2}C_{2}}$ $ (|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{n}\rangle ^{P}_{A_{2}B_{2}C_{2}})$ are the same one $g_{l}g_{q}(f_{n}f_{m})$, that is, the group of the three-photon systems is ambiguous about which one of the three-photon systems $A_{1}B_{1}C_{1}$ and $A_{2}B_{2}C_{2}$ has the spatial-mode (polarization) bit-flip error. However, these cross combinations can be used to distill a high-fidelity two-photon hyperentangled state, with which Alice, Bob, and Charlie can reproduce a subset of high-fidelity hyperentangled three-photon systems again, called hyperentanglement link (HL). The relation between the cross combinations $|\psi ^{+}_{l}\rangle ^{S}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{q}\rangle ^{S}_{A_{2}B_{2}C_{2}}(l \neq q \in \{0, 1, 2, 3\})$ in spatial-mode DOF and $|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{n}\rangle ^{P}_{A_{2}B_{2}C_{2}}(m\neq n \in \{0, 1, 2, 3\})$ in polarization DOF, and the hyperentangled two-photon systems is shown in Table 1, where $\vert \phi ^+_1\rangle _{MN}^{S} =\frac {1}{\sqrt {2}}(|s_{1}\rangle |\bar {s}_{1}\rangle +|s_{2}\rangle |\bar {s}_{2}\rangle )$, $\vert \phi ^+_2\rangle _{MN}^{S} =\frac {1}{\sqrt {2}}(|s_{1}\rangle |\bar {s}_{2}\rangle +|s_{2}\rangle |\bar {s}_{1}\rangle )$, $|\phi _{1}^{\pm }\rangle _{MN}^{P} =1/\sqrt {2}~(|RR\rangle \pm |LL\rangle )$, and $|\phi _{2}^{\pm }\rangle _{MN}^{P} =1/\sqrt {2}~(|RL\rangle \pm |LR\rangle )$ $(M\neq N\in \{A_{1},B_{1},C_{1}\}; s=a,b,c)$. In Table 1, the hyperentangled two-photon systems shared by two of the three parties can be described with the following density matrices

Table 1. The states of the two-photon systems obtained from cross-combinations and their probabilities (suppose $x=A_{1}B_{1}C_{1}$ and $y=A_{2}B_{2}C_{2}$ for simplification).

View Table | View all tables in this article

In instance (3), the states of the photons $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ are all of the same parity in the spatial-mode DOF but of the different parities in the polarization DOF, which corresponds to the identity combinations $|\psi ^{+}_{l}\rangle ^{S}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{q}\rangle ^{S}_{A_{2}B_{2}C_{2}}(l = q \in \{0, 1, 2, 3\})$ with the probability of $g_{l}^{2}$ in the spatial-mode DOF and the cross combinations $|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{n}\rangle ^{P}_{A_{2}B_{2}C_{2}}(m\neq n \in \{0, 1, 2, 3\})$ with the probability of $f_{n}f_{m}$ in the polarization DOF. For the spatial-mode DOF, the states of the six-photon system $A_{1}B_{1}C_{1}A_{2}B_{2}C_{2}$ in this case are the same as those in instance (1), possessing the same results as those in instance (1), which means that Alice, Bob, and Charlie would regain the three-photon system ${A_{1}B_{1}C_{1}}$ to perform the second step of our hyper-EPP.

In instance (4), the states of the photons $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ are of the different parities in the spatial-mode DOF, but of the same parity in the polarization DOF, which corresponds to the cross combinations $|\psi ^{+}_{l}\rangle ^{S}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{q}\rangle ^{S}_{A_{2}B_{2}C_{2}}(l \neq q \in \{0, 1, 2, 3\})$ with the probability of $g_{l}g_{q}$ in the spatial-mode DOF and the identity combinations $|\psi ^{+}_{m}\rangle ^{P}_{A_{1}B_{1}C_{1}} \otimes |\psi ^{+}_{n}\rangle ^{P}_{A_{2}B_{2}C_{2}}(m= n \in \{0, 1, 2, 3\})$ with the probability of $f_{m}^{2}$ in the polarization DOF. Similarly, the states of the six-photon system $A_{1}B_{1}C_{1}A_{2}B_{2}C_{2}$ in this case are the same as those in instance (1) for the polarization DOF, possessing the same results as those in instance (1), which means that Alice, Bob, and Charlie would regain the three-photon system ${A_{1}B_{1}C_{1}}$ to perform the second step of our hyper-EPP.

4.2 The second step of hyper-EPP with swap gates and HL

4.2.1 The second step of hyper-EPP with S-S-swap or P-P-swap gates

For some three-photon systems obtained from instances (3) and (4), the hyperentangled photons have the higher fidelity in the spatial-mode and polarization DOFs, respectively. One can perform the second step of the hyper-EPP with S-S-swap (or P-P-swap) gates in Fig. 3, to change the higher-fidelity components from various three-photon systems into the same three-photon system accompanied by higher fidelities in two DOFs. This step will extremely improve the efficiency of our hyper-EPP. Suppose Alice, Bob, and Charlie share two three-photon systems $A_{1}B_{1}C_{1}$ and $A'_{1}B'_{1}C'_{1}$(only substituting $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$, $c_{1}$, and $c_{2}$ of three-photon system $A_{1}B_{1}C_{1}$ for $a'_{1}$, $a'_{2}$, $b'_{1}$, $b'_{2}$, $c'_{1}$, and $c'_{2}$, respectively, in the spatial mode DOF) after the first step. Here, $A_{1}B_{1}C_{1}$ is created from case (3) with a higher fidelity in the spatial-mode DOF and $A'_{1}B'_{1}C'_{1}$ is created from case (4) with a higher fidelity in the polarization DOF.

Generally, there are two possible purification groups. The first one is to purify the three-photon system $A_{1}B_{1}C_{1}$ with the three-photon system $A'_{1}B'_{1}C'_{1}$. Alice, Bob, and Charlie perform the fidelity-robust P-P-swap gates on the photon pairs $A_{1}A'_{1}$, $B_{1}B'_{1}$, and $C_{1}C'_{1}$, respectively, as shown in Fig. 5(a). After detecting the photons $A'_{1}$, $B'_{1}$, and $C'_{1}$ with the SPMDs, Alice, Bob, and Charlie finish the purification of the three-photon system $A_{1}B_{1}C_{1}$, by replacing the lower-fidelity polarization state of three-photon system $A_{1}B_{1}C_{1}$ with the higher-fidelity polarization state of three-photon system $A'_{1}B'_{1}C'_{1}$. Now, the three-photon system $A_{1}B_{1}C_{1}$ in two DOFs are both of higher fidelities, which are the same as those in instance (1) in principle. The other one is to purify the three-photon system $A'_{1}B'_{1}C'_{1}$ with the three-photon system $A_{1}B_{1}C_{1}$. This group is much similar to the first one, while three parties need to use the S-S swap gates shown in Fig. 3(a). The performances of both purification groups, in principle, are the same as each other.

 

Fig. 5. (a) Schematic diagram of the second step of our hyper-EPP with P-P-swap gates. (b) Schematic diagram of the second step of our hyper-EPP with hyperentanglement link (HL).

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4.2.2 The second step of hyper-EPP with hyperentanglement link (HL)

For the two-photon systems obtained in instance (2), as the original three-photon hyperentangled states are symmetric to each other, we use the states $\rho _{A_{1}B_{1}}$ and $\rho _{A_{2}C_{1}}$ as an example to describe the principle of hyperentangled three-photon production from a subset of hyperentangled two-photon subsystems with HL, shown in Fig. 5(b), and assume that $g_{1} = g_{2} = g_{3}$ and $f_{1} = f_{2} = f_{3}$. The density matrices in Eq. (13) become

Table 2. The states of the three-photon hyperentangled systems obtained from two two-photon systems and their probabilities with HL (suppose $x=A_{1}B_{1}$ , $y=A_{2}C_{1})$.

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4.3 The efficiency of our hyper-EPP

After the above two purification steps, the first round of the hyper-EPP process is accomplished. The fidelity of the three-photon system $A_{1}B_{1}C_{1}$ can be improved further by repeating multiple rounds of the hyper-EPP in both two DOFs. Furthermore, the efficiency of obtaining the purified three-photon system $A_{1}B_{1}C_{1}$ after the first purification step is $\eta _{1}$, while the efficiency of obtaining the three-photon system $A_{1}B_{1}C_{1}$ after introducing the second purification step with swap gates (i.e., the efficiency $\eta _{S}$) and HLs (i.e., the efficiency $\eta _{H}$) changes into $\eta _{2}$,

 

Fig. 6. (a) The efficiency $\eta _{1}$ and (b) The efficiency $\eta _{2}$ versus the initial spatial-mode fidelity $g_{0}$ and polarization fidelity $f_{0}$, respectively.

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5. Discussion and conclusion

In our hyper-EPP, the S-P-PCGs finish the first step by picking out the useful combinations from the original mixed state contributing to the final entanglement, and then the S-S-swap (or P-P-swap) gates and HL, fully utilizing the cross combinations discarded in previous MEPPs for only one DOF [30,31], will be used to greatly improve the efficiency of the hyper-EPP. Obviously, the S-P-PCG is also a core component of the HL. Therefore, the S-P-PCGs and the S-S-swap (or P-P-swap) gates are two essential constituents for implementing the high-efficiency hyper-EPP in both the spatial-mode and polarization DOFs. As above discussed in the sections 2 and 3, the S-P-PCGs and the S-S-swap (or P-P-swap) gates can be constructed with the interaction between the photon and the NV union. The nonideal scattering effects, the nonzero side leakage of the cavity and the finite coupled strength between the cavity and the NV, will inevitably reduce the operation of the hyper-EPP for original photon systems, such as indistinguishable parities in two DOFs and swapping process imperfectly. Fortunately, we transfer the infidelity coming from practical scattering into a predictable failure that triggers the single-photon detectors. In other words, the fidelity of the S-PCG (or P-PCG) and one of the S-S-swap (or P-P-swap) gate, in principle, approach unity when the single-photon detectors are nonresponsive, without regard to the nonradiative decay of excited state of the NV center, photon loss in propagation and detector failure, mode mismatch and crosstalk [42,43].

However, the efficiencies of the S-PCG (or P-PCG) and the S-S-swap (or P-P-swap) gate are $\eta _{p} = |P_{2}|^{4}$ and $\eta _{s} = |P_{2}|^{8}$, respectively, where $P_{2} =\frac {1}{2}(t + r -t_{0 }- r_{0})$. Considering the dipole resonant with the cavity mode $\omega _c=\omega _k=\omega _0=\omega$, the reflection and transmission coefficients become $t = -(2F_{ p }+ 1 +\lambda /2 )^{-1}$ and $r = (4F _{p }+\lambda )/(4F _{p }+\lambda +2)$ for $g\neq 0$, and they are $t_{ 0} = -(1 +\lambda /2 )^{-1}$ and $r_{ 0} =\lambda /(2 +\lambda )$ for $g = 0$, where the Purcell factor $F_{ p }= g^{ 2} /(\mu \gamma )$ and the cavity decay rate $\lambda =\kappa /\eta$. As shown in Figs. 7(a)–7(b), the efficiencies $\eta _{p}$ and $\eta _{s}$ are largely affected by the Purcell factor $F_{ P}$ and the cavity decay rate $\lambda$ from the practical NV unions. In the case $F_{ P}=4$ and $\lambda = 0.1$ [42], the efficiencies are $\eta _{p}=80.38\%$ and $\eta _{s}=64.60\%$, respectively. Obviously, the efficiencies $\eta _{p}$ and $\eta _{s}$ could be further improved by increasing the Purcell factor $F_{ P}$ and reducing the cavity decay rate $\lambda$.

 

Fig. 7. (a) The efficiency $\eta _{p}$ of our fidelity-robust S-PCG (or P-PCG) vs the Purcell factor $F_{ P}$ and the cavity decay rate $\lambda$ with the condition $\omega =\omega _{c}=\omega _{X^{-}}$. (b) The efficiency $\eta _{s}$ of our fidelity-robust S-S-swap (or P-P-swap) gate vs the same condition.

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Our efficient hyper-EPP is assisted by the NV unions, in which the transmission-reflection property of circularly polarized photon interacting with the double-sided cavity-NV-center system can be used for photon-photon interaction in quantum information processing in view of both the spatial-mode and polarization DOFs. The NV center in diamond is an appropriate dipole emitter in cavity QED to obtain the high-fidelity transmission-reflection property in the Purcell regime, with nanosecond manipulation time [45] and long spin coherence time (ms) by using dynamical decoupling techniques [46]. Moreover, the transmission-reflection property of double-sided-cavity-NV systems having two spatial modes is more flexible to perform the quantum information processing for photon systems in two DOFs, as the large transmittance and reflectance difference between the coupled cavity and the uncoupled one are symmetrical than the reflection property of one-sided-cavity condition [27]. As the phonon sideband of NV center is quite wide even under 1.6 K [47], to increase the efficiency of the NV union, the single-mode photonic device [48] can suppress the emission. Besides, the assumption of weak excitation limit inevitably restricts the working frequency of our protocol [39].

In summary, we have proposed the hyper-EPP for three-photon systems hyperentangled in the spatial-mode and polarization DOFs, which has a higher efficiency optimizing quantum resources based on the fidelity-robust S-P-PCGs, S-S-swap (or P-P-swap) gates, and HLs. One pumps the higher-fidelity hyperentanglement of different three-photon systems into the same one by quantum swap gates, and simultaneously reproduces the three-photon hyperentangled systems from hyperentangled two-photon subsystems by HLs. Moreover, the current quantum circuits make our hyper-EPP operates faithfully, which is different from the previous hyper-EPP resorting to the reflection property of one-sided-cavity system [27], as the errors from practical scattering are transferred into the click of the single-photon detector with the modified NV union. Furthermore, our hyper-EPP is suitable to purify the multiphoton systems entangled in one DOF, and purify the photon systems hyperentangled in multiple DOFs.