Abstract
Entanglement concentration protocol (ECP) is used to extract the maximally entangled states from less entangled pure states. Here we present a general hyperconcentration protocol for twophoton systems in partially hyperentangled Bell states that decay with the interrelation between the timebin and the polarization degrees of freedom (DOFs), resorting to an inputoutput process with respect to diamond nitrogenvacancy centers coupled to resonators. We show that the resource can be utilized sufficiently and the success probability is largely improved by iteration of the hyperECP process. Besides, our hyperECP can be directly extended to concentrate nonlocal partially hyperentangled Nphoton GreenbergerHorneZeilinger states, and the success probability remains unchanged with the growth of the number of photons. Moreover, the timebin entanglement is a useful DOF and it only requires one path for transmission, which means it not only economizes on a large amount of quantum resources but also relaxes from the pathlength dispersion in longdistance quantum communication.
Introduction
Quantum entanglement is the key important resource for quantum communication, such as quantum teleportation^{1}, quantum dense coding^{2,3}, quantum key distribution^{4,5}, quantum secret sharing^{6}, and quantum secure direct communication^{7,8} as entangled photons are generally considered as the ideal information carriers and are used to connect distant quantum nodes in longdistance quantum communication on account of its highspeed transmission and striking lownoise features. Usually entangled photon pairs are produced locally. The photon loss and decoherence caused by the interaction between the photonic quantum system and its environment will inevitably decrease its entanglement during the practical entanglement distribution in longdistance quantum communication.
After passing through a noisy channel, the maximally entangled photon states decay into less entangled pure states or mixed states, leading to the destruction on the fidelity and the security of longdistance quantum communication protocols. In order to depress the decoherence effect on the entangled systems, two interesting quantum techniques, entanglement purification and entanglement concentration, could be exploited to obtain highfidelity entangled photon systems. In detail, entanglement purification^{9,10,11,12,13,14} is used to distill a subset of highly entangled states from a set of mixed entangled states, while entanglement concentration^{15,16,17,18} is to extract the maximally entangled states from less entangled pure states. Since Bennett et al.^{15} proposed the first entanglement concentration protocol (ECP) for twophoton systems relying on the Schmidt projection method, some good ECPs^{16,17,18} have been presented.
Hyperentanglement, the entanglement simultaneously in multiple degrees of freedom (DOFs) of a quantum system^{19}, has some important applications in quantum communication. It can increase the channel capacity of quantum communication^{20,21,22,23,24}, achieve the complete Bellstate analysis for the quantum states in the polarization DOF^{11}, be used to teleport the unknown quantum state in two DOFs^{21} and complete the hyperentanglement swapping between two photonic quantum systems without entanglement^{22}, and help to design the deterministic hyperentanglement purification^{11,12,13,14} which solves the troublesome problem that the parties in quantum repeaters should sacrifice a large amount of quantum resources with conventional entanglement purification protocols (EPPs)^{9,10} as the deterministic EPPs^{11,12,13,14} work in a completely deterministic way^{25,26,27}. Recently, some interesting hyperentanglement concentration protocols (hyperECPs)^{28,29,30,31,32,33,34,35} were proposed. For example, in 2013, Ren et al.^{28} proposed the first hyperECP for twophoton systems in polarizationspatial lesshyperentangled states with linear optical elements only, including the cases for the nonlocal photonic quantum systems with known and unknown parameters, respectively. More interestingly, they proposed the parametersplitting method^{28}, a fascinating method, to extract the maximally entangled photons when the coefficients of the initial partially entangled state are known, and this method is very efficient and simple in terms of concentrating partially entangled state as it can be achieved with the maximum success probability by performing the protocol only once. In 2014, Li and Ghose^{30} proposed a hyperconcentration scheme for nonlocal Nphoton hyperentangled GreenbergerHorneZeilinger (GHZ) states via linear optics. Sequentially, they^{31} presented two efficient schemes for concentration of nonlocal Nphoton hyperentanglement based on the crossKerr nonlinearity. In 2016, Liu et al.^{34} presented a hyperECP for the partially hyperentangled Nparticle GHZ state assisted by a lessentangled Nparticle GHZ state and three single photons. So far, most of the existing hyperECPs^{28,29,30,31,32,33,34} focus on lesshyperentangled states in the polarization and spatial modes DOFs. In 2015, Li and Ghose^{35} presented two hyperECPs for twophoton states that are partially entangled in the polarization and timebin DOFs with linear optics.
The electronic spin associated with a diamond nitrogen vacancy (NV) center is an exceptional solidstate spin qubit system due to optical controllability^{36,37,38,39} and exceeding 10 ms coherence time by using dynamical decoupling techniques^{40}. The electron spin of the NV center can be exactly initialized^{41}, manipulated^{36,37,38,39,41} and read out^{42,43}. Therefore, the diamond NV center is an attractive platform for quantum information processing due to its longlived coherence time at room temperature. Many interesting approaches for quantum computation and quantum communication have been proposed based on the NV center in diamond coupled to an optical cavity in theory^{24,29,44,45,46} and implemented in experiment^{40,41,47,48,49,50,51,52}. For example, Ren et al.^{29} presented the spatialpolarization photonic hyperentanglement purification and concentration resorting to the nonlinear optics of the NV center embedded in a photonic crystal cavity coupled to a waveguide in 2013. In 2015, Liu and Zhang^{24} presented two interesting schemes for the generation and complete nondestructive analysis of hyperentanglement assisted by nitrogenvacancy centers in resonators. Jelezko et al.^{41} have experimentally demonstrated a conditional controlled quantum gate on electronnuclear spins of an NV center in 2004. The creation of an entanglement between two distant NV electron spins^{40}, a single photon and an NV electron spin^{47}, the electron and nearby nuclear spins^{48} have been experimentally demonstrated.
In this paper, we investigate a general hyperECP for twophoton systems in an arbitrary partially hyperentangled unknown Bell state that decays with the interrelationship between the timebin and the polarization DOFs. Our hyperECP is achieved by the Schmidt projection method and two paritycheck gates that are constructed with the optical property of the NV center coupled to a resonator. By iteration of the hyperECP process, the success probability of our hyperECP becomes much higher than that in the hyperECP with linear optics. At last, we show that our hyperECP is suitable for arbitrary partially hyperentangled Nphoton GHZ states, and the success probability is still unchanged with the growth of the number of photons.
Results
The optical property of an NVcavity platform
An NV center in diamond is created by a replaceable nitrogen atom substituting for a carbon atom and an adjacent vacancy in the diamond lattice. The ground state of the NV center is an electronic spin triplet state holding a 2.8 GHz zerofield splitting between the magnetic sublevels m_{s} = 0〉 and m_{s} = ±1〉 in virtue of the spinspin interaction^{53}. One of the six excited states^{54} is very robust against the relatively low strain, which is lower than the spinorbit splitting, and magnetic fields possessing the stable symmetry properties protected by an energy gap retaining the polarization properties of its optical transitions^{55,56}. Here the electric triple states of the ground m_{s} = 0 and m_{s} = ±1 represent 0〉 and ±1〉, respectively. E_{±}〉 indicate the orbital states with the angular momentum projections ±1 along the NV axis. In the presence of the small external magnetic field (2π × 200 MHz) which has little effect on the symmetry properties of the A_{2}〉 state, the twofold degenerace m_{s} = ±1〉 sublevel is split in two levels. The transitions frequency between ±1〉 and A_{2}〉 is in the optical regime, i.e., ±1〉 ↔ A_{2}〉 are driven by the σ_{−} (left − L) and σ_{+} (right − R) circularly polarized photon at ~637 nm (shown in Fig. 1), respectively.
The schematic diagram of a diamond NV center coupled to a resonator is shown in Fig. 1. An incident single photon with frequency ω_{p} enters a singlesided cavity with frequency ω_{c}, which traps a ^type threelevel diamond NV center with frequency difference ω_{0} between −1〉 and A_{2}〉. The cavity mode is driven by the input field . By solving the Heisenberg equations of motion for the annihilation operation of cavity mode and the lowing operation σ_{−} of the NV center^{57},
In the weak excitation limit 〈σ_{z}〉 = −1, one can obtain the reflection coefficient for the NVcavity system^{58,59}
Here the cavity output field is connected with the input field by the inputoutput relation . The vacuum input field b_{in}(t) has the standard commutation relation . κ and γ are the decay rates of the cavity and the spontaneous emission rate of the NV center, respectively. g is the coupling rate of the NVcavity system. In the case of g = 0 in which the NV center is uncoupled from the cavity, Eq. (2) could convert into for an empty cavity.
It is usually not difficult to find that if the photon feels a hot cavity (g ≠ 0), it will get a phase e^{iφ} after reflection. Otherwise, if the photon feels a cold cavity, it will obtain a phase shift . Supposed that the NV center is initially prepared in the state −1〉, the only possible transition is −1〉 ↔ A_{2}〉. The L polarized photon feels a hot microtoroidal resonator (MTR), while R polarized photon would sense a bare MTR due to a polarization mismatch, and the corresponding output states of the L and R photons can be obtained as^{45}
In contrast, if the NV is prepared in the state +1〉, the input pulse L (R) always feel a bare cavity due to the polarization mismatch (the large detuning), and the corresponding output states of the L and R photons can be obtained as
The phase shifts are the function of the frequency detuning (ω_{p} − ω_{c}) under the resonant condition ω_{c} = ω_{0}. By adjusting ω_{p} = ω_{c} = ω_{0}, one can see that the reflection coefficients for the hot cavity r(ω_{p}) and the cold cavity r_{0}(ω_{p}) can be written as
If the condition satisfies , and r_{0}(ω_{p}) = −1 can be obtained. That is, the spinselective optical transition rules can be described as
This inputoutput property of a cavityNVcenter system can be used to construct the paritycheck gates (PCGs) for a twophoton system in both the timebin (NV_{1} in Fig. 2(a)) and the polarization (NV_{2} in Fig. 2(b)) DOFs.
Paritycheck gates for a twophoton system
A twophoton system AC has four Bell states in the timebin DOF and the four Bell states in the polarization DOF. After passing through a noisy channel, the maximally entangled photon states decay into less entangled pure states denoted as
where α^{2} + β^{2} = γ^{2} + δ^{2} = 1. The Bell states () and () stand for the evenparity and the oddparity modes in the timebin (polarization) DOF of the two photon system, respectively. Here the superscripts T and P represent the timebin and the polarization DOFs of the twophoton system, respectively. s and l are the two different timebins, the early (s) and the late (l), which possesses the time interval Δt between the two timebins. R and L denote the rightcircular and leftcircular polarizations of photons, respectively.
The paritycheck gate (TPCG) for a twophoton system in the timebin DOF is constructed with circularly polarizing beam splitters (CPBSs), PC_{s(l)}, NV_{1}, and SWs, shown in Fig. 2(a). CPBS_{i} (i = 1, 2, …) represents a polarizing beam splitter in the circular basis, which transmits the photon in the rightcircular polarization R〉 and reflects the photon in the leftcircular polarization L〉, respectively. The PC_{s} (PC_{l}) in the spatialmode a_{1} and c_{1} (a_{2} and c_{2}) is a Pockels cell (PC)^{60} which performs a bitflip operation () on the polarization DOF of the photon A (C) at the specific time only when the s(l)path component appears. That is,
It is worth pointing out that our PC_{s} (PC_{l}) is consisted of two halfwave plates (HWPs) (indicating the mutual transformation between the circular polarization and linear polarization) and the PC′_{s} (PC′_{l}) shown in ref. 60. Taken the transformation L〉^{s}(R〉^{s}) as an example, it can be accomplished with the processes . SW_{j} (j = 1, 2) is an optical switch which makes the wavepacket of the photons A and C successively enter into (or keep away from) the NV_{1} center. Suppose that the initial state of the twophoton system AC is a Bell state and that of the auxiliary NV_{i} is with i = 1, 2, Alice lets the photon A (C) pass through CPBS_{1} (CPBS_{2}), PC_{s(l)}, SW_{1}, NV_{1}, SW_{2}, PC_{s(l)}, and CPBS_{3} (CPBS_{4}) in sequence, and the state of the complicated system composed of NV_{1} and the twophoton system AC evolves into
where . Similarly, after interaction, the evolutions of the other states of the system can be described as follows:
It is quite clear that the polarization states of the two photons AC have not be affected. By measuring the auxiliary NV_{1} in the orthogonal basis {+〉,−〉}, one can distinguish the evenparity timebin Bell states from the oddparity ones. That is, if the auxiliary NV_{1} is projected into the state +〉_{1}, the timebin state is the evenparity one; otherwise, it is the oddparity one.
The principle of our polarization paritycheck gate (PPCG) is shown in Fig. 2(b) and it is used to distinguish the parity of the hyperentangled Bell states in polarization DOF. Similar to our TPCG, if one lets the photons A and C pass through the quantum circuit shown in Fig. 2(b) in sequence, the rule for the evolutions of quantum states of the complicated system composed of two photons AC and the auxiliary NV_{2} is
By measuring the auxiliary NV_{2} in the orthogonal basis {+〉,−〉}, one can distinguish the hyperentangled Bell states with the evenparity mode from those with the oddparity mode in the polarization DOF without influencing the states of the photons in the timebin DOF. In detail, if the auxiliary NV_{2} is projected into the state +〉_{2}, the polarization state is the evenparity mode; otherwise, it is the oddparity one.
Highefficiency hyperECP for arbitrary twophoton systems
In a longdistance quantum communication, the maximally hyperentangled Bell state in both the timebin and polarization DOFs may decay to the partially hyperentangled Belltype state by the independent decoherence of the entanglement in two DOFs^{35}. Also, the maximally hyperentangled Bell state will decay to an arbitrary partially hyperentangled Bell state if the interrelation between the timebin and the polarization DOFs is taken into account. Here
Here the subscripts p and q present the photons held by two distant parties, Alice and Bob, respectively. The four unknown parameters α_{1}, α_{2}, α_{3}, and α_{4} satisfy the normalization condition α_{1}^{2} + α_{2}^{2} + α_{3}^{2} + α_{4}^{2} = 1, and three unknown parameters are independent. The principle of our general hyperECP for twophoton systems in both the polarization and the timebin DOFs is shown in Fig. 3. It includes two steps which can be described in detail as follows.
(1) The first step of our hyperECP for the twophoton systems. To realize the first step of our hyperECP with the Schmidt projection method, a pair of the twophoton systems AB and CD from a set of the twophoton systems in the state is required for each process in hyperECP. That is, the two photons A, and C belong to Alice, and the other two photons B and D belong to Bob.
The principle of the first step of our hyperECP for the photon pairs AB and CD is shown in Fig. 3. The initial state of the fourphoton system ABCD can be written as . Alice performs the TPCG and PPCG on the photon pair AC. The outcomes can be divided into four groups, and they are discussed in detail as follows.
(1.1) If the outcomes of the TPCG and PPCG are in an evenparity timebin mode and an oddparity polarization mode, respectively, the fourphoton system is projected into the state Φ_{1eo}〉_{ABCD} with the probability of p_{1eo} = 2(α_{1}α_{2}^{2} + α_{3}α_{4}^{2}). Here
Subsequently, Alice (Bob) performs the singlephoton measurement (SPM) on the photon C (D). The SPM setup is composed of linear optical elements, shown in Fig. 3. The effect of the unbalanced interferometer (UI) can be described as and . Here the length difference between the two arms s and l is set exactly to cΔt, where c is the speed of the photons. After passing through two PCs and two UIs, the state Φ_{1eo}〉_{ABCD} is transformed into the state Φ′_{1eo}〉_{ABCD}. Here
Obviously, both the two photons C and D will arrive at their SPMs at the same time, respectively, i.e., in the middle time slot. However, there are two potential spatial modes for each photon, the up mode c_{1}(d_{1}) and the down mode c_{2}(d_{2}), which makes the two photons C and D be measured in both the polarization DOF and the spatial mode DOF. The effect of a 50:50 beam splitter (BS) can be described as and (m = c, d). R_{45} is used to perform a Hadamard operation on the polarization DOF of the photons, that is, and . At last, the two photons C and D are detected by the singlephoton detectors. The relationship between the measurement results of the detectors and the shared states ϕ_{1eo}〉_{AB} is shown in Table 1. If two detectors (, , , or ) are clicked, the twophoton system AB is projected into the state , which is the partially hyperentangled Belltype state with the polarization DOF in a maximally entangled Bell state. For the other three cases of the two clicked detectors, a phase flip operation , in sequence on the photon B is required to obtain the partially hyperentangled Belltype state ϕ_{1eo}〉_{AB}.
(1.2) If the outcomes of the TPCG and PPCG are in an oddparity timebin mode and an evenparity polarization mode, respectively, the fourphoton system is projected into the state Φ_{1oe}〉_{ABCD} with the probability of p_{1oe} = 2(α_{1}α_{3}^{2} + α_{2}α_{4}^{2}). Here
Similar to the discussion above, after two parters perform the SPMs on the photons C and D, twophoton system AB is projected into the hyperentangled Bell state with the timebin DOF in a maximally entangled Bell state by assisting the conditional phase flip operation Y ( or ) on photon B depending on the measurement results of the detectors shown in Table 1.
(1.3) If the outcomes of the TPCG and PPCG are in an oddparity timebin mode and an oddparity polarization mode, respectively, the fourphoton system is projected into the state Φ_{1oo}〉_{ABCD} with the probability of p_{1oo} = 2(α_{1}α_{4}^{2} + α_{2}α_{3}^{2}). Here
Alice and Bob make the photons A, B, C, and D pass through the PC_{l}, PC_{l}, PC_{s}, and PC_{s}, respectively, which realize the polarization bitflip operations on the timebin modes l_{A}, l_{B}, s_{C}, and s_{D}, respectively. Then the state Φ_{1oo}〉_{ABCD} is transformed into . Similar to the case (1.2), after replacing α_{1}α_{3} and α_{2}α_{4} with α_{1}α_{4} and α_{2}α_{3}, respectively, the state of the twophoton system AB is transformed into .
(1.4) If the outcomes of the TPCG and PPCG are in an evenparity timebin mode and an evenparity polarization mode, respectively, the fourphoton system is projected into the state Φ_{1ee}〉_{ABCD} with the probability of . Here
Alice and Bob perform the SPMs on the photons C and D, respectively, and then the state Φ_{1ee}〉_{ABCD} collapses to , which is the partially hyperentangled Bell state with less entanglement than the state .
(2) The second step of our hyperECP for twophoton systems. In this step, another two photon pairs A′B′ and C′D′ in the partially hyperentangled Bell state are required, which are identical to the two photon pairs AB and CD. Here the two photons A′C′ belong to Alice, and the two photons B′D′ belong to Bob. Alice and Bob perform the same operations on the photon pairs A′B′ and C′D′ as those on the photon pairs AB and CD, and the same results can be obtained. That is, the four cases ϕ_{1eo}〉_{A′B′}, ϕ_{1oe}〉_{A′B′}, ϕ_{1oo}〉_{A′B′}, and ϕ_{1ee}〉_{A′B′} are obtained with the probabilities p_{1eo}, p_{1oe}, p_{1oo}, and p_{1ee}, respectively, by replacing the four photons ABCD with A′B′C′D′. Alice and Bob will distill a maximally hyperentangled Bell state from the partially hyperentangled Belltype states obtained by the above three cases of the first step. The principle of this step is the same as the first one shown in Fig. 3 by replacing the photons ABCD with ABA′B′. That is, Alice performs the TPCG and PPCG on the photon pairs AA′.
(2.1) For the case in (1.1), the state of the fourphoton system ABA′B′ is Ψ_{1}〉_{ABA′B′} = ϕ_{1eo}〉_{AB} ⊗ ϕ_{1eo}〉_{A′B′}. Alice picks up the case when the outcome of the TPCG is in an oddparity timebin mode, and a maximally hyperentangled Bell state can be obtained whether the outcome of the PPCG is in an oddparity polarization mode or in an evenparity polarization mode. That is, the fourphoton system is projected into the states Ψ_{1oe}〉_{ABA′B′} and Ψ_{1oo}〉_{ABA′B′} with the same probability p_{1} = p_{1oy}/p_{1eo} (p_{1oy} = 4α_{1}α_{2}α_{3}α_{4}^{2}), respectively. Here
After performing the SPMs on the photons A′ and B′ and assisting the conditional phaseflip operation on the photon B, the twophoton system AB is projected into the maximally hyperentangled Bell state with the probability 2p_{1}.
If the outcome of the TPCG is in an evenparity timebin mode, the fourphoton system is projected into the states Ψ_{1eo}〉_{ABA′B′} and Ψ_{1ee}〉_{ABA′B′} with the same probability p′_{1}/p_{1ey} = p_{1eo}, respectively. Here
where . After measuring the photons A′ and B′, the twophoton system AB is projected into the partially hyperentangled Belltype state with the probability 2p′_{1}, which can be used in the second step in the second round of the hyperECP process.
(2.2) For the case in (1.2), the state of the fourphoton system ABA′B′ is Ψ_{2}〉_{ABA′B′} = φ_{1oe}〉_{AB} ⊗ φ_{1oe}〉_{A′B′}. Likewise, Alice and Bob pick up the case when the outcome of the PPCG is in an oddparity polarization mode, and a maximally hyperentangled Bell state with the probability 2p_{2}(p_{2} = p_{1yo}/p_{1eo}, p_{1yo} = p_{1oy}) can be obtained whether the outcome of the TPCG is in an oddparity timebin mode or in an evenparity polarization mode. If the outcome of the PPCG is in an evenparity timebin mode, the twophoton system AB is projected into the partially hyperentangled Belltype state with the same probability 2p′_{2} .
(2.3) For the case in (1.3), the same operations are performed on the state Ψ_{3}〉_{ABA′B′} = φ_{1oo}〉_{AB}⊗φ_{1oo}〉_{A′B′} as those on the state Ψ_{2}〉_{ABA′B′} in case (2.2). The maximally hyperentangled Bell state can be obtained with the probability 2p_{3}(p_{3} = p_{1xo}/p_{1oo,} p_{1xo} = 4α_{1}α_{2}α_{3}α_{4}^{2}) and the state can be obtained with the probability 2p′_{3}(p′_{3} = p_{1xe}/p_{1oo}, .
In the hyperECP for the twophoton systems in the partially hyperentangled Bell state with linear optics in ref. 35, only one of the three cases (1.1), (1.2), and (1.3) can be preserved, and only one of the two timebin (polarization) parity cases (the even parity or odd parity of the case (2.1), (2.2), or (2.3)) is preserved in the second step. These six cases can all be preserved in our hyperECP with the NV center, so the success probability P_{1} of the first round of the hyperECP process is almost five times larger than that in the hyperECP with linear optical elements. The success probability in ref. 35 is P = m, m ∈ (p_{1}, p_{2}, p_{3}). After the first round of our hyperECP, the total success probability of the maximally hyperentangled Bell state is P_{1} = 2(p_{1} + p_{2} + p_{3}). The left of Fig. 4 shows the procedure of the first round of our ECP for twophoton systems in an arbitrary partially hyperentangled Bell state with TPPCG in detail.
Now, let us discuss the second round of our ECP for twophoton systems in an arbitrary partially hyperentangled Bell state with TPPCG shown in the right of Fig. 4. For the partially hyperentangled Bell state ϕ_{1ee}〉_{AB} preserved in the case (1.4), requiring four copies of the twophoton systems to complete the two steps in the second round of the hyperECP process. While for the hyperentangled Belltype states ϕ_{1ey}〉_{AB}, ϕ_{1ye}〉_{AB}, and ϕ_{1xe}〉_{AB} preserved in the second step, two copies of the photon systems in each state are required to complete only the second step in the second round of the hyperECP process. After the second round of the hyperECP, the success probability of the maximally hyperentangled Bell state is
where . Again, the partially hyperentangled Bell states preserved in the nth round of the hyperECP process can be used to distill the maximally hyperentangled Bell state in the (n + 1)th round. The success probability of the hyperentanglement concentration process in nth (n > 2) round can be described as follows
where
After the nth round iteration of the hyperECP process, the total success probability is . The success probabilities equal to P_{1} = 0.225, P_{2} = 0.312, and P_{5} = 0.338 when α_{1} = 0.3, α_{3} = 0.4, and α_{2} = α_{4}. Further, they can be increased to P_{1} = 0.353, P_{2} = 0.536, and P_{5} = 0.665 with α_{1} = α_{3} = 0.45, and α_{2} = α_{4}. If the conditions become α_{1} = α_{3} = 0.5, α_{2} = 0.51, and , the success probabilities raise to P_{1} = 0.375, P_{2} = 0.585, and P_{5} = 0.762. Obviously, with the iteration of our hyperECP process, the total success probability P will be increased largely.
Discussion
It is clear that our hyperECP can be generalized to distill a maximally hyperentangled Nphoton GHZ state from an arbitrary partially hyperentangled GHZclass state that decays with the interrelation between the timebin and polarization DOFs. Suppose an arbitrary partially hyperentangled Nphoton GHZ state is described as
The subscripts A, B, …, and Z represent the photons that are kept by the remote users, called Alice, Bob, …, and Zach, respectively. As there are also three independent parameters in the state , which is similar to the state , our hyperECP can be directly extended to distill maximally hyperentangled Nphoton GHZ states with the operations performed by Alice only with the TPCG and PPCG on her own photon pairs. The success probability of our hyperECP for Nphoton systems in an arbitrary partially hyperentangled GHZ state is the same as the one for twophoton systems in an arbitrary partially hyperentangled Bell state. That is because that only Alice is required to perform the concentration operations, and the remaining N − 1 parties do nothing, which can be viewed as Bob with a complicate system in essence (In detail, ss ... s〉_{AB...Z} ⇒ s_{A}s_{B}〉,ll ... l〉_{AB...Z} ⇒ l_{A}l_{B}〉, RR ... R〉_{AB...Z} ⇒ R_{A}R_{B}〉, and LL ... L〉_{AB...Z} ⇒ L_{A}L_{B}〉). Therefore, the success probability remains unchanged with the growth of the number of photons. When the number of photons to be concentrated is large, our scheme may be more efficient and more practical.
The polarization DOF and the spatial mode DOF are the two most popular DOFs of the photon as they are easy to manipulate and measure with linear optical elements. However, using the spatial mode of each photon to carry information requires two paths during the transmission, which may introduce the pathlength dispersion in longdistance multiphoton communication. The timebin states, also as a conventional classical DOF, can be simply discriminated by the time of arrival. The timebin DOF is very helpful for quantum communication as it can be used to accomplish the faithful qubit transmission without ancillary qubits^{61}, the deterministic twophoton entanglement purification^{62}, the arbitrary multiphoton entanglement sharing^{63}, and the complete error correction for nonlocal spatialpolarization hyperentangled photon pairs^{64}.
In summary, we have proposed a general hyperECP for improving the entanglement of the twophoton systems in an arbitrary partially hyperentangled Bell state that decays with the interrelationship between the timebin and the polarization DOFs, resorting to the TPCG and PPCG that are constructed by the optical property of NVcavity systems. Our hyperECP is different from the hyperECP^{35} with unknown parameters, which is focused on the partially hyperentangled pure states accompanied by the independent decoherence in two DOFs. We show that the resource can be utilized sufficiently and the success probability is largely improved by iteration of the hyperECP process. The success probability of the first round of our hyperECP is almost five times than that in the hyperECP^{35} with linear optical elements. In addition, our hyperECP can be straightforwardly generalized for arbitrary partially hyperentangled Nphoton GHZ states, especially for the case with the interrelation between the two DOFs of multiphoton systems, and the success probability remains unchanged with the growth of the number of photons. Besides hyperECP, the basic paritycheck gate elements, including PPCG and TPCG can also be used to construct the highefficiency hyperentanglement purification protocol for obtaining highfidelity hyperentangled states from mixed hyperentangled states. Moreover, the timebin entanglement is a useful DOF and it only requires one path during the transmission process, which means that it not only economizes on a large amount of quantum resources but also relaxes the pathlength dispersion in longdistance quantum communication.
Methods
Average fidelities and efficiencies of the paritycheck gates
In this part, we give a brief discussion about the experimental implementation of our scheme. The NV center in diamond has attracted much attention with millisecond coherence time^{37}, and its ground state spin coherence time can be extended to 480 μs with Gaussian decay using a Hahn echo sequence^{38}, which may be made further efforts extend much longer by coupling with an optical cavity. The individual diamond NVcenter has reached nanosecond occupancy time^{39}, and its spin states can be read out nondestructively with spindependent photoluminescence. However, it is known that only very little of the total NV spontaneous optical emission is the direct transitions between the ground and the excited states^{42,53}, and this weak zero phonon line (ZPL) emission presents an experimental challenge to our proposals. In 2009, Barclay et al.^{49} showed that it is possible to enhance the ZPL emission rate by 47% if the Q of the microdisk can be increased to 2.5 × 10^{4}. In recent years, the ZPL emission rate has been enhanced from 3% to 70%^{65,66}. When the NV center is coupled to the resonator, the spontaneous emission into the ZPL can be largely enhanced, and the interaction of the NV center and photons is also strengthened.
Generally speaking, the reflection rule may be not perfect in experiment. The main factors that reduce the efficiency and fidelity of our scheme are the cavity field decay rate γ, cavity side leakage rate κ and the coupling strength g in the coupled reflection coefficients r(ω_{p}) (r_{0}(ω_{p}) = −1). Defining the efficiency as the yield of the photons, that is, η = η_{output}/η_{input}. Here η_{input} is the number of the input photon, whereas η_{output} is the number of the output photon. The fidelity is defined as F = 〈ψ_{ideal}ψ_{real}〉^{2}, i.e., the overlap of the output states of the system in the ideal case ψ_{ideal}〉 and the realistic case ψ_{real}〉. The fidelities of the TPCG and PPCG in both the evenparity mode and the oddparity mode are
Apparently, the fidelity of the even (odd)parity mode of the TPCG equals to the one of the PPCG. The even (odd)parity mode in the timebin DOF can not be directly distinguished, which is different from the ones in the polarization and spatial DOFs. Therefore, the effect of the TPCG first transfers the even (odd)parity mode in the timebin DOF into the same parity mode in the polarization DOF, and then the parity measurement results of the polarization DOF feedback to the ones of the timebin DOF in essence.
Our hyperECP only requires Alice to perform the local paritycheck (TPCG and PPCG) operations on her own photon pair. Let us define the average fidelity of the TPPCG as , where represents the fidelity of the twophoton system paralleling in two DOFs (either the evenparity mode or the oddparity mode). The average fidelities and efficiencies of our proposal depended mainly on the effect of , which are shown in Fig. 5(a,b), respectively. For the diamond NV center in the MTR with whisperinggallery microresonator mode system, the research^{45} shows that r(ω_{p}) ~ 0.95 when with ω_{c} = ω_{p} = ω_{0}; when with ω_{c} = ω_{p} = ω_{0}, provided that there is a MTR with (corresponding to κ ~ 1 GHz) or (corresponding to κ~10 GHz) according to the experimental results^{49}, the coupling strength should be on the order of hundreds of megahertz in order to reach r(ω_{p})~1. Our TPPCG and TPCG with fidelities and efficiencies greater than , , , and can be achieved when , which makes our hyperECP easier to be realized. It is not difficult to find that higher fidelity and efficiency can be acquired in the condition of the stronger coupling strength and the lower cavity decay rate.
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How to cite this article: Du, F.F. et al. General hyperconcentration of photonic polarizationtimebin hyperentanglement assisted by nitrogenvacancy centers coupled to resonators. Sci. Rep. 6, 35922; doi: 10.1038/srep35922 (2016).
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Acknowledgements
F.G. is supported by the Fundamental Research Funds for the Central Universities under Grant No. 2015KJJCA01. G.L. is supported by the National Natural Science Foundation of China under Grant Nos 11175094 and 91221205, Fund of Key Laboratory 9140C75010215ZK65001, and the National Basic Research Program of China under Grant No. 2015CB921002. G.L. is a member of the Center of Atomic and Molecular Nanosciences, Tsinghua University.
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F.F.D. completed the calculation and prepared the figures. F.F.D., F.G.D. and G.L.L. wrote the main manuscript text. G.L.L. supervised the whole project. All authors reviewed the manuscript.
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Du, FF., Deng, FG. & Long, GL. General hyperconcentration of photonic polarizationtimebin hyperentanglement assisted by nitrogenvacancy centers coupled to resonators. Sci Rep 6, 35922 (2016). https://doi.org/10.1038/srep35922
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