Abstract

High-dimensional quantum system provides a higher capacity of quantum channel, which exhibits potential applications in quantum information processing. However, high-dimensional universal quantum logic gates is difficult to achieve directly with only high-dimensional interaction between two quantum systems and requires a large number of two-dimensional gates to build even a small high-dimensional quantum circuits. In this paper, we propose a scheme to implement a general controlled-flip (CF) gate where the high-dimensional single photon serve as the target qudit and stationary qubits work as the control logic qudit, by employing a three-level Λ-type system coupled with a whispering-gallery-mode microresonator. In our scheme, the required number of interaction times between the photon and solid state system reduce greatly compared with the traditional method which decomposes the high-dimensional Hilbert space into 2-dimensional quantum space, and it is on a shorter temporal scale for the experimental realization. Moreover, we discuss the performance and feasibility of our hybrid CF gate, concluding that it can be easily extended to a 2n-dimensional case and it is feasible with current technology.

© 2016 Optical Society of America

1. Introduction

Higher-dimensional photon system is of great interest owing to the outstanding features exhibited in higher capacity [1–3], violations of locality [4–6] and resilience [4, 7] to noise. Its applications in the field of high-dimensional quantum information processing (QIP) have attracted much attention recently, e.g., increasing security against eavesdropping [8, 9] and simplify the implementation of quantum logic [10–12]. Meanwhile, the high-dimensional quantum system [13] permits an exponential computation speedup over classical machines which is another hint at the fundamental relevance of the concept. So far, high-dimensional quantum coding has been implemented in various physical system. In photon systems, the time-energy [14] or position-momentum degrees of freedom (DOF) [15, 16], spatial modes [17–19], continuous variables [20], as well as orbital-angular momentum (OAM) DOF [21–25] are all promising information carriers for high-dimensional QIP. In 2014, Krenn et al. managed to create an entanglement of (100×100) dimensions with only two photons [26]. In solid-state systems, energy levels of atoms [27] and artificial atoms [28] which are generally anharmonically spaced can also offer a promising medium for realizing quantum control in large dimensional Hilbert spaces. In 2015, Anderson et al. design and implement unitary maps in a 16-dimensional Hilbert space associated with the 6S1/2 ground state of 133 Cs [27].

The information of a high-dimensional photon can be stored into a multimode atomic ensembles [29, 30] while the high-dimensional interaction between two atomic ensembles is difficult to achieve in theory and experiment. Currently, many theoretical and experimental works have been devoted to the small-circuit structure for the few-qubit systems, and it has been well solved. However, the case for high-dimensional few-qudit systems is quite complex. High-dimensional quantum logic gate is a key element in scalable quantum computation [31] and quantum communication. Implementing scalable quantum algorithms requires a high level of control over multiple quantum systems. Although any d-dimensional quantum gates can be achieved by decomposing into sequentially implemented single- and two-qubit gates, it requires much longer time and yields lower overall fidelities for the synthesis procedure for this complex quantum circuit. For example, in a 2n-dimensional quantum system in which a logic qudit à = A1A2 ⊗ ... ⊗ An ( = B1B2 ⊗ ... ⊗ Bn) is constructed with n 2-dimensional physical qubits A1(B1), A2(B2),... and An(Bn), a 2n-dimensional controlled-flipped (CF) gate operation [32] can be identified by means of the gate operation sequence as

UA˜B˜=CB1A1CB2A2CBnAnTB1A2B2TB2A3B3TBn1AnBnTB1A3B3B2TB2A4B4B3TBn2AnBnBn1TB1An1Bn1Bn2B2TB2AnBnBn1B3TB1AnBnBn1B2,
in which one needs n two-qubit CNOT gates CBiAi(i=1,2,n), (n − 1) 2-control-bit Toffoli gates [33] TBj1AjBj(j=2,3,n), (n − 2) 3-control-bit Toffoli gates TBk2AkBkBk1(k=3,n), one n-control-bit Toffoli gate TB1AnBnBn1B2 and the rest can be deduced by analog. The simplest decomposition of a n-control-bit Toffoli gate requires (2n − 1) 2-qubit gates [34] and in general, the number of two-qubit gates required for constructing a 2n-dimensional CF gate is i=0n1(ni)(2i+1) which will increase rapidly at a rate proportional to n2. This obviously raises the level of complication for constructing high-dimensional quantum gates, not to speak of a scalable high-dimensional quantum computer in experiment. Therefore, it is significant to seek a simpler scheme for implementing high-dimensional quantum gates.

In practice, the photons are often regarded as the best option for flying qubits due to the fast speed, but photonic qubits are hardly scalable [35] on account of the weak interaction between photons. However, by exploiting a hybrid systems with the interaction between photons and solid-state system, the scalability of the photon system can be improved. The quantized whispering-gallery mode (WGM) resonator is required to be of a high-Q factor and a small mode volume [36, 37] and it provides a good platform for studies of hybrid quantum system. By using WGM resonators, such as microtoroidal resonators [38], WGM bottle microresonators [39], microspheres [40] and diamond-GaP microdisks [41], the strong coupling between a atom or an artificial atom and a photon has been experimentally demonstrated. Interestingly, when a WGM microresonator couples to a fiber, its Q factor is surely degraded [38], allowing the photons to be inserted and transmitted on demand through fibers. In experiment, the single-photon input-output process from a microresonator which is coupled to a single atom has been demonstrated [38,39]. Theoretically, similar physics has been shown useful for entanglement generation [42–45], entanglement concentration for the nitrogen-vacancy (NV) centers in decoherence-free subspace [46], universal quantum computation [47–49] and so on.

In this article, we present some deterministic schemes to construct universal quantum gates between high-dimensional single photons and stationary qubits by employing the three-level Λ-type system and WGM microresonator coupled unit. A Λ system is hereafter referred to as an atom, although it can be implemented by artificial atom systems such as nitrogen-vacancy centers [42,46]. In these gates, the Λ-type atom system which works as a logical qudit can perform the CF operation on a d-dimensional single photon in the spatial-mode DOF. Compared with the traditional method in which a d-dimensional quantum gate is implemented by a synthesis procedures of single- and two-qubit gates, in our scheme, the required interaction times between the photon and solid state system reduce greatly, and it is on a shorter temporal scale for the experimental realization. We show that our scheme can be extended to a 2n-dimensional case and it provides a simpler construction of key quantum circuits enabling a more feasible optical realization for high-dimensional QIP.

The remainder of this article is structured as follows. For the sake of simplicity and clarity, we first explain the physical model of our scheme with the simplest case. In section 2.1, a atom-WGM-waveguide platform is briefly reviewed, and then we exploit this unit to design 4-dimensional and 8-dimensional hybrid CF gate in section 2.2 and 2.3. Subsequently, we show how to extend our scheme to a 2n-dimensional with the basic Λ-type atom and WGM-waveguide coupled units in section 2.4. Finally, the results and the feasibility are discussed and a summary is given in section 3.

2. The deterministic 2n-dimensional universal quantum gates using WGM microresonators

Among the multiple DOFs available for high-dimensional quantum information coding, the spatial-mode DOF of photons emerges as a promising alternative as it is more stable against the noise and easily manipulated. The probability that the bit-flip error takes place in the spatial-mode DOF is negligible and it is easy to perform the high-dimensional quantum fast Fourier transform on the single photon in the spatial-mode DOF with only linear optics [50]. Furthermore, based on the OAM-to-polarization coupling effect induced by a modified Mach-Zehnder interferometer [51], the OAM state of a single-photon can be sorted in different paths efficiently in experiment, and with the polarization-independent wavelength division multiplexers (WDMs), photons can be used to guide to different paths according to their frequencies or time-bin states [52]. Therefore, the information encoded in these DOFs can be transformed into the spatial-mode DOF, and one can perform the high-dimensional QIP in these DOFs for photon system with the help of the precise knowledge in high-dimensional spatial-mode DOF QIP.

The basic building block of our high-dimensional hybrid quantum gate is the 4-dimensional (4-d) controlled-X (CX) gate between a 4-dimensional photon in spatial-mode DOF and a atomic qubit of a WGM-waveguide coupled unit. The essence of our architecture is the reflection and the transmission rules of circularly polarized lights interacting with a WGM-waveguide system. Let us first explain our proposal with the basic component step by step.

2.1. Interaction between a polarized photon and a Λ-type atom-WGM resonator coupled system

The basic building block of the present scheme is a Λ-type atom-WGM-waveguide unit, that is, a Λ-type atom fixed on the exterior surface of a WGM resonator side-coupled to four tapered fibers which is shown in Fig. 1. The atom has two degenerate ground states (|± 1〉) and an excited state (|A2〉). The qubits involved in the atom system are encoded on the states |± 1〉 and the excited one |A2〉 works as an ancillary state. As demonstrated in Fig. 1, a σ-polarized photon with frequency ω0 can only induce the transition |+ 1〉 → |A2〉, where ω0 is the transition frequency. On the other hand, the σ+-polarized photon can only trigger the transition |− 1〉 → |A2〉. The WGM microresonator interfaced with more than two tapered fibers can be structured as an add-drop system [53, 54], as depicted in Fig. 1. Here, the ports a0a3 are input ports, and ports a′0a′3 are output ports, respectively. In the case where the resonator is strongly coupled with a particle, the resonance frequency of the resonator is modified, thus preventing the buildup of the resonator field. As a result, the photon off-resonant with the resonator is transmitted through the bus waveguide and remains in the initial spatial mode. Otherwise, the input single-photon in port aj on-resonant with the bare resonator couples inside the resonator and is dropped to the field a′j+1 (j = 0, 1, 2, 3).

 figure: Fig. 1

Fig. 1 The Λ-type atom is fixed on the exterior surface of the WGM microresonator, which is coupled with four tapered fibers (add-drop structure). The waveguides with the ports a′0a′3 are the bus waveguides and that with ports a0a3 are the drop waveguides, respectively. The effective energy-level diagram of an atom is shown in the dashed frame. The possible cavity-mode-induced transitions are |+ 1〉 → |A2〉 driven resonantly by absorbing a σ-circular polarized photon and |− 1〉 → |A2〉 by absorbing a σ+-circular polarized photon.

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The Λ-type atom-WGM resonator coupled system exhibits similar features with the Jaynes-Cummings model [55]. Considering an incident single-photon pulse with frequency ωp from the input port a1, the Heisenberg equations of motion for the cavity-field annihilation operator ak driven by the input field ain and the lowering operator σ of the atom can be described as [39, 42]

dadt=[i(ωcωp)+κ12+κ22]a(t)gσ(t)+κ1ain,dσdt=[i(ω0ωp)+γ2]σ(t)gσz(t)a(t).
where ωc and ω0 represent the frequency of the resonator field and the atom, respectively. σ and σz denote the lowering operator and the inversion operator of the atom. κ1 and κ2 denote the coupling losses of WGM -bus and WGM-drop waveguides, respectively. γ stands for the spontaneous emission rate of the atom and g is the coupling strength between the MTR and the atom. Referring to Ref. [56], drop and add efficiencies are maximized when κ1 = κ2 = κ, and it increases as κ increases. Thus, we have the same coupling loss to both the drop and bus waveguides in our present scheme.

To make sure a weak excitation by the single-photon pulse on the Λ-type atom initially prepared in the ground state, we have 〈σz〉 = −1 through the whole processing. Based on the input and output relation aout=ainka, r(ωp) = |aout/ain| and t(ωp) = 1 + r(ωp), where aout is the output field operator, we can solve the steady-state solution of the reflection and transmission coefficients by setting the time varying part to zero and adiabatically eliminating the cavity mode, expressed as

t(ωp)=i(ωcωp)[i(ω0ωp)+γ2]+g2[i(ωcωp)+κ][i(ω0ωp)+γ2]+g2,r(ωp)=κ[i(ω0ωp)+γ/2][i(ω0ωp)+γ/2][i(ωcωp)+κ]+g2.
The dynamics of the composite system can be described as follows: if the Λ-type atom is initially prepared in |+ 1〉, the only possible transition is |+ 1〉 → |A2〉, while only the σ circularly polarized single-photon pulse will interact with the atom-WGM resonator system. In this case, the σ photon remains in its input path and acquires a phase shift ϕ determined by Eq. (3). Otherwise, when the atom is initially prepared in |− 1〉, the σ photon senses a bare resonator as the polarization mismatch and the state of the photon is transformed from |aj〉 to |a′j+1〉 with an acquired phase shift ϕ0 that is different from ϕ.

Under the resonance condition with ω0 = ωc = ωp, the transmission and reflection coefficients t(ωp) and r(ωp) for the coupled cavity system can be written as

t(ωp)=2g2γκ+2g2,r(ωp)=κγγκ+2g2
In the case g = 0, in which the atom is uncoupled from the cavity, whereas the photon is in resonance with the bare resonator, the reflection coefficient is changed to r0 = −1 (and t0 = 0). Ultimately, under the resonance condition and strong-coupling regime where we approximately have ϕ = 0, ϕ0 = π, the spin-selective optical transition rule of the basic 4-dimensional hybrid quantum gate composed of a atom and a WGM-waveguide (shown in Fig. 1) can be summarized as
|ajσ,+1|ajσ,+1,|ajσ,1|aj+1σ,1,
where i = 0, 1, 2, 3 and here j + 1 in the |aj+1〉 means modding 4. Form Eq. (5), one can see that, if the state of the electron-spin in atom is |− 1〉, the spatial-mode state of a σ photon will flip with an added π phase; otherwise, the spatial-mode state of the σ photon will remain unchanged when the state of the electron-spin in the Λ-type atom is |+ 1〉. We define this spin-selective optical transition rule described in Eq. (5) as the 4-d CX gate.

One of the physical requirements is the access to the WGM microresonator interfaced with tapered fibers. Fortunately, the high Q microtoroid resonators have been demonstrated by Armani et al. in 2003 [57]. Photonic qubits could be coupled into the resonators through the tapered fiber waveguide. Here the tapered fiber waveguides could be fabricated by placing the single-mode fiber on the moveable brackets and heated by the flame of a hydrogen torch. Note that it has been demonstrated that waveguide-to-waveguide power transfer efficiency of 93% is measured and the insertion loss remains less than 0.02% in the optical four port resonant coupler (add-drop geometry), using ultrahigh Q (> 108) toroidal microcavities [53]. Motivated by this technology, our proposed system could be realized by using several tapered fibers coupled with the resonator on different sides, and it has been demonstrated experimentally [39, 53, 58].

2.2. Implementation of the CF gate operations on the 4-dimensional photon and the Λ system

In this section, we show how to build a quantum circuit for 4-dimensional CF gate, in which the stationary qubits in the Λ-type atom 1 (Λ1) and the Λ-type atom (Λ2) work as the control qubits, while the 4-dimensional spatial-mode state of the photon A serve as the target qubit, based on the spin-selective optical transition rules of the atom summarized above. Here, |ĩ〉 (i = 0, 1, 2, 3) denotes the 4-dimensional logical quantum state for the 2-electron-spin system composed of Λ1 and Λ2, which is correspondingly given by

|0˜=|+1,+1Λ2Λ1,|1˜=|+1,1Λ2Λ1,|2˜=|1+1Λ2Λ1,|3˜=|1,1Λ2Λ1.
The |aj〉 (j = 0, 1, 2, 3) denotes the 4-dimensional spatial-mode state for the photon A, and the 4 × 4 flip matrix for the spatial-mode state of the photon is
X=[0100001000011000]
under the basis expanded by {|a0〉, |a1〉, |a2〉, |a3〉}.

The 2-dimensional qubit states of the atom controlling the states of a 4-dimensional photon in the spatial-mode DOF flipped operations, such as

CX=[I400X4],CX2=[I400X42]
can be used to construct a 4-dimensional CF gate operating on the 2 atoms (Λ1 and Λ2) and a 4-dimensional single-photon A. The 4-dimensional hybrid CF gate can be characterized by
CF4=CXCX2=[I40000X0000X20000X3]
under the basis expanded by {|0̃a0〉, |0̃a1〉, |0̃a2〉, |0̃a3〉, |1̃a0〉, |1̃a1〉, |1̃a2〉, |1̃a3〉, |2̃a0〉, |2̃a1〉, |2̃a2〉, |2̃a3〉, |3̃a0〉, |3̃a1〉, |3̃a2〉, |3̃a3〉}, from which one can see that if the state of 2-atom system is |ĩ〉, after the gating operation the spatial-mode of the photon A is changed as |aj〉 → |aj+i〉, where i + j in the |aj+i〉 means modding 4 and I4 is the 4 × 4 identity matrix.

The construction of our hybrid 4-dimensional CF gate is shown in Fig. 2. Suppose the flying photon A is prepared in the state |φA0 = |σ〉(α|a0〉 + β|a1〉 + γ|a2〉 + ξ|a3〉), where the arbitrary complex parameters α, β, γ and ξ satisfy the relations |α|2 + |β|2 + |γ|2 + |ξ|2 = 1. The state of 2 atoms (Λ1 and Λ2) is initially prepared in the state |φe0 = (α̃|0̃〉 + β̃|1̃〉 + γ̃|2̃〉 + ξ̃|3̃〉)Λ2Λ1 and |α̃|2 + |β̃|2 + |γ̃|2 + |ξ̃|2 = 1. As illustrated in Fig. 2(a), the incident photon A first flies through a set of half-wave plates(HWPs), which serves as a π-phase-add operation on the |σ〉 photon. Then the photon A passes through the 4-dimensional CX gate, and the state of the photon A-Λ12 system is changed as

|φA0|φe0|σA[α˜|0˜Λ2Λ1(α|a0+β|a1+γ|a2+ξ|a3)A+β˜|1˜Λ2Λ1(α|a1+β|a2+γ|a3+ξ|a0)A+γ˜|2˜Λ2Λ1(α|a0+β|a1+γ|a2+ξ|a3)A+ξ˜|3˜Λ2Λ1(α|a1+β|a2+γ|a3+ξ|a0)A].

 figure: Fig. 2

Fig. 2 Schematic diagram for a 4-dimensional controlled-flip gate (a 4-dimensional CF gate) operating on the 2 atomic qubits and a 4-dimensional single-photon system. The HWP represents a half-wave plate which serves as a π-phase-add operation on the |σ〉 photon. c1, c2, c3, and c4 are four circulators, and the transmission direction of the c1 and c4 is clockwise while the transmission direction of the c2 and c3 is counter clockwise.

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The quantum circuit shown in Fig. 2(b) can be used to construct the other controlled quantum gate, the 4-d CX2 gate, between a 2-dimensional atom (Λ2) and a 4-dimensional single-photon system. As shown in Fig. 2(b), c1, c2, c3, and c4 are four circulators, and the transmission direction of the c1 and c4 is clockwise while the transmission direction of the c2 and c3 is counter clockwise. When a σ photon is injected into the input-port a′1 (a′0, a′2, or a′3), if the state of the Λ2 is |− 1〉, the input σ photon on-resonant with the bare resonator passes through the circulator c2 (c4, c1, or c3), couples inside the resonator twice and is dropped to the field a″3 (a″2, a″0, or a″1) with an acquired phase shift 2ϕ0; otherwise, if the Λ2 is |+ 1〉 (in the case where the resonator is strongly coupled with a particle, the resonance frequency of the resonator is modified, thus preventing the buildup of the resonator field), the σ photon remains in the initial spatial mode with an acquired phase shift ϕ.

Therefore, under the resonance condition and strong-coupling regime where we approximately have ϕ = 0, ϕ0 = π, the spin-selective optical transition rule of the 4-d CX2 quantum gate performed on a NV center and a WGM resonator [shown in Fig. 2(b)] can be summarized as

|ajσ,1|aj+2σ,1,|ajσ,+1|ajσ+,+1,
where j = 0, 1, 2, 3 and here j + 2 in the |a″j+2〉 means modding 4. Form Eq. (11), one can see that, providing that the state of Λ2 is |− 1〉, the spatial-mode state of a σ photon will flip twice; otherwise, the spatial-mode state of the σ photon will remain unchanged. We define this spin-selective optical transition rule described in Eq. (11) as the 4-d CX2 gate.

According to the input-output relation shown in Eq. (11), after the interaction with the Λ2 atom-WGM resonator coupled system [the 4-d CX2 gate shown in Fig. 2(b)], the state of the complicated system A12 transforms to

|σA[α˜|0˜Λ2Λ1(α|a0+β|a1+γ|a2+ξ|a3)A+β˜|1˜Λ2Λ1(α|a1+β|a2+γ|a3+ξ|a0)A+γ˜|2˜Λ2Λ1(α|a2+β|a3+γ|a0+ξ|a1)A+ξ˜|3˜Λ2Λ1(α|a3+β|a0+γ|a1+ξ|a2)A].
From Eq. (12), one can see that the photon A in the spatial mode |aj〉 will be flipped into the spatial mode |a″j+i〉 when the state of Λ1 and Λ2 is in the state |ĩ〉, and we successfully perform the hybrid 4-dimensional CF gating with only twice interaction between the photon and the atoms.

2.3. Implementation of the controlled-flip gate operations on the 8-dimensional photon and the Λ system

In this section, we show how to build a quantum circuit for 8-dimensional CF gate, in which the stationary qubits in the atom 1 (Λ1), the atom 2 (Λ2), and the atom 3 (Λ3) work as the control qubits, while the 8-dimensional spatial-mode state of the photon A serves as the target qudit. Here, |ĩ〉 (i = 0, 1, 2, 3, 4, 5, 6, 7) denotes the 8-dimensional logical quantum state for the 3-qubit system composed of the Λ1, Λ2, and Λ3, correspondingly given by

|0˜=|+1,+1,+1Λ3Λ2Λ1,|1˜=|+1,+1,1Λ3Λ2Λ1,|2˜=|+1,1,+1Λ3Λ2Λ1,|3˜=|+1,1,1Λ3Λ2Λ1,|4˜=|1,+1,+1Λ3Λ2Λ1,|5˜=|1,+1,1Λ3Λ2Λ1,|6˜=|1,1,+1Λ3Λ2Λ1,|7˜=|1,1,1Λ3Λ2Λ1.
The |aj〉 (j = 0, 1, 2, 3, 4, 5, 6, 7) denotes the 8-dimensional spatial-mode state for the photon A, and the 8 × 8 flip matrix for the spatial-mode state of the photon is given by
X8=[0I710]
which can flip the photonic state as |ajX8|aj+1, and I7 is the 7 × 7 identity matrix. The operation of a 8-dimensional CF gate operating on the 3 atoms (Λ1, Λ2 and Λ3) and a 8-dimensional single-photon A can be identified by means of the gate operation sequence as
CF8=CX8CX82CX84=[I800X8][I800X82][I800X84]=[I800000000X800000000X8200000000X8300000000X8400000000X8500000000X8600000000X87]
One can see that if the state of 3-NV-center system is |ĩ〉, after the gate operation the spatial-mode of the photon A is changed as |aj〉 → |aj+i〉, where i + j in the |aj+i〉 means modding 8 and I8 is the 8 × 8 identity matrix. Here, CX8, CX82 and CX84 represent the 2-dimensional electron-spin state controlled the 8-dimensional photonic state in the spatial-mode flipped operations, in which the photon A in the spatial mode |aj〉 will be flipped into the spatial mode |a′j+1〉, |a′j+2〉, and |a′j+4〉, respectively, on condition that the electron-spin state of the atom is in the state |− 1〉; otherwise, the photon remains unchanged when the state of the atom is |+ 1〉.

The architecture of our hybrid 8-dimensional CF gate is demonstrated in Fig. 3. We assume that the flying photon A is prepared in the state |φA8 = |σ〉 (α0|a0〉 + α1|a1〉 + α2|a2〉 + α3|a3〉 + α4|a4〉 + α5|a5〉 + α6|a6〉 + α7|a7〉), where the arbitrary complex parameters αj satisfy the relations ∑j |αj|2 = 1. The state of 3 atoms is initially prepared in the state |φe3 = (β̃0|0̃〉 + β̃1|1̃〉 + β̃2|2̃〉 + β̃3|3̃〉 + β̃4|4̃〉 + β̃5|5̃〉 + β̃6|6̃〉 + β̃7|7̃〉)Λ3Λ2Λ1 and ∑i |β̃i|2 = 1.

 figure: Fig. 3

Fig. 3 Schematic diagram for a 8-dimensional controlled-flip gate (a 8-d CF gate) operating on the 3-qubit atoms and a 8-dimensional single-photon system. The 50:50 beam splitter (BS) works as the Hadamard gate on the 2-dimensional spatial-mode photonic state.

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The incident photon A will fly through the Λ1-WGM resonator unit twice as one can see in Fig. 3(a). Here, the aj is input port and the a′j is output port for the spatial-mode state |aj〉 of the photon, respectively. First, after passing through a set of circulators [the transmission direction of the circulators are illustrated in the Fig. 3(a)] and HWPs, the photon A interacts with the Λ1-WGM resonator unit, and the state of the photon A-Λ123 system evolves as

|φA8|φe3[(β˜0|0˜+β˜2|2˜+β˜4|4˜+β˜6|6˜)Λ3Λ2Λ1(α0|a0+α1|a1+α2|a2+α3|a3+α4|a4+α5|a5+α6|a6+α7|a7)+(β˜1|1˜+β˜3|3˜+β˜5|5˜+β˜7|7˜)Λ3Λ2Λ1(α0|a1+α1|a2+α2|a3+α3|a0+α4|a5+α5|a6+α6|a7+α7|a4)]|σA.
Subsequently, the spatial-mode component |a′0〉 (|a′4〉) of the photon A passes through the 50:50 beam splitter (BS) which works as the Hadamard gate on the spatial-mode states of a single photon and changes the spatial-mode component |a′0〉 (|a′4〉) into
|a0BS12(|a01+|a02),|a4BS12(|a41+|a42).
Afterwards, the spatial-mode component |a01, |a02, |a41, and |a42 of the photon A are sent into the input ports a01, a02, a41, and a42 of the Λ1-WGM resonator unit, respectively. Given the input-output relation shown in Eq. (5), the state of the complicated system A3Λ2Λ1 transforms to
|σA[(β˜0|0˜+β˜2|2˜+β˜4|4˜+β˜6|6˜)Λ3Λ2Λ1(α02|a01+α02|a02+α1|a1+α2|a2+α3|a3+α42|a41+α42|a42+α5|a5+α6|a6+α7|a7)A+(β˜1|1˜+β˜3|3˜+β˜5|5˜+β˜7|7˜)Λ3Λ2Λ1(α0|a1+α1|a2+α2|a3+α32|a41+α32|a42+α4|a5+α5|a6+α6|a7+α72|a01+α72|a02)A].
Then, the spatial-mode component |a01, |a02, |a41, and |a42 of the photon A are injected into the input ports a01, a02, a41, and a42 of the 2 − BS′. Referring to the inversion of Eq. (17), the state of the complicated system A3Λ2Λ1 becomes
[(β˜0|0˜+β˜2|2˜+β˜4|4˜+β˜6|6˜)Λ3Λ2Λ1(α0|a0+α1|a1+α2|a2+α3|a3+α4|a4+α5|a5+α6|a6+α7|a7)+(β˜1|1˜+β˜3|3˜+β˜5|5˜+β˜7|7˜)Λ3Λ2Λ1(α0|a1+α1|a2+α2|a3+α3|a4+α4|a5+α5|a6+α6|a7+α7|a0)]|σA.
From Eq. (19), one can see that the photon A in the spatial mode |aj〉 will be flipped into the spatial mode |a′j+1〉 when the state of Λ1 is in the state |− 1〉, and we successfully perform the hybrid 8-dimensional CX8 (the 8-d CX) gating with only twice interaction between the photon and the Λ1 atom.

After performing the hybrid 8-d CX gate on the photon A and the Λ1, the spatial-mode component |aj〉 of the photon A is injected into the Λ2-WGM resonator unit from the input port aj as shown in Fig. 3(b), and according to the Eq. (5), when the electron-spin state of Λ2 is in the state |− 1〉, the spatial-mode states of the photon will evolve as

|a0|a2,|a2|a4,|a4|a6,|a6|a0,|a1|a3,|a3|a5,|a5|a7,|a7|a1,
where a″j represents the output port of the Λ2-WGM-waveguide unit. From Eq. (20) and Eq. (5), one can see that the photon A in the spatial mode |a′j〉 will be flipped into the spatial mode |a′j+2〉 when the electron-spin state of Λ2 is in the state |− 1〉; otherwise, the photon remains unchanged. Such is the hybrid 8-dimensional CX82 (the 8-d CX2) gating with only one time’s interaction between the photon and the atom. With the 8-d CX2 gate, the whole state of the hybrid system A3Λ2Λ1 becomes
[(β˜0|0˜+β˜4|4˜)Λ3Λ2Λ1j=07αj|ajA+(β˜2|2˜+β˜6|6˜)Λ3Λ2Λ1j=07αj|aj+2A+(β˜1|1˜+β˜5|5˜)Λ3Λ2Λ1j=07αj|aj+1A+(β˜3|3˜+β˜7|7˜)Λ3Λ2Λ1j=07αj|aj+3A].
Here, j + 1, j + 2, and j + 3 in the spatial-mode states |a″j+1〉, |a″j+2〉, and |a″j+3〉, respectively, mean modding 8.

Finally, after performing the hybrid 8-d CX2 gate on the photon A and the Λ2, the spatial-mode component |aj〉 of the photon A is injected into the Λ3-WGM resonator unit from the input port aj as shown in Fig. 3(c), and according to the Eq. (12), when the state of atom Λ3 is in the state |− 1〉, the spatial-mode states of the photon will evolve as

|a0|ao4,|a1|ao5,|a2|ao6,|a3|ao7,|a4|ao0,|a5|ao1,|a6|ao2,|a7|ao3.
Here, aoj stands for the output port of the Λ3-WGM resonator unit. From Eq. (22) and Eq. (6), one can see that the photon A in the spatial mode |a″j〉 will be flipped into the spatial mode |aoj+4 when the electron-spin state of Λ3 is in the state |− 1〉; otherwise, the photon remains unchanged. Such is the hybrid 8-dimensional CX84 (the 8-d CX4) gating with only two times’ interaction between the photon and the atom. With the 8-d CX4 gate, the whole state of the hybrid system A3Λ2Λ1 becomes
|σAi=07β˜i|i˜Λ3Λ2Λ1j=07αj|aoj+iA.
Here, j + i in the spatial-mode states |aoj+i means modding 8. From Eq. (23), one can see that, with the optical circuit illustrated in Fig. 3, the photon A in the spatial mode |aj〉 will be flipped into the spatial mode |aoj+i when the state of 3-atom system (Λ3Λ2Λ1) is |ĩ〉, and we successfully perform the hybrid 8-dimensional CF gating with five times of interaction between the photon and the Λ system.

2.4. Extending to 2n-dimensional CF gate operation on the photon and the Λ system

With a good knowledge of 4-dimensional and 8-dimensional CF gate operations in the hybrid system, we are able to extend our scheme to a 2n-dimensional case with the basic atom and WGM resonator coupled units. A hybrid 2n-dimensional CF gate operation can be identified by means of the gate operation sequence as

CF2n=CX2nCX2n2CX2n4CX2n2n1,
where CX2nk (k = 1, 2, 4,...2n−1) represents the controlled flip operation between a 2-dimensional electron-spin states of the i-th atom (Λi, i=log2k+1) and a 2n-dimensional photon in the spatial-mode DOF, by which the photon in the spatial mode |aj〉 will be flipped into the spatial mode |aj+k〉 when the electron-spin state of Λi center is in the state |− 1〉; otherwise, the photon remains unchanged.

Any 2n-dimensional CX2nk operation optical circuit can easily be constructed by the 8-d CX gate, the 8-d CX2 gate or 4-d CX2 gate. For example, according to Eq. (12), one can directly exchange the spatial-mode states of a single photon as |aj〉 ↔ |aj+2n−1〉 when the state of the n-th atom is |− 1〉 by repeating 4-d CX2 gate for 2n−2 times, thus a CX2n2n1 operation between the n-th atom and the 2n-dimensional photon can be achieved. Moreover, according to Eq. (20), one can directly exchange the spatial-mode states of a single photon as |aj〉 → |aj+2n−2〉 → |aj+2n−1〉 → |aj+3×2n−2〉 → |aj〉 when the state of the (n − 1)-th atom is |− 1〉 by repeating 8-d CX gate for 2n−3 times, implementing a CX2n2n2 operation between an atom and a 2n-dimensional photon. As for other CX2nk operations, the spatial-mode states of a single photon will change as |aj〉 → |aj+k〉 → |aj+2k〉 → .... → |a2n+ jk〉 → |aj〉 when the electron-spin state of Λi center is in the state | − 1〉, and the corresponding CX2nk operation optical circuit can be recursively assembled out of 8-d CX gates and 8-d CX2 gates. For instance, one should repeat the 8-d CX2 gate operation for 4 times and perform a 8-d CX gate operation to achieve a CX321 operation on the atom and a 32-dimensional photon.

3. Discussion and conclusion

As described in Eq. (1), a 2n-dimensional CF gate can be achieved by decomposing into sequentially implemented single- and two-qubit gates, and the number of two-qubit gates required is i=0n1(ni)(2i1). For the case n=2, UA˜B˜(4×4)=CB1A1CB2A2TB1A2B2, while in the situation of n = 3, UA˜B˜(8×8)=CB1A1CB2A2CB3A3TB1A2B2TB2A3B3TB1A3B3B2. That is, the fundamental requirements for constructing a 4-dimensional CF gate and a 8-dimensional CF gate are 5 times’ 2-qubit interaction and 14 times’ 2-qubit interaction, respectively, by decomposing 4- and 8-dimensional Hilbert space into 2-dimensional quantum space. However, in our scheme, the single photons only need to interact with atoms 2 times or 5 times for separately implementing a 4-dimensional CF gate or a 8-dimensional CF gate operation. To vividly demonstrate the merits of our scheme, we calculate the required interaction times of the present schemes as pictured in Fig. 4. The black curve S1 stands for the interaction time between the high-dimensional photon and atom by decomposing d-dimensional Hilbert space into 2-dimensional quantum space, whereas the line S2 represents that of the our scheme. Apparently, in our scheme, the interaction times of a d-dimensional (d < 128) two-qudit CF quantum gate can be remarkably reduced, especially when d < 64, to less than half the interaction times required in a conventional way. Meanwhile, it is on a shorter temporal scale for the experimental realization. Furthermore, compared with the Eq. (1), the form of Eq. (25) is more simple and easily to extend. In order to improve the dimension of QIP, one only needs to add an atom-WGM resonator coupled unit shown in Fig. 3(b) rather than a complex quantum circuit composed of (n + 1) multi-qubit gating operations. In addition, the order of the interaction between the photon and the atoms can be exchanged arbitrarily in our scheme, but in traditional scheme described in Eq. (1), it must to be implemented in a certain sequence. Obviously, our scheme provides a simpler construction of key quantum circuits enabling a more feasible optical realization for high-dimensional QIP.

 figure: Fig. 4

Fig. 4 The required interaction times between the photon and atoms of the present schemes for constructing a d-dimensional CF gate. The black curve S1 stands for the traditional method in which a d-dimensional quantum gates can be achieved by decomposing into sequentially implemented single- and two-qubit gates, whereas the line S2 represents that of the our scheme.

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The level configuration under our consideration in Fig. 1 can be found in 87Rb; for example, the level with F = 1 (e.g., 52S1/2 of Rb) acts as the ground state and the excited state could be 52P3/2. And the high efficiency switching of optical signals between two optical fibers controlled by a single Rb atom trapped at the center of a bottle cavity has been demonstrated experimentally [39]. Besides the atom, the artificial atoms such as NV centers can also be considered as a good carrier of stationary qubits in our scheme. The NV centers are negatively charged with six electrons from the nitrogen and three carbons surrounding the vacancy. The ground state is a spin triplet with the zero field splitting at 2.87 GHz between levels |0〉 (ms = 0) and |± 1〉 (ms = ±1) owing to spin-spin interactions [59]. The six excited states are defined by group theory [60] as |A1〉 = |E〉| + 1〉 − |E+〉| − 1〉, |A2〉 = |E〉| + 1〉 + |E+〉| − 1〉, |E1〉 = |E〉| − 1〉 − |E+〉| + 1〉, |E2〉 = |E〉| − 1〉 + |E+〉| + 1〉, |Ex〉 = |X〉|0〉, |Ey〉 = |Y〉|0〉, with |E〉, |E+〉, |X〉, and |Y〉 being orbital states. The six excited states are eigenstates of the full Hamiltonian including spin-orbit and spin-spin interactions in the absence of any perturbation, such as by an external magnetic field or crystal strain. The spin-orbit interaction splits the pair (A1, A2) from the others by at least about 5.5 GHz. The spin-spin interaction increases the energy gap and produces a gap of 3.3 GHz between |A2〉 and |A1〉 [42, 60]. Thus, in the limit of low strain, the |A2〉 state is robust with the stable symmetric properties, and decays to the ground state sublevels |− 1〉 and |+ 1〉, respectively, and, with a low probability, decays nonradiatively to |0〉.

As mentioned above, the key elements of our high-dimensional controlled-flip operation scheme are the 4-dimensional logic gates. To demonstrate our theoretical results, we now propose an experimentally accessible quantum device which uses a diamond NV center coupling to a resonator which have been studied in 2-dimensional universal quantum computation [47]. In experiment, by using microspheres [40] and diamond-GaP microdisks [41], strong coupling between the NV center and WGM resonator has been demonstrated. With its exceeded second-scale storage time [61] and nanosecond-scale manipulating time [62], NV center exhibits good properties as a promising candidate for QIP. By defining the gate fidelity as F = 〈φf|φ0〉, we can obtain the fidelities of the two gate operations. Here |φ0〉 is the ideal final state, and |φf〉 is the finial state in realistic by considering experimental factors. In the resonant case, the evolution of an initial state |ϕ=12(|+1+|1)N1(|+1+|1)N2|σA|a0A is |ϕ12|σA{|1N1[|1N2|a3A+|+1N2(t|a1+r2|a3)A]+|+1N1[|1N2(t|a2r|a3)A+|+1N2(t2|a0+tr2|a2tr|a1)r3|a3)A]}, Thus, the fidelity of the 4-dimensional CF gate operation is

F=(1+2t+t2)24[1+2t2+(1+t2)(r2+r4)+t4+r6],
which is also the fidelity of the storage information about the spatial-mode DOF of the photon A on the spins of Λ1 and Λ2. To evaluate the performance of the gate operation, we numerically simulated the results of the fidelity as a function of g/κγ in the case of ωc = ωp = ω0, as shown in Fig. 5. From Fig. 5, one can see that the higher fidelities and efficiencies of the gate depend on the higher NV-cavity coupling strength. When g/κγ is larger than 1.5, the fidelity and efficiency of the is higher than 0.96 and 0.7, respectively. As showed by Chen et al. [42], when g/κγ3 in the resonance situation with κγ2π×0.1GHz, t(ωp) ≈ 0.95 which is not difficult to achieve in experiments. As demonstrated in the Ref. [41], the coupling of a NV center and a GaP microdisk has reached g/2π : 0.3 ∼ 1GHz, and the experimental cavity-QED parameters [g, κ, γtotal, γ]/2π = [0.3, 26, 0.013, 0.0004] GHz of an NV center coupled to a microdisk with Q ∼ 104 have been demonstrated. It is known that only 4% of the total NV spontaneous optical emission is direct transitions between the ground and the excited states [60] within the narrow-band zero phonon line (ZPL). And it is an experimental challenge to isolate this weak ZPL emission in our proposals. Fortunately, Barclay et al. [63] showed that the ZPL emission rate can be enhanced to 47% if the Q of the microdisk reaches 2.5×104, and recently, the ZPL emission rate has been enhanced to 70% [64–67].

 figure: Fig. 5

Fig. 5 The fidelity and efficiency of our scheme in 4-dimensional case as a function of the parameter g/κγ. The blue solid curve represents the fidelity of the 4-dimensional CF gate while the red dashed line corresponds to the efficiency. Here, g/κγ0.5.

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Meanwhile, since the mixing among the excited states of the NV center caused by the strain effect in the diamond [60], we briefly discuss our scheme by considering a negligible energy shift due to a relatively low strain. We suppose that for the N–V center prepared in |+ 1〉, the strain induces an imperfect dipolar transition between the excited state and the state |+ 1〉, and σ would mix in the transition with ξ2 probability due to the involvement of an σ-polarized component [42, 60]. As a result, the fidelity of our scheme in this case solves as F=(2ξ2)24, and we have F = 0.985(0.993) in the case of ξ = 0.12(0.08). The above results manifest that the higher fidelity depends on the smaller spontaneous emission rate of NV center and the higher NV center-resonator coupling strength. The result of our simulation is in good qualitative consistency with the theoretical speculations.

In summary, we present and demonstrate a general technique that harnesses high-dimensional information carriers to implement the deterministic quantum logic gates. The technique that we describe is independent of the physical encoding of quantum information and the way in which the elemental gates have been constructed in Section II. Consequently, it has the potential to be integrated in conjunction with existing technology. Clearly the tools are available to exploit our technique, the benefits of which lie at the increased capacity of single photons for carrying information, enabling a simpler construction of key quantum circuits and a more feasible optical realization for high-dimensional QIP. The performance of our scheme was analyzed in this hybrid system. Our scheme has the ability to map from the atomic state to photonic system for communication across a quantum network, as well as the ability to map from the photonic system to atomic state (atomic systems offer very long coherence times) for information storage after quantum communication [68]. Furthermore, it can also be exploited to implement some essential two-photon quantum operation to connect the two neighboring nodes in a network, such as high-dimensional Bell-state analysis. Thus, the hybird physical system proposed in this paper is a good candidate for realizing high-capacity long-distance quantum communication. Moreover, Our proposed scheme is an efficient way to implement quantum gates allowing high-dimensional quantum computing tasks to be demonstrated on a shorter time scale. It provides a platform for further studies of hybrid quantum information processing and high-capacity quantum communication.

Acknowledgments

This work was supported by the National Natural Science Foundation of China through Grants (No. 11404031 and No. 61471050), Beijing Higher Education Young Elite Teacher Project (No. YETP0456). The project was supported by Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P. R. China.

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53. H. Rokhsari and K. J. Vahala, “Ultralow Loss, High Q, Four Port Resonant Couplers for Quantum Optics and Photonics,” Phys. Rev. Lett. 92, 253905 (2004). [CrossRef]   [PubMed]  

54. F. Monifi, J. Friedlein, S. K. Ozdemir, and L. Yang, “A robust and tunable addCdrop filter using whispering gallery mode microtoroid resonator,” J. Lightwave Technol. 30, 3306 (2012). [CrossRef]  

55. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994). [CrossRef]  

56. F. Monifi, S. K. Ozdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode micro-cavity,” Appl. Phys. Lett 103, 181103 (2013). [CrossRef]  

57. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef]   [PubMed]  

58. F. Monifi, S. K. Özdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode microcavity,” Appl. Phys. Lett. 103, 181103 (2013). [CrossRef]  

59. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006). [CrossRef]  

60. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730(2010). [CrossRef]   [PubMed]  

61. G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789 (2011). [CrossRef]  

62. G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520 (2009). [CrossRef]   [PubMed]  

63. P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009). [CrossRef]  

64. R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stör, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “WaveCparticle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470 (2009). [CrossRef]  

65. S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-Enhanced Single Photon Emission from a Nanoassembled Metal-Diamond Hybrid Structure at Room Temperature,” Nano. Lett. 9, 1694 (2009). [CrossRef]   [PubMed]  

66. P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

67. A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011). [CrossRef]  

68. The relevant application of our scheme proposed here is going to be discussed in our following work.

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    [Crossref] [PubMed]
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    [Crossref]
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  68. The relevant application of our scheme proposed here is going to be discussed in our following work.

2015 (6)

D. Y. Cao, B. H. Liu, Z. Wang, Y. F. Huang, C. F. Li, and G. C. Guo, “Multiuser-to-multiuser entanglement distribution based on 1550 nm polarization-entangled photons,” Sci. Bull. 60, 1128 (2015).
[Crossref]

B. E. Anderson, H. Sosa-Martinez, C.A. Riofrío, Ivan H. Deutsch, and Poul S. Jessen, “Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System,” Phys. Rev. Lett. 114, 240401 (2015).
[Crossref] [PubMed]

Ehsan Zahedinejad, Joydip Ghosh, and Barry C. Sanders, “High-Fidelity Single-Shot Toffoli Gate via Quantum Control,” Phys. Rev. Lett. 114, 200502 (2015).
[Crossref] [PubMed]

D. S. Ding, W. Zhang, Z. Y. Zhou, S. Shi, G. Y. Xiang, X. S. Wang, Y. K. Jiang, B. S. Shi, and G. C. Guo, “Quantum Storage of Orbital Angular Momentum Entanglement in an Atomic Ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

J. S. Xu and C. F. Li, “Quantum integrated circuit: classical characterization,” Sci. Bull. 60, 141 (2015).
[Crossref]

L. Zhou and Y. B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
[Crossref]

2014 (8)

L. Zhou and Y. B. Sheng, “Detection of nonlocal atomic entanglement assisted by single photon,” Phys. Rev. A 90, 024301(2014).
[Crossref]

C. Wang, T. J. Wang, Y. Zhang, R. Jiao, and G. Jin, “Concentration of entangled nitrogen-vacancy centers in decoherence free subspace,” Opt. Express 22, 1551 (2014).
[Crossref] [PubMed]

W. H. Zhang, Q. Q. Qi, J. Zhou, and L. X. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112, 153601 (2014).
[Crossref] [PubMed]

T. J. Wang and C. Wang, “Universal hybrid three-qubit quantum gates assisted by a nitrogen-vacancy center coupled with a whispering-gallery-mode microresonator,” Phys. Rev. A 90, 052310 (2014).
[Crossref]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilingera, “Generation and confirmation of a (100 100)-dimensional entangled quantum system,” PNAS 111, 6243 (2014).
[Crossref]

V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
[Crossref]

R. Fickler, R. Lapkiewicz, M. Huber, M. P. J. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014).
[Crossref] [PubMed]

C. Zheng and G.F. Long, “Quantum secure direct dialogue using Einstein-Podolsky-Rosen pairs,” Sci. China Phys. Mech. Astron. 57, 1238 (2014).
[Crossref]

2013 (7)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

D. O’Shea, C. Junge, J. Volz, and A. Rauschenbeutel, “Fiber-Optical Switch Controlled by a Single Atom,” Phys. Rev. Lett. 111, 193601 (2013).
[Crossref]

S. P. Liu, J. H. Li, R. Yu, and Y. Wu, “Achieving maximum entanglement between two nitrogen-vacancy centers coupling to a whispering-gallery-mode microresonator,” Opt. Express 21, 3501(2013).
[Crossref] [PubMed]

L. Y. Cheng, H. F. Wang, S. Zhang, and K. H. Yeon, “Quantum state engineering with nitrogen-vacancy centers coupled to low-Q microresonator,” Opt. Express 21, 5988 (2013).
[Crossref] [PubMed]

H. R. Wei and F.G. Deng, “Compact quantum gates on electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities,” Phys. Rev. A 88, 042323 (2013);;H. R. Wei and G. L. Long, “Universal photonic quantum gates assisted by ancilla diamond nitrogen-vacancy centers coupled to resonators,” Phys. Rev. A 91, 032324 (2015).
[Crossref]

F. Monifi, S. K. Ozdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode micro-cavity,” Appl. Phys. Lett 103, 181103 (2013).
[Crossref]

F. Monifi, S. K. Özdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode microcavity,” Appl. Phys. Lett. 103, 181103 (2013).
[Crossref]

2012 (5)

F. Monifi, J. Friedlein, S. K. Ozdemir, and L. Yang, “A robust and tunable addCdrop filter using whispering gallery mode microtoroid resonator,” J. Lightwave Technol. 30, 3306 (2012).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional timeCenergy entanglement,” Appl. Phys. B 106(3), 543 (2012).
[Crossref]

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett 108, 143603 (2012).
[Crossref] [PubMed]

2011 (6)

A. Dada, J. Leach, G. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011).
[Crossref]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

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A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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2010 (4)

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730(2010).
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Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
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2009 (8)

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J. S. Kim, A. Das, and B. C. Sanders, “Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity,” Phys. Rev. A 79, 012329 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-Enhanced Single Photon Emission from a Nanoassembled Metal-Diamond Hybrid Structure at Room Temperature,” Nano. Lett. 9, 1694 (2009).
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2008 (3)

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. I. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062 (2008).
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Ronen Barak and Yacob Ben Aryeh, “Quantum fast Fourier transform and quantum computation by linear optics,” J. Opt. Soc. Am. B 24, 231 (2007).
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2006 (3)

Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075(2006).
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2005 (4)

L. Neves, G. Lima, J. G. Aguirre Gomez, C. H. Monken, C. Saavedra, and S. Padua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A 72, 031801(R) (2005).
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2004 (2)

L. M. Duan and H. J. Kimble, “Scalable Photonic Quantum Computation through Cavity-Assisted Interactions,” Phys. Rev. Lett. 92, 127902 (2004).
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H. Rokhsari and K. J. Vahala, “Ultralow Loss, High Q, Four Port Resonant Couplers for Quantum Optics and Photonics,” Phys. Rev. Lett. 92, 253905 (2004).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
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D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418 (2000).
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S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-Enhanced Single Photon Emission from a Nanoassembled Metal-Diamond Hybrid Structure at Room Temperature,” Nano. Lett. 9, 1694 (2009).
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B. E. Anderson, H. Sosa-Martinez, C.A. Riofrío, Ivan H. Deutsch, and Poul S. Jessen, “Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System,” Phys. Rev. Lett. 114, 240401 (2015).
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A. Dada, J. Leach, G. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. I. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062 (2008).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789 (2011).
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G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520 (2009).
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R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stör, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “WaveCparticle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470 (2009).
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P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
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Benson, O.

S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-Enhanced Single Photon Emission from a Nanoassembled Metal-Diamond Hybrid Structure at Room Temperature,” Nano. Lett. 9, 1694 (2009).
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V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angleCorbital angular momentum variables,” Science 329, 662–665 (2010).
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Malcolm N. O’Sullivan-Hale, Irfan Ali Khan, Robert W. Boyd, and John C. Howell, “Pixel entanglement: experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
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A. Dada, J. Leach, G. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011).
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M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789 (2011).
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V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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Childress, L.

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730(2010).
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Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075(2006).
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V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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A. Dada, J. Leach, G. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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J. S. Kim, A. Das, and B. C. Sanders, “Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity,” Phys. Rev. A 79, 012329 (2009).
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B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. I. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062 (2008).
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B. E. Anderson, H. Sosa-Martinez, C.A. Riofrío, Ivan H. Deutsch, and Poul S. Jessen, “Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System,” Phys. Rev. Lett. 114, 240401 (2015).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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L. M. Duan and H. J. Kimble, “Scalable Photonic Quantum Computation through Cavity-Assisted Interactions,” Phys. Rev. Lett. 92, 127902 (2004).
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M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
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A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

Feng, M.

Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett 89, 240401 (2002).
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Weinfurter, H.

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional timeCenergy entanglement,” Appl. Phys. B 106(3), 543 (2012).
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B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nature Physics 5, 134–140 (2009).
[Crossref]

Wilcut, E.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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Worhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
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Wrachtrup, J.

R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stör, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “WaveCparticle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470 (2009).
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D. S. Ding, W. Zhang, Z. Y. Zhou, S. Shi, G. Y. Xiang, X. S. Wang, Y. K. Jiang, B. S. Shi, and G. C. Guo, “Quantum Storage of Orbital Angular Momentum Entanglement in an Atomic Ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
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J. S. Xu and C. F. Li, “Quantum integrated circuit: classical characterization,” Sci. Bull. 60, 141 (2015).
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F. Monifi, S. K. Ozdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode micro-cavity,” Appl. Phys. Lett 103, 181103 (2013).
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F. Monifi, S. K. Özdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode microcavity,” Appl. Phys. Lett. 103, 181103 (2013).
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F. Monifi, J. Friedlein, S. K. Ozdemir, and L. Yang, “A robust and tunable addCdrop filter using whispering gallery mode microtoroid resonator,” J. Lightwave Technol. 30, 3306 (2012).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angleCorbital angular momentum variables,” Science 329, 662–665 (2010).
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Yeon, K. H.

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S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
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S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
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Ehsan Zahedinejad, Joydip Ghosh, and Barry C. Sanders, “High-Fidelity Single-Shot Toffoli Gate via Quantum Control,” Phys. Rev. Lett. 114, 200502 (2015).
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R. Fickler, R. Lapkiewicz, M. Huber, M. P. J. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett 89, 240401 (2002).
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D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418 (2000).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilingera, “Generation and confirmation of a (100 100)-dimensional entangled quantum system,” PNAS 111, 6243 (2014).
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L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev.Lett. 100, 110504 (2008).

Zhang, S.

Zhang, W.

D. S. Ding, W. Zhang, Z. Y. Zhou, S. Shi, G. Y. Xiang, X. S. Wang, Y. K. Jiang, B. S. Shi, and G. C. Guo, “Quantum Storage of Orbital Angular Momentum Entanglement in an Atomic Ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
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Zhang, W. H.

W. H. Zhang, Q. Q. Qi, J. Zhou, and L. X. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112, 153601 (2014).
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Zhang, Y.

Zheng, C.

C. Zheng and G.F. Long, “Quantum secure direct dialogue using Einstein-Podolsky-Rosen pairs,” Sci. China Phys. Mech. Astron. 57, 1238 (2014).
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Zhou, J.

W. H. Zhang, Q. Q. Qi, J. Zhou, and L. X. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112, 153601 (2014).
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L. Zhou and Y. B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
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L. Zhou and Y. B. Sheng, “Detection of nonlocal atomic entanglement assisted by single photon,” Phys. Rev. A 90, 024301(2014).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
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D. S. Ding, W. Zhang, Z. Y. Zhou, S. Shi, G. Y. Xiang, X. S. Wang, Y. K. Jiang, B. S. Shi, and G. C. Guo, “Quantum Storage of Orbital Angular Momentum Entanglement in an Atomic Ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
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D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418 (2000).
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Appl. Phys. B (1)

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F. Monifi, S. K. Ozdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode micro-cavity,” Appl. Phys. Lett 103, 181103 (2013).
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S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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L. Zhou and Y. B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett 89, 240401 (2002).
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Phys. Rev. Lett. (14)

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W. H. Zhang, Q. Q. Qi, J. Zhou, and L. X. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112, 153601 (2014).
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Phys. Rev. X (1)

P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

Phys. Rev.Lett. (1)

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PNAS (1)

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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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Other (3)

The relevant application of our scheme proposed here is going to be discussed in our following work.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
[Crossref]

A d-dimensional controlled-flip gate is used to perform a flip operation on a target qudit or not, depending on the states of the control qudit. That is, if the state of the control qudit is |i〉, after the gating operation the the state of the target qudit is changed as |j〉 → |j + i〉, where i + j in the |j + i〉 means modding d.

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Figures (5)

Fig. 1
Fig. 1 The Λ-type atom is fixed on the exterior surface of the WGM microresonator, which is coupled with four tapered fibers (add-drop structure). The waveguides with the ports a′0a′3 are the bus waveguides and that with ports a0a3 are the drop waveguides, respectively. The effective energy-level diagram of an atom is shown in the dashed frame. The possible cavity-mode-induced transitions are |+ 1〉 → |A2〉 driven resonantly by absorbing a σ-circular polarized photon and |− 1〉 → |A2〉 by absorbing a σ+-circular polarized photon.
Fig. 2
Fig. 2 Schematic diagram for a 4-dimensional controlled-flip gate (a 4-dimensional CF gate) operating on the 2 atomic qubits and a 4-dimensional single-photon system. The HWP represents a half-wave plate which serves as a π-phase-add operation on the |σ〉 photon. c1, c2, c3, and c4 are four circulators, and the transmission direction of the c1 and c4 is clockwise while the transmission direction of the c2 and c3 is counter clockwise.
Fig. 3
Fig. 3 Schematic diagram for a 8-dimensional controlled-flip gate (a 8-d CF gate) operating on the 3-qubit atoms and a 8-dimensional single-photon system. The 50:50 beam splitter (BS) works as the Hadamard gate on the 2-dimensional spatial-mode photonic state.
Fig. 4
Fig. 4 The required interaction times between the photon and atoms of the present schemes for constructing a d-dimensional CF gate. The black curve S1 stands for the traditional method in which a d-dimensional quantum gates can be achieved by decomposing into sequentially implemented single- and two-qubit gates, whereas the line S2 represents that of the our scheme.
Fig. 5
Fig. 5 The fidelity and efficiency of our scheme in 4-dimensional case as a function of the parameter g / κ γ . The blue solid curve represents the fidelity of the 4-dimensional CF gate while the red dashed line corresponds to the efficiency. Here, g / κ γ 0.5 .

Equations (25)

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U A ˜ B ˜ = C B 1 A 1 C B 2 A 2 C B n A n T B 1 A 2 B 2 T B 2 A 3 B 3 T B n 1 A n B n T B 1 A 3 B 3 B 2 T B 2 A 4 B 4 B 3 T B n 2 A n B n B n 1 T B 1 A n 1 B n 1 B n 2 B 2 T B 2 A n B n B n 1 B 3 T B 1 A n B n B n 1 B 2 ,
d a d t = [ i ( ω c ω p ) + κ 1 2 + κ 2 2 ] a ( t ) g σ ( t ) + κ 1 a in , d σ d t = [ i ( ω 0 ω p ) + γ 2 ] σ ( t ) g σ z ( t ) a ( t ) .
t ( ω p ) = i ( ω c ω p ) [ i ( ω 0 ω p ) + γ 2 ] + g 2 [ i ( ω c ω p ) + κ ] [ i ( ω 0 ω p ) + γ 2 ] + g 2 , r ( ω p ) = κ [ i ( ω 0 ω p ) + γ / 2 ] [ i ( ω 0 ω p ) + γ / 2 ] [ i ( ω c ω p ) + κ ] + g 2 .
t ( ω p ) = 2 g 2 γ κ + 2 g 2 , r ( ω p ) = κ γ γ κ + 2 g 2
| a j σ , + 1 | a j σ , + 1 , | a j σ , 1 | a j + 1 σ , 1 ,
| 0 ˜ = | + 1 , + 1 Λ 2 Λ 1 , | 1 ˜ = | + 1 , 1 Λ 2 Λ 1 , | 2 ˜ = | 1 + 1 Λ 2 Λ 1 , | 3 ˜ = | 1 , 1 Λ 2 Λ 1 .
X = [ 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ]
CX = [ I 4 0 0 X 4 ] , CX 2 = [ I 4 0 0 X 4 2 ]
CF 4 = CX CX 2 = [ I 4 0 0 0 0 X 0 0 0 0 X 2 0 0 0 0 X 3 ]
| φ A 0 | φ e 0 | σ A [ α ˜ | 0 ˜ Λ 2 Λ 1 ( α | a 0 + β | a 1 + γ | a 2 + ξ | a 3 ) A + β ˜ | 1 ˜ Λ 2 Λ 1 ( α | a 1 + β | a 2 + γ | a 3 + ξ | a 0 ) A + γ ˜ | 2 ˜ Λ 2 Λ 1 ( α | a 0 + β | a 1 + γ | a 2 + ξ | a 3 ) A + ξ ˜ | 3 ˜ Λ 2 Λ 1 ( α | a 1 + β | a 2 + γ | a 3 + ξ | a 0 ) A ] .
| a j σ , 1 | a j + 2 σ , 1 , | a j σ , + 1 | a j σ + , + 1 ,
| σ A [ α ˜ | 0 ˜ Λ 2 Λ 1 ( α | a 0 + β | a 1 + γ | a 2 + ξ | a 3 ) A + β ˜ | 1 ˜ Λ 2 Λ 1 ( α | a 1 + β | a 2 + γ | a 3 + ξ | a 0 ) A + γ ˜ | 2 ˜ Λ 2 Λ 1 ( α | a 2 + β | a 3 + γ | a 0 + ξ | a 1 ) A + ξ ˜ | 3 ˜ Λ 2 Λ 1 ( α | a 3 + β | a 0 + γ | a 1 + ξ | a 2 ) A ] .
| 0 ˜ = | + 1 , + 1 , + 1 Λ 3 Λ 2 Λ 1 , | 1 ˜ = | + 1 , + 1 , 1 Λ 3 Λ 2 Λ 1 , | 2 ˜ = | + 1 , 1 , + 1 Λ 3 Λ 2 Λ 1 , | 3 ˜ = | + 1 , 1 , 1 Λ 3 Λ 2 Λ 1 , | 4 ˜ = | 1 , + 1 , + 1 Λ 3 Λ 2 Λ 1 , | 5 ˜ = | 1 , + 1 , 1 Λ 3 Λ 2 Λ 1 , | 6 ˜ = | 1 , 1 , + 1 Λ 3 Λ 2 Λ 1 , | 7 ˜ = | 1 , 1 , 1 Λ 3 Λ 2 Λ 1 .
X 8 = [ 0 I 7 1 0 ]
CF 8 = CX 8 CX 8 2 CX 8 4 = [ I 8 0 0 X 8 ] [ I 8 0 0 X 8 2 ] [ I 8 0 0 X 8 4 ] = [ I 8 0 0 0 0 0 0 0 0 X 8 0 0 0 0 0 0 0 0 X 8 2 0 0 0 0 0 0 0 0 X 8 3 0 0 0 0 0 0 0 0 X 8 4 0 0 0 0 0 0 0 0 X 8 5 0 0 0 0 0 0 0 0 X 8 6 0 0 0 0 0 0 0 0 X 8 7 ]
| φ A 8 | φ e 3 [ ( β ˜ 0 | 0 ˜ + β ˜ 2 | 2 ˜ + β ˜ 4 | 4 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 0 + α 1 | a 1 + α 2 | a 2 + α 3 | a 3 + α 4 | a 4 + α 5 | a 5 + α 6 | a 6 + α 7 | a 7 ) + ( β ˜ 1 | 1 ˜ + β ˜ 3 | 3 ˜ + β ˜ 5 | 5 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 1 + α 1 | a 2 + α 2 | a 3 + α 3 | a 0 + α 4 | a 5 + α 5 | a 6 + α 6 | a 7 + α 7 | a 4 ) ] | σ A .
| a 0 B S 1 2 ( | a 0 1 + | a 0 2 ) , | a 4 B S 1 2 ( | a 4 1 + | a 4 2 ) .
| σ A [ ( β ˜ 0 | 0 ˜ + β ˜ 2 | 2 ˜ + β ˜ 4 | 4 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 2 | a 0 1 + α 0 2 | a 0 2 + α 1 | a 1 + α 2 | a 2 + α 3 | a 3 + α 4 2 | a 4 1 + α 4 2 | a 4 2 + α 5 | a 5 + α 6 | a 6 + α 7 | a 7 ) A + ( β ˜ 1 | 1 ˜ + β ˜ 3 | 3 ˜ + β ˜ 5 | 5 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 1 + α 1 | a 2 + α 2 | a 3 + α 3 2 | a 4 1 + α 3 2 | a 4 2 + α 4 | a 5 + α 5 | a 6 + α 6 | a 7 + α 7 2 | a 0 1 + α 7 2 | a 0 2 ) A ] .
[ ( β ˜ 0 | 0 ˜ + β ˜ 2 | 2 ˜ + β ˜ 4 | 4 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 0 + α 1 | a 1 + α 2 | a 2 + α 3 | a 3 + α 4 | a 4 + α 5 | a 5 + α 6 | a 6 + α 7 | a 7 ) + ( β ˜ 1 | 1 ˜ + β ˜ 3 | 3 ˜ + β ˜ 5 | 5 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 1 + α 1 | a 2 + α 2 | a 3 + α 3 | a 4 + α 4 | a 5 + α 5 | a 6 + α 6 | a 7 + α 7 | a 0 ) ] | σ A .
| a 0 | a 2 , | a 2 | a 4 , | a 4 | a 6 , | a 6 | a 0 , | a 1 | a 3 , | a 3 | a 5 , | a 5 | a 7 , | a 7 | a 1 ,
[ ( β ˜ 0 | 0 ˜ + β ˜ 4 | 4 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j A + ( β ˜ 2 | 2 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j + 2 A + ( β ˜ 1 | 1 ˜ + β ˜ 5 | 5 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j + 1 A + ( β ˜ 3 | 3 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j + 3 A ] .
| a 0 | a o 4 , | a 1 | a o 5 , | a 2 | a o 6 , | a 3 | a o 7 , | a 4 | a o 0 , | a 5 | a o 1 , | a 6 | a o 2 , | a 7 | a o 3 .
| σ A i = 0 7 β ˜ i | i ˜ Λ 3 Λ 2 Λ 1 j = 0 7 α j | a o j + i A .
CF 2 n = CX 2 n CX 2 n 2 CX 2 n 4 CX 2 n 2 n 1 ,
F = ( 1 + 2 t + t 2 ) 2 4 [ 1 + 2 t 2 + ( 1 + t 2 ) ( r 2 + r 4 ) + t 4 + r 6 ] ,

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