Quantum entanglement lies at the heart of quantum information science and technology and is thought to be the key resource to realize quantum computation and quantum information processing (QIP) tasks, ^{[1– 4]} such as quantum cryptography, ^[1] quantum teleportation, ^[3] quantum secure direct communication, ^[5] quantum cloning machine, ^[6] and so on. For the case of a tripartite system, it is well known that there are two kinds of entanglement, one is the Greenberger– Horne– Zeilinger (GHZ) state,

the other is the W state,

The GHZ state is non-equivalent to W state owing to the fact that they cannot be converted to each other by local operations and classical communication. Great efforts have been taken to research both the GHZ and W states since these two states have been shown to have valuable applications in quantum information science.^{[7– 12]} Lots of theoretical and experiment schemes have been proposed and reported to generate these two states.^{[13– 23]}

Generating remote entanglement is critical for constructing a quantum communication network and realizing long-range quantum information processing.^{[24, 25]} Many meaningful schemes have been proposed to generate remote entangled states^[26] or implement quantum computation^{[27, 28]} in distant participants. For the theoretical long distance quantum information processing case, the noise is not seriously considered in general. But it is one of the most important problems in experiments and practical applications since the quantum system would inevitably couple with its environment. This interaction will decrease the reliability of the quantum channel or even cause the quantum states to be changed.^{[29, 30]} Some schemes, such as entanglement purification, ^{[31– 34]} error correction, ^{[35– 38]} and quantum repeaters^{[39, 40]} have been proposed to solve this problem. In contrast to the above schemes, Song et al. proposed a robust scheme^[41] to generate a remote atomic W state in a polarization noisy channel through an interesting method, i.e. time-bin encoding, which was first proposed by Brendel et al.^[42] and used widely in QIP tasks.^{[43– 53]} The polarization noise in their scheme had no influence on the fidelity of generating the intended state but it would reduce the total success probability to (1/2)^{n– 1} for n nodes. In addition, Huang et al. proposed a fault tolerant quantum secure direct communication against collective noise.^[54] Wu et al. also proposed three quantum dialogue protocols under the condition of collection noise.^[55]

Recently, a novel solid-state QIP system, the NV center in diamond, has been examined intensively^[56] because it has optical controllability and good long-lived spin triplets, even at room temperature.^[57] To generate entanglement, the NV center is always designed to couple to one of the following resonators, i.e. nanomechanical resonators (NAMRs)^[58] and MTRs^{[59, 60]} with a quantized whispering-gallery mode (WGM), ^{[19, 61]} or embed in cavities.^{[62, 63]} During these apparatus, the NV center coupled to MTRs with a quantized WGM is highly competitive because the required Q factor of the MTR can be surely degraded when it couples to a fiber, which allows the photons to be inputted and outputted on demand through fibers. This feature enables researchers to deterministically generate entanglement of different NV centers in a scalable fashion. Experimentally, by using microspheres, ^{[64, 65]} diamond-GaP microdisks, ^[66] and SiN photonic crystals, ^[67] a strong coupling has been demonstrated between the NV center and the WGM. Besides, MTRs or MTRs-like equipment have been paid more attention since single-photon input– output process with the MTR coupling to atom has been reported.^[59] Moreover, the quantum non-demolition measurement for the electronic spin state of diamond NV centers has been proposed^[68] and experimentally demonstrated.^[69] These works have shown that an NV center coupled to MTR is a suitable platform for QIP tasks.

Inspired by the above researches, we propose to generate remote an n-qubit GHZ state and a W state of NV centers coupled to MTRs through noisy channels by using time-bin encoding processes and fast optical switches. The polarization and phase noise induced by noisy channels generally affect the time of state generation but not its success probability and fidelity. The structure of this paper is arranged as follows. In Section 2, we briefly introduce the rationale of the basic model. In Section 3, we describe the process to generate the four-qubit GHZ state and W state through a noisy channel in two nodes. In Section 4, the method to generalize the proposals to n-qubit is given through regular noise-changeable channels among n nodes. A discussion and conclusion are given in Section 5.

The NV centers considered here are negatively charged with six electrons from the nitrogen and three carbons surrounding the vacancy. The ground state are electronic spin triplet states | 0⟩ (m_s = 0), | + 1⟩ (m_s = 1), and | − 1⟩ (m_s = − 1). The six excited states are recorded as | A₁⟩ , | A₂⟩ , | E_x⟩ , | E_y⟩ , | E₁⟩ , and | E₂⟩ .^{[61, 70]} Ground states | 0⟩ and | ± 1⟩ are split with a gap of 2.87 GHz as a result of spin– spin interactions. The excited states are eigenvectors of the full Hamiltonian, which consists of a spin– spin interaction and a spin– orbit interaction without any perturbation. The pair (A₁, A₂) is split from the others at about 5.5 GHz owing to the spin– orbit interaction. The gap between states | A⟩ ₁ and | A⟩ ₂ would be increased to 3.3 GHz through the spin– spin interaction. Consequently, state | A⟩ ₂ would decay to sublevels | − 1⟩ and | + 1⟩ with emitting left (L) and right (R) polarization photons because state | A⟩ ₂ is robust with the stable symmetric properties in the limit of low strain, which has been demonstrated in Ref. [70]. With the aim of detecting ground state information conveniently and obtaining appropriate state transformation process, an external magnetic field B₀ along the NV center symmetry axis should be introduced to the split states | − 1⟩ and | + 1⟩ . The effective energy levels of the NV centers used throughout our proposal are | A⟩ ₂, | + 1⟩ , and | – ⟩ , respectively. A sketch drawing is shown in Fig. 1. Considering a process where a single-photon pulse with frequency ω _in inputs in an MTR cavity with the cavity mode frequency ω _c, and interacts with the NV center through input– output process. Because the qubits are encoded in states | − 1⟩ and | + 1⟩ , the weak excitation condition, i.e. ⟨ σ _z⟩ = − 1, should be satisfied throughout this paper. By combining the Heisenberge motion equations of the NV center lowing operators (σ _– ) and cavity field operator (a), and the input– output relationship , we can achieve the reflection coefficients^{[61, 71– 73]}

where ω ₀, κ , γ , and g denote the transition frequency between states | A₂⟩ and | − 1⟩ , cavity damping rate, NV center dipolar decay rate, and the coupling coefficient between cavity and NV center, respectively. Under the condition ω _c = ω ₀ = ω _in, the reflection coefficients would be simplified to

Fig. 1.

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Fig. 1. A schematic diagram of the single photon input– output process. (a) The NV center fixing on the exterior surface of MTR and interacting with the single photon through input– output process. (b) The effective configuration of the NV center. The transformation from state | − 1⟩ to | A2⟩ is driven resonantly by the input L-polarized single photon. The transformation from state | + 1⟩ to | A2⟩ will not happen in the present scheme due to large detuning caused by the level splitting. B0 is an external magnetic field along the NV center symmetry axis to split | ± 1⟩ state. γ e is the electron gyromagnetic ratio.

After choosing coupling strength , the reflection coefficient in Eq. (4a) is reduced to r(ω _in) = 1. Based on the above analysis, the controlled phase flip gate could be realized between the input pulse and NV center energy levels, which could be expressed as

after applying a phase shifter with π phase on the output pulse.

Next, we shift our attention to illustrate the principle for generalizing the above proposals to n-qubit among two nodes or n (n !⩾ 3) nodes. For the case of n (n ⩾ 3) nodes, the proposals are robust for the regular changeable noise. This merit enables our proposals to have broadened applications in large scale QIP through noisy channels.

4.1. GHZ state

In this subsection, we illustrate the process of n-qubit GHZ state generation among n nodes, which are schematically drawn in Fig. 5. The case to generate n-qubit GHZ states between two nodes can be easily derived by following the process in Subsection 3.1 by adding NV centers behind NV 2 or NV 4 in Fig. 2. Thus, we will focus on describing the n nodes case in this subsection.

Fig. 5.

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Fig. 5. A schematic diagram to generalize GHZ state generation to n-qubit case among n nodes. The role of HWP1-HWP4n− 4 is to perform a σ x operation on the photon pulse, while the role of HWP4n− 3 and HWP4n− 2 is to realize the transformation . The control sequences of optical switches are shown in Fig. 7. Subscript k denotes the sequence number of the node where the optical switches are located. It should be noted that the optical path and the elements from node 2 to node n − 1 are all same, except the the delay lines (marked with green boxes) under the optical switches S9k− 9, whose length is designed as LDj = 2j− 1LD(1 ⩽ j ⩽ n− 1).

We first consider the 3-node case, based on which the n-node case can be understood and derived. The initial state is . Assuming that the noisy channel 1 has the effect , and noisy channel 2 has the effect on the pulse with arriving time inside interval (2L₀+ L₁+ L)/c– (L₀+ 2L₁+ L)/c and on the pulse with arriving time inside interval (2L₀+ L₁+ L+ L_D)/c– (L₀+ 2L₁+ L+ L_D)/c, respectively. The control sequences of optical switches S₁– S₉ and S₁₀– S₁₈ are shown in Fig. 3 and Fig. 6, respectively. The state evolution process can be expressed as

where and . From the above expression (24), we can obtain the following information, that is, a | GHZ⟩ ₊ (| GHZ⟩ ₋ ) state will be achieved if the detector D₁ or D₄ (D₂ or D₃) clicks at any one of the four following times, i.e., (2L₀+ 2L₁+ 2L)/c, (2L₀+ 2L₁+ 2L+ L_D)/c, (2L₀+ 2L₁+ 2L+ 2L_D)/c, and (2L₀+ 2L₁+ 2L+ 3L_D)/c. That is to say, the steady noise introduced by the noisy channel 1 and the regular changeable noise introduced by the noisy channel 2 have no effect on the state generation.

Fig. 6.

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Fig. 6. Control sequences of optical switches S10– S18. The channels following optical switch S11 should have the same influence on the photon with arriving time inside the red-dotted-line-formed rectangle, but can have a different influence on the photon with arriving time inside a different red-dotted-line-formed rectangle. The control sequences of optical switches S1– S9 are identical to Fig. 3. Δ t2⋍ (LD+ L0− L1)/c.

Fig. 7.

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	Fig. 7. Time control sequences of optical switches for n-node (n⩾ 3) GHZ and W states generation. The regular noise-changeable channels following optical switch S9k– 7 should have the same influence on the photon with arriving time inside the red-dotted-line-formed rectangle but can have a different influence on the photon with arriving time inside a different red-dotted-line-formed rectangle.

We now discuss the n-node (n > 3) case. For simplicity, we assume that the noises in different channels are the same and can be expressed as | R⟩ → e^iφ μ | R⟩ + e^iω ν | L⟩ . Initially, suppose that the photon pulse is in state , and each of the NV centers is in state | + ⟩ . According to Fig. 5 and Fig. 7, the process of state evolution could be written as

From the last term in Eq. (25), one can see that if D₄ clicks at the time [(n− 1)L₀+ (n− 1)L₁+ (n− 1)L+ (2^{n− 1}− 1)L_D]/c, then the n-qubit GHZ state would be generated. Similar results would be achieved if other detectors click at the time [(n− 1)L₀+ (n− 1)L₁+ (n− 1)L+ ℓ L_D]/c (ℓ is an integer which belongs to the interval [0, (2^{n− 1}– 1)]).

4.2. W state

The case of the n-qubit W state is schematically drawn in Fig. 8. The case to generate a n-qubit W state between two nodes can be derived similarly according to Subsection 3.2 and Fig. 8(a). Thus, we focus on the case among n nodes. For simplicity, we first consider the three-node case, based on which the n-node case can be deduced. The initial state of the whole system is | ψ ⟩ ₁ = | R⟩ | + ⟩ ₁| + ⟩ ₂| + ⟩ ₃. Assuming that the influence of noisy channels is the same as in Subsection 4.1, then we can have the following process

From Eq. (26), we can see that the steady noise caused by channel 1 and the regular changeable noise caused by channel 2 would influence the time of state generation but have no effect on the success probability and fidelity. The n-node case can also be derived in a similar way based on the sketch in Fig. 8(b) and the control sequences of the optical switches in Fig. 7.

We now discuss the n-node (n > 3) case. For simplicity, supposing that the noises in different channels are the same and can be expressed as | R⟩ → e^iφ μ | R⟩ + e^iω ν | L⟩ . Initially, the photon pulse is in state | R⟩ , and each of the NV centers is in state | + ⟩ . According to Fig. 7 and Fig. 8, the process of state evolution could be written as

From Eq. (27), one can see that the n-qubit W state can be generated if detector D₂ clicks at any one of the times [(n– 1)L₀+ (n– 1)L₁+ (n– 1)L+ ℓ L_D]/c (ℓ is an integer which belongs to the interval [0, (2^{n– 1}– 1)]). Moreover, the noises have no influence on state generation since the overall phase could be eliminated.

Fig. 8.

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Fig. 8. A schematic diagram to generalize W state generation to n-qubit case. The photon pulses are both prepared in | R⟩ state and the NV centers are all prepared in | + ⟩ state. Hl denotes the HWP with the action and HWPk denotes the HWP with the action | R⟩ ↔ | L⟩ . (a) To generate n qubit W state between two nodes. The control sequences are the same as those in Fig. 3. (b) To generate an n qubit W state among n nodes. The control sequences are the same as those in Fig. 6. The optical path and elements from node 2 to node n – 1 are all the same, except for the delay lines (marked with green boxes) under the optical switches S9k– 9, whose length is designed as LDj = 2j– 1LD.

The feasibility of our proposals mainly depends on the time-bin encoding and the performance of our NV center, it is thus essential to discuss the experimental feasibility of these two primary components. The accuracy of the time is seriously dependent on the length of the delay lines. Thus, the delay line lengths L₀, L₁, and L_D are critical in our proposals. The time Δ t, Δ t₁, and Δ t₂ should be big enough to make sure that the status of optical switches has enough time to be changed, which can be achieved by adjusting the lengths of L₀, L₁, and L_D. Meanwhile, the time interval Δ t should be less than L/c to ensure that the noisy channel has a steady influence on the photon pulse with arriving time inside the red-dotted-line-formed rectangle, as shown in Fig. 7. In addition, L_D should have a similar value as L₁ to ensure L₀+ L_D> L₁, this is the prerequisite condition to unmix the pulses with different noises and arriving times. In addition, The reason why we have not considered the interaction time between photon pulses and NV centers is that it has a smaller level. Sometimes it may even be bigger than (L₁– L₀)/c. Since the | L⟩ component of photon pulse with different noises and arriving time will inevitably interact with the NV centers, we can add extra delay lines in the paths which | R⟩ component passes through to make the two paths balanced. The only influence of these balanced paths is to shift the control sequences in Fig. 7 to the right for some definite distance, this is the reason why we have not considered the interaction time. Experimentally, time-bin encoding has been widely researched and demonstrated. Brendel et al. have demonstrated the time-bin encoding with time difference (L₁– L₀)/c in ns orders of magnitude.^{[42, 43]} Recently, Wang et al. have realized the high-speed tomography of time-bin-entangled photons with the time difference 21.3 ns^[44] and Takeda et al. have demonstrated it in a time difference 242 ns.^[45] Besides, time-bin encoding has been verified as interacting with semiconductor quantum wells, ^[46] quantum dots, ^{[47– 49]} collective atomic excitation in a rare-earth-doped solid, ^[50] and it has been realized in other QIP tasks.^{[45, 51]} Furthermore, Humphreys et al. have recently demonstrated a robust heralded controlled-phase gate with higher fidelity by using single-spatial time-bin encoding.^[52] Donohue et al. have recently reported an effective technique using a nonlinear interaction between chirped entangled time-bin photons and shaped laser pulses to perform projective measurements on arbitrary time-bin states with ps-scale separations to readout the information of time-bin encoding.^[53] That is, recent advances in Refs. [52] and Ref. [53] have provided powerful supporting for time-bin-encoding-based QIP tasks.

On the other hand, many efforts have been paid to investigate the performance of the NV centers and the coupled resonators. Although the ideal coupling strength between the NV center and optical resonator is supposed to be on the order of hundreds of megahertz, it is inevitably diminished due to some out-out-control effects.^[75] Research on composite microcavity systems where NV centers in bulk diamond couple evanescently to optical resonators^{[65– 67]} makes it possible to retain the excellent properties of NV centers. The experiment in Ref. [66] provided the datum (g, κ , γ )/2π ≈ (0.3, 26, 0.0004) with a relative low-Q value 10⁴. This datum is very close to the requirement of our proposal , which is the same as Ref. [61]. It deserves special mention that the NV centers used here can be replaced by any system which can realize a controlled phase flip gate.

The efficiency of our proposals is defined as the probability of the photons to be detected after the generation process. Thus, similar to the scheme proposed in Ref. [76], we use to denote the efficiency for GHZ state and W state, in which n denotes number of NV centers. The fidelity of the present proposal is defined as F = | ⟨ ψ _ideal| ψ _prac⟩ | ², where | ψ _ideal⟩ is the ideal final state of the quantum system after processing for state generation, and | ψ _prac⟩ is the final state of the system by considering practical external reservoirs. In Fig. 9, we plot the efficiency of n-qubit state generation and fidelity of 4-qubit state generation, from which we can see that the fidelity would be higher than 92% if the condition g²/κ γ > 15 is satisfied. However, it should be noted that the fidelity would decrease as the number of NV centers grow. The efficiency would be higher than 90% if the cavity parameters g²/κ γ are greater than 50 and the number of NV centers is less than 10. Nevertheless, in practical experiments, the efficiency would also be influenced by other factors; such as, the photon loss induced by the fiber and the insensitivity of the photon detector.

Fig. 9.

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	Fig. 9. (a) Efficiency of n-qubit GHZ and W-states generation. (b) Fidelity of 4-qubit GHZ and W states generation.

In conclusion, we have designed theoretical proposals to generate GHZ and W states of NV centers coupled to microresonator through noisy channels with the help of time-bin encoding processes and optical switches. The polarization and phase noise induced by noisy channels generally affect the time of state generation but not its success probability and fidelity, which is a big progress compared with the success probability 1/2 in Ref. [41]. The proposals can also be generalized to n-qubit between two nodes or among n nodes with n – 1 different noisy channels. In addition, this proposal is robust for the regular changeable noise. For the n-qubit state generation, we here only consider the two extreme cases, i.e. two-node case and n-node case. In fact, the n-qubit state can also be generated in more than two and less than n nodes if the operators place more than one NV center in any node. In other words, our proposals provide an alternative avenue for remote large scale QIP through noisy channels via time-bin encoding.