momentum dofs. our hyperentanglement puriﬁcation …

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Hyperentanglement purification for two-photon six-qubit quantum systems∗

Guan-Yu Wang, Qian Liu, and Fu-Guo Deng†

Department of Physics, Applied Optics Beijing Area Major Laboratory,

Beijing Normal University, Beijing 100875, China

(Dated: October 2, 2018)

Recently, two-photon six-qubit hyperentangled states were produced in experiment and they canimprove the channel capacity of quantum communication largely. Here we present a scheme forthe hyperentanglement purification of nonlocal two-photon systems in three degrees of freedom(DOFs), including the polarization, the first-longitudinal-momentum, and the second longitudinalmomentum DOFs. Our hyperentanglement purification protocol (hyper-EPP) is constructed withtwo steps resorting to parity-check quantum nondemolition measurement on the three DOFs andSWAP gates, respectively. With these two steps, the bit-flip errors in the three DOFs can becorrected efficiently. Also, we show that using SWAP gates is a universal method for hyper-EPPin the polarization DOF and multiple longitudinal momentum DOFs. The implementation of ourhyper-EPP is assisted by nitrogen-vacancy centers in optical microcavities, which could be achievedwith current techniques. It is useful for long-distance high-capacity quantum communication withtwo-photon six-qubit hyperentanglement.

PACS numbers: 03.67.Bg, 03.67.Pp, 03.65.Yz, 03.67.Hk

I. INTRODUCTION

Quantum entanglement plays a critical role in quantuminformation processing [1]. It is the key resource in quan-tum communication, such as quantum teleportation [2],quantum dense coding [3, 4], quantum key distribution[5, 6], quantum secret sharing [7], quantum secure directcommunication [8, 9], and so on. Maximally entangledstates can be used as the quantum channel for teleportingan unknown state of a quantum particle, without mov-ing the particle itself [2]. It can also be used to carry twobits of information by moving only one two-level parti-cle [3], not two or more. The two parties of quantumcommunication can create a private key with entangledphoton pairs in an absolutely secure way [5, 6]. More-over, they can exchange the secret message directly andsecurely without distributing the private key if they sharemaximally entangled photon pairs [8, 9].

Hyperentanglement, a state of a quantum system be-ing simultaneously entangled in multiple degrees of free-dom (DOFs), has attracted much attention as it canimprove both the channel capacity and the security ofquantum communication largely, beat the channel ca-pacity of linear photonic superdense coding [10], as-sist the complete analysis of Bell states [11–14], andso on. With the β barium borate (BBO) crystal, pho-ton pairs produced by spontaneous parametric downconversion (PDC) can be in a hyperentangled state.Many theoretical and experimental schemes for the gen-eration of hyperentangled states have been proposedand implemented in optical systems [15–23], such as inpolarization-momentum DOFs [18], polarization-orbital-

∗Published in Phys. Rev. A 94, 032319 (2016)†Corresponding author: [email protected]

angular momentum DOFs [19], multipath DOFs [20], andso on. In 2009, Vallone et al. [23] demonstrated experi-mentally the generation of a two-photon six-qubit hyper-entangled state in three DOFs.

Although the interaction between a photon and its en-vironment is weaker than other quantum systems, it stillinevitably suffers from channel noise, such as thermalfluctuation, vibration, imperfection of an optical fiber,and birefringence effects. The interaction between thephotons and the environment will make the entangledphoton pairs in less entangled states or even in mixedstates, which will decrease the security and the efficiencyof quantum communication. Entanglement purificationis used to obtain a subset of high-fidelity nonlocal en-tangled quantum systems from a set of those in mixedentangled states [24–28]. In 1996, Bennett et al. [24] in-troduced the entanglement purification protocol (EPP)for quantum systems in a Werner state [29] with quantumcontrolled-NOT gates. In 2001, Pan et al. [26] proposedan EPP with linear optical elements. In 2002, Simonand Pan [27] presented an EPP for a PDC source, notan ideal entanglement source. In 2008, Sheng et al. [28]proposed an efficient EPP for polarization entanglementfrom a PDC source, assisted by nondestructive quantumnondemolition detectors (QND) with cross-Kerr nonlin-earity. In 2010, Sheng and Deng [30] introduced the orig-inal deterministic EPP for two-photon systems, whichis far different from the conventional EPPs [24–28] asit works in a deterministic way, not a probabilistic one.Subsequently, some interesting deterministic EPPs wereproposed [31–34]. In 2003, Pan et al. [35] demonstratedthe entanglement purification of photon pairs using lin-ear optical elements. Recently, Ren and Deng [36] pro-posed the original hyperentanglement purification pro-tocol (hyper-EPP) for two-photon four-qubit systemsin mixed polarization-spatial hyperentangled Bell stateswith polarization bit-flip errors and spatial-mode bit-flip

http://arxiv.org/abs/1607.00082v3

2

errors assisted by nonlinear optical elements.A nitrogen vacancy (NV) center in diamond is an at-

tractive candidate for quantum information processingbecause of its long-lived coherence time at room temper-ature and optical controllability. The coherence time ofa diamond NV center can continue for 1.8 ms [37]. Ina diamond NV center, the electron spin can be exactlypopulated by the optical pumping with 532-nm light, andit can be easily manipulated [38–41] and readout [42, 43]by using the microwave excitation. Many interesting ap-proaches based on an NV center in diamond coupled toan optical cavity have been proposed in theory [44–48]and implemented in experiment [49–53].In this paper, we present a hyper-EPP for non-

local photon systems entangled in three DOFs as-sisted by nitrogen-vacancy centers in optical mi-crocavities, including the polarization DOF, thefirst-longitudinal-momentum DOF, and the second-longitudinal-momentum DOF. Our protocol is completedby two steps. The first step resorts to parity-checkquantum nondemolition measurement on the polariza-tion DOF (P-QND) and the two longitudinal-momentumDOFs (S-QND). The second step resorts to the SWAPgate between the polarization states of two photons (P-P-SWAP gate) and the SWAP gate between the polariza-tion state and the spatial state of one photon (P-F-SWAPgate and P-S-SWAP gate). Also, we show that using theSWAP gates is a universal method for hyper-EPP in thepolarization DOF and multiple longitudinal-momentumDOFs. Our hyper-EPP can effectively improve the en-tanglement of photon systems in long-distance quantumcommunication.This paper is organized as follows: We give the inter-

face between a circularly polarized light and a diamondnitrogen-vacancy center confined in an optical microcav-ity in Sec. II. Subsequently, we present the polarization-spatial parity-check QND of two-photon six-qubit sys-tems in Sec. III, and then, we give the principle of ourSWAP gate between the polarization states of two pho-tons in Sec. IVA and that of our SWAP gate betweenthe polarization state and the spatial state of one photonin Sec. IVB. In Sec.V, we propose an efficient hyper-EPP for mixed two-photon six-qubit hyperentangled Bellstates. In Sec. VI, we discuss the expansion for puri-fying the two-photon systems in the polarization DOFand multiple longitudinal momentum DOFs. A discus-sion and a summary are given in Sec. VII. In addition,some other cases for our hyper-EPP are discussed in theAppendix.

II. THE INTERFACE BETWEEN ACIRCULARLY POLARIZED LIGHT AND ADIAMOND NITROGEN-VACANCY CENTERCONFINED IN AN OPTICAL MICROCAVITY

A diamond NV center consists of a vacancy adjacentto a substitutional nitrogen atom. The NV center is neg-

atively charged with two unpaired electrons located atthe vacancy. The energy-level structure of the NV centercoupled to the cavity mode is shown in Fig. 1(a). Theground state is a spin triplet with the splitting at 2.87GHz in a zero external field between levels |0〉 ≡ |m = 0〉and | ± 1〉 ≡ |m = ±1〉 owing to spin-spin interaction.The excited state, which is one of the six eigenstates ofthe full Hamiltonian including spin-spin and spin-orbitinteractions in the absence of any perturbation, is labeledas |A2〉 = |E−〉| + 1〉+ |E+〉| − 1〉, where |E+〉 and |E−〉are the orbital states with the angular momentum pro-jections +1 and −1 along the NV axis, respectively. Weencode the qubits on the sublevels | ± 1〉 as the groundstates, and take |A2〉 as an auxiliary excited state. Thetransitions | + 1〉 ↔ |A2〉 and | − 1〉 ↔ |A2〉 in the NVcenter are resonantly coupled to the right (R) and the left(L) circularly polarized photons with the identical tran-sition frequency, respectively. The two transitions takeplace with equal probability.

1� 1�

0

R L

(a) (b)

2A

FIG. 1: (a) The energy-level diagram of an NV-cavity system.The qubit is encoded on the ground states | ± 1〉 and theexcited state |A2〉. The transitions |±1〉 ↔ |A2〉 are driven bythe right and light circularly polarized photons, respectively.(b) Schematic diagram for an NV center confined inside asingle-sided optical resonant microcavity.

Let us consider the composite unit, a diamond NV cen-ter confined inside a single-sided resonant microcavity, asshown in Fig. 1(b). The Heisenberg equations of motionfor this system can be written as [54]

a(t) = −[

i(ωc − ωp) +κ

2

]

a(t)− gσ−(t)

−√κ ain,

σ−(t) = −[

i(ω0 − ωp) +γ

2

]

σ−(t)− gσz(t) a(t)

+√γ σz(t)bin(t),

aout(t) = ain(t) +√κ a(t).

(1)

Here a(t) and σ−(t) are the annihilation operator of thecavity mode and the transition operator of the diamondNV center. σz(t) represents the inversion operator of theNV center. ωc, ωp, and ω0 are the frequencies of thecavity mode, the photon, and the diamond NV center

3

level transition, respectively. g, γ, and κ are the couplingstrength between a diamond NV center and a cavity, thedecay rate of a diamond NV center, and the damping rateof a cavity, respectively. bin(t) is the vacuum input field

with the commutation relation [bin(t), b†in(t

′)] = δ(t− t′).In the weak excitation limit, 〈σz〉 = −1, the reflection

coefficient for the NV-cavity unit is [55, 56]

r(ωp) =[i(ωc − ωp)− κ

2 ][i(ω0 − ωp) +γ2 ] + g2

[i(ωc − ωp) +κ2 ][i(ω0 − ωp) +

γ2 ] + g2

. (2)

When the diamond NV center is uncoupled to the cav-ity or coupled to an empty cavity, that is, g = 0, thereflection coefficient can be written as

r0(ωp) =i(ωc − ωp)− κ

2

i(ωc − ωp) +κ2

. (3)

If ω0 = ωc = ωp, the reflection coefficients can be writtenas

r =−κγ + 4g2

κγ + 4g2, r0 = −1. (4)

That is, when ω0 = ωc = ωp, the change of the inputphoton can be summarized as

|R〉|+ 1〉 → r|R〉| + 1〉, |R〉| − 1〉 → −|R〉| − 1〉,|L〉|+ 1〉 → −|L〉|+ 1〉, |L〉| − 1〉 → r|L〉| − 1〉. (5)

Considering the condition g ≥ 5√κγ, we can obtain the

approximate evolution of the system composed of a dia-mond NV center and a photon as follows [45]:

|R〉|+ 1〉 → |R〉|+ 1〉, |R〉| − 1〉 → −|R〉| − 1〉,|L〉|+ 1〉 → −|L〉|+ 1〉, |L〉| − 1〉 → |L〉| − 1〉. (6)

!"#$#% &!"#$#%

'(')*)

*('(

')*)*(

FIG. 2: The source of two-photon six-qubit hyperentangle-ment. The detailed description is shown in [23].

III. POLARIZATION-SPATIAL PARITY-CHECKQND OF TWO-PHOTON SIX-QUBIT SYSTEMS

A two-photon six-qubit hyperentangled Bell state canbe described as follows:

|Ψ〉AB =1√2(|H〉A|H〉B + |V 〉A|V 〉B)

⊗ 1√2(|l〉A|r〉B + |r〉A|l〉B)

⊗ 1√2(|I〉A|I〉B + |E〉A|E〉B).

(7)

Here the subscripts A and B represent the two photons.H and V represent the horizontal and the vertical po-larizations of photons, respectively. r and l representthe right and the left spatial modes of a photon in thefirst-longitudinal-momentum DOF, respectively. E andI denote the external and the internal spatial modes of aphoton in the second-longitudinal-momentum DOF, re-spectively, shown in Fig. 2. This two-photon six-qubithyperentangled Bell state can be produced by two 0.5-mm-thick type I BBO crystal slabs aligning one be-hind the other and an eight-hole screen [23], shown inFig. 2. When a continuous-wave (cw) vertically polarizedAr+ laser beam interacts through spontaneous paramet-ric downconversion (SPDC) with the two BBO crystalslabs, the nonlinear interaction between the laser beamand the BBO crystal leads to the production of the de-generate photon pairs which are entangled in polariza-tion and belong to the surfaces of two emission cones.As shown in Fig. 2, the insertion of an eight-hole screenallows us to achieve the double longitudinal-momentumentanglement.

Generally speaking, an arbitrary two-photon six-qubithyperentangled Bell state for a photon pair AB can bewritten as

|Ψ6〉AB = |ψ〉P ⊗ |ψ〉F ⊗ |ψ〉S , (8)

where |ψ〉P is one of the four Bell states for a two-photonsystem in the polarization DOF,

|φ±〉p =1√2(|RR〉 ± |LL〉)AC ,

|ψ±〉p =1√2(|RL〉 ± |LR〉)AC .

(9)

|ψ〉F is one of the four Bell states for a two-photon systemin the first-longitudinal-momentum DOF,

|φ±〉F =1√2(|rr〉 ± |ll〉)AC ,

|ψ±〉F =1√2(|rl〉 ± |lr〉)AC ,

(10)

and |ψ〉S is one of the four Bell states for a two-photon

4

system in the second-longitudinal-momentum DOF,

|φ±〉S =1√2(|EE〉 ± |II〉)AC ,

|ψ±〉S =1√2(|EI〉 ± |IE〉)AC .

(11)

The polarization-spatial parity-check QND is usedto distinguish the odd-parity mode (|ψ±〉p, |ψ±〉F , and|ψ±〉S) from the even-parity mode (|φ±〉p, |φ±〉F , and|φ±〉S) of the hyperentangled Bell states in both the po-larization and the two longitudinal-momentum DOFs.

lI

lE

rI

rI

lI

C

2Z

'4Z

'3Z

'1Z

'2Z

1CPBS

2CPBS

3CPBS

4CPBS

5CPBS

6CPBS

7CPBS

8CPBS

'1CPBS

'2CPBS

'3CPBS

'4CPBS

'5CPBS

'6CPBS

'7CPBS

'8CPBS

rE

rI

lE

lI

rE

rE

lE

rE

rI

lE

lI

A

1Z

4Z

3Z

NV

SW SW

FIG. 3: Schematic diagram of the polarization parity-checkQND. CPBSi (CPBSi′) (i = 1, 2, 3, 4) is a circularly polar-izing beam splitter which transmits the photon in the right-circular polarization |R〉 and reflects the photon in the left-circular polarization |L〉, respectively. Zi (Zi′) (i = 1, 2, 3, 4)is a half-wave plate which performs a polarization phase-flipoperation σP

z = |R〉〈R|−|L〉〈L| on the photon. SW is an opti-cal switch which lets the wave packets of a photon in differentspatial modes pass into and out of the NV center sequentially.

The schematic diagram of the polarization parity-checkQND is shown in Fig. 3, which consists of some circularlypolarizing beam splitters CPBSi, an NV center, and somehalf-wave plates Zi. The NV center is prepared in theinitial state |ϕ+〉 = 1√

2(|+1〉+ |− 1〉). If the two-photon

system AC is in a hyperentangled Bell state, one caninject the photons A and C into the quantum circuitsequentially. The evolutions of the system consisting ofthe two photons and the NV center are described as

|φ±〉p|ψ〉F |ψ〉S ⊗ |ϕ+〉 → |φ±〉p|ψ〉F |ψ〉S ⊗ |ϕ+〉,|ψ±〉p|ψ〉F |ψ〉S ⊗ |ϕ+〉 → |ψ±〉p|ψ〉F |ψ〉S ⊗ |ϕ−〉,

(12)

where |ϕ−〉 = 1√2(|+1〉−|−1〉). By measuring the state of

the NV center in the orthogonal basis {|ϕ+〉, |ϕ−〉}, one

can distinguish the even-parity Bell states |φ±〉p from theodd-parity Bell states |ψ±〉p of the two-photon system inthe polarization DOF without affecting the states of thetwo-photon systems in the spatial-mode DOFs. That is,if the NV center is projected into the state |ϕ+〉, the po-larization state of the hyperentangled two-photon systemis |φ±〉p. Otherwise, the polarization state of the hyper-entangled two-photon system is |ψ±〉p.

C

rE

rI

lE

lI

SW

SW

rE

rI

lE

lI

rE

rI

lE

lI

rE

rI

lE

lI

1Z

2Z 3Z

4Z

'1Z

'2Z

'4Z

SW

SW

1NV2NV

A

'3Z

FIG. 4: Schematic diagram of the spatial-mode parity-checkQND.

The schematic diagram of the spatial-mode parity-check QND is shown in Fig. 4. It consists of twoNV centers and some half-wave plates Zi (Zi′). Eachof the two NV centers is prepared in the initial state|ϕ+〉i = 1√

2(|+ 1〉+ | − 1〉) with i = 1, 2. One can inject

the photons A and C into the quantum circuit sequen-tially. The evolutions of the system composed of the twophotons and the two NV centers are described as:

|ψ〉P |φ±〉F |φ±〉S⊗|ϕ+〉1|ϕ+〉2 → |ψ〉P |φ±〉F |φ±〉S⊗|ϕ+〉1|ϕ+〉2,|ψ〉P |φ±〉F |ψ±〉S⊗|ϕ+〉1|ϕ+〉2 → |ψ〉P |φ±〉F |ψ±〉S⊗|ϕ+〉1|ϕ−〉2,|ψ〉P |ψ±〉F |φ±〉S⊗|ϕ+〉1|ϕ+〉2 → |ψ〉P |ψ±〉F |φ±〉S⊗|ϕ−〉1|ϕ+〉2,|ψ〉P |ψ±〉F |ψ±〉S⊗|ϕ+〉1|ϕ+〉2 → |ψ〉P |ψ±〉F |ψ±〉S⊗|ϕ−〉1|ϕ−〉2.

(13)

By measuring the states of two NV centers in the or-thogonal basis {|ϕ+〉, |ϕ−〉}, one can distinguish the even-parity states |φ±〉F from the odd-parity states |ψ±〉Fin the first-longitudinal-momentum DOF and the even-parity states |φ±〉S from the odd-parity states |ψ±〉S inthe second longitudinal momentum DOF, without af-fecting the states of the two-photon system in the po-larization DOF. That is, if NV1 is projected into thestate |ϕ+〉1, the two-photon system is in the state |φ±〉Fin the first-longitudinal-momentum DOF and if NV2 isprojected into the state |ϕ+〉2, the two-photon system isin the state |φ±〉S in the second-longitudinal-momentumDOF, respectively. If NV1 is projected into the state|ϕ−〉1, the two-photon system is in the state |ψ±〉F inthe first-longitudinal-momentum DOF and if NV2 is pro-jected into the state |ϕ−〉2, the two-photon system is

5

in the state |ψ±〉S in the second-longitudinal-momentumDOF, respectively.

'Ak

Ak

2CPBS

1PH

'1PH

2PH

'2PH

DL

NV

'DL

'Ak

Ak

1CPBS

3CPBS

4CPBS

5CPBS

'2CPBS'5CPBS

'1CPBS

'3CPBS

'4CPBS

6CPBS

'6CPBS

FIG. 5: Schematic diagram of the SWAP gate between thepolarization states of photon A and photon A′ (P-P SWAPgate). Hpi (i = 1, 2, 1′, 2′) is a half-wave plate which per-forms the Hadamard operation |R〉 → 1√

2(|R〉 + |L〉), |L〉 →

1√2(|R〉 − |L〉) on the polarization DOF of a photon. DL

(DL′) is a time-delay device which makes the two wave pack-ets in different paths interfere with each other at last. k

(k = rE, rI, lE, lI) represents any spatial modes of onephoton.

IV. SWAP GATES

A. SWAP gate between the polarization states oftwo photons

The SWAP gate between the polarization states of twophotons (P-P-SWAP gate) is used to swap the polariza-tion states of photon A and photon A′ without affect-ing their spatial-mode states. The initial states of twophotons in the polarization DOF and two longitudinal-momentum DOFs are

|Φ〉A = |Φp〉A ⊗ |Φk〉A,|Φ〉A′ = |Φp〉A′ ⊗ |Φk〉A′ .

(14)

Here |Φp〉A = α1|R〉 + β1|L〉 and |Φp〉A′ = α2|R〉 +β2|L〉 are the polarization states of the photons Aand A′, respectively. |Φk〉A = (γ1|r〉 + δ1|l〉) ⊗(ε1|E〉 + ξ1|I〉) and |Φk〉A′ = (γ2|r〉 + δ2|l〉) ⊗ (ε2|E〉 +ξ2|I〉) represent the states of the photons A and A′

in both the first-longitudinal-momentum DOF and thesecond-longitudinal-momentum DOF, respectively. Theschematic diagram of the P-P-SWAP gate is shown inFig. 5. Suppose that the NV center is prepared in theinitial state |Φ〉NV = 1√

2(|+1〉+ | − 1〉), the SWAP gate

works with the following steps.

First, one injects the photons A and A′ into the quan-tum circuit sequentially, and lets photon A (A′) passthrough the circularly polarizing beam splitter CPBS1

(CPBS1′), CPBS2 (CPBS2′), NV center, CPBS3

(CPBS3′), and CPBS4 (CPBS4′) in sequence. Af-ter performing a Hadamard operation on the NV center[|+1〉 → 1√

2(|+1〉+|−1〉), |−1〉 → 1√

2(|+1〉−|−1〉)], the

whole state of the system composed of two photons andone NV center is transformed from |Φ〉0 to |Φ〉1. Here

|Φ〉0 =|Φ〉A ⊗ |Φ〉A′ ⊗ |Φ〉NV ,

|Φ〉1 =[α1α2|R〉A|R〉A′ |+ 1〉 − β1α2|L〉A|R〉A′ | − 1〉− α1β2|R〉A|L〉A′ | − 1〉+ β1β2|L〉A|L〉A′ |+ 1〉]⊗ |Φk〉A ⊗ |Φk〉A′ .

(15)

Second, one performs Hadamard operations on pho-tons A and A′ with the half-wave plates Hp1 and Hp1′ .The state of the whole system is changed into

|Φ〉2 =1

2[α1α2(|R〉+ |L〉)A(|R〉+ |L〉)A′ |+ 1〉

− β1α2(|R〉 − |L〉)A(|R〉+ |L〉)A′ | − 1〉− α1β2(|R〉+ |L〉)A(|R〉 − |L〉)A′ | − 1〉+ β1β2(|R〉 − |L〉)A(|R〉 − |L〉)A′ |+ 1〉]⊗ |Φk〉A ⊗ |Φk〉A′ .

(16)

Third, one lets photon A (A′) pass though CPBS5

(CPBS5′), CPBS2 (CPBS2′), NV center, CPBS3

(CPBS3′), CPBS6 (CPBS6′), and Hp2 (Hp2′), and thestate of the whole system is changed from |Φ〉2 to |Φ〉3.Here

|Φ〉3 =[α1α2|R〉A|R〉A′ |+ 1〉 − β1α2|R〉A|L〉A′ | − 1〉− α1β2|L〉A|R〉A′ | − 1〉+ β1β2|L〉A|L〉A′ |+ 1〉]⊗ |Φk〉A ⊗ |Φk〉A′ .

(17)

Finally, by performing a Hadamard operation on theNV center, the state of the whole system is transformedinto

|Φ〉4 =1√2(α2|R〉 − β2|L〉)A|Φk〉A

⊗ (α1|R〉 − β1|L〉)A′ |Φk〉A′ ⊗ |+ 1〉

+1√2(α2|R〉+ β2|L〉)A|Φk〉A

⊗ (α1|R〉+ β1|L〉)A′ |Φk〉A′ ⊗ | − 1〉.

(18)

By measuring the NV center in the orthogonal basis{| + 1〉, | − 1〉}, the polarization state of photon A isswaped with the polarization state of photon A′ with-out disturbing their states in the spatial-mode DOFs. Ifthe NV center is projected into state |+1〉, phase-flip op-erations σP

z = |R〉〈R| − |L〉〈L| are performed on the po-larization DOF of photons A and A′. After the phase-flip

6

operations, the state of the two photons is transformedinto

|Φ〉f1 =(α2|R〉+ β2|L〉)A|Φk〉A⊗ (α1|R〉+ β1|L〉)A′ |Φk〉A′ .

(19)

Here, |Φ〉f1 is the objective state of the P-P-SWAP gate.If the NV center is projected into state | − 1〉, the ob-jective state can be obtained directly without phase-flipoperations.

lI lI

)(a

1CPBS

2CPBS

3CPBS

4CPBS

1X

2X

rE rE

rI rI

lE lE

lI lI

)(b

1CPBS

2CPBS

3CPBS

4CPBS

1X

2X

rE rE

rI rI

lE lE

FIG. 6: (a) Schematic diagram of the SWAP gate betweenthe polarization state and the spatial-mode state in the first-longitudinal-momentum DOF of one photon. (b) Schematicdiagram of the SWAP gate between the polarization state andthe spatial-mode state in the second-longitudinal-momentumDOF of one photon. Xi (i = 1, 2) is a half-wave plate whichperforms a polarization bit-flip operation σP

x = |R〉〈L| +|L〉〈R| on the photon.

B. SWAP gate between the polarization state andthe spatial state of one photon

The schematic diagram of our SWAP gate betweenthe polarization state and the spatial-mode state in thefirst-longitudinal-momentum DOF of one photon (P-F-SWAP) is shown in Fig. 6(a), which is constructed withlinear optical elements such as CPBSi (i = 1, 2, 3, 4) andhalf-wave plate Xi (i = 1, 2). One can inject photon A,which is in the state described as Eq. (14), into the quan-tum circuit shown in Fig. 6(a). After photon A passesthrough the quantum circuit shown in Fig. 6(a), the stateof photon A is transformed into

|Φ〉f2 = (γ1|R〉+ δ1|L〉)⊗ (α1|r〉 + β1|l〉)⊗ (ε1|E〉+ ξ1|I〉).

(20)

Here |Φ〉f2 is the objective state of the P-F-SWAP gate.The schematic diagram of our SWAP gate between thepolarization state and the spatial-mode state in the sec-ond longitudinal-momentum DOF of one photon (P-S-SWAP gate) is shown in Fig. 6(b). After photon A, whoseinitial state is described as Eq. (14), passes through thecircuit shown in Fig. 6(b), the state is changed into

|Φ〉f3 = (ε1|R〉+ ξ1|L〉)⊗ (γ1|r〉+ δ1|l〉)⊗ (α1|E〉+ β1|I〉).

(21)

Here, |Φ〉f3 is just the objective state of the P-S-SWAPgate.

V. EFFICIENT HYPER-EPP FOR MIXEDTWO-PHOTON SIX-QUBIT

HYPERENTANGLED BELL STATES

In the practical transmission of photons in hyperentan-gled Bell states for high-capacity quantum communica-tion, both the bit-flip error and the phase-flip error willoccur on the photon systems. Although a phase-flip er-ror cannot be directly purified, it can be transformedinto a bit-flip error using a bilateral local operation [24–28]. If a bit-flip error purification has been successfullysolved, phase-flip errors also can be solved perfectly. Inthis way the two parties in quantum communication, sayAlice and Bob, can purify a general mixed hyperentan-gled state. Below, we only discuss the purification of thetwo-photon six-qubit hyperentangled mixed state withbit-flip errors in the three DOFs.Two identical two-photon six-qubit systems in mixed

hyperentangled Bell states in the polarization DOF andtwo longitudinal-momentum DOFs with bit-flip errorscan be described as follows:

ρAB = [F1|φ+〉pAB〈φ+|+ (1− F1)|ψ+〉pAB〈ψ+|]⊗ [F2|φ+〉FAB〈φ+|+ (1− F2)|ψ+〉FAB〈ψ+|]⊗ [F3|φ+〉SAB〈φ+|+ (1− F3)|ψ+〉SAB〈ψ+|],

ρCD = [F1|φ+〉pCD〈φ+|+ (1 − F1)|ψ+〉pCD〈ψ+|]⊗ [F2|φ+〉FCD〈φ+|+ (1− F2)|ψ+〉FCD〈ψ+|]⊗ [F3|φ+〉SCD〈φ+|+ (1− F3)|ψ+〉SCD〈ψ+|].

(22)

Here the subscripts AB and CD represent two photonpairs shared by the two parties in quantum communica-tion , say Alice and Bob. Alice holds the photons A andC, and Bob holds the photons B and D. F1, F2, and F3

represent the probabilities of states |φ+〉pAB (|φ+〉pCD),|φ+〉FAB (|φ+〉FCD), and |φ+〉SAB (|φ+〉SCD) in the mixedstates, respectively.The initial state of the system composed of the two

identical two-photon six-qubit subsystems ABCD canbe expressed as ρ0 = ρAB ⊗ ρCD. It can be viewedas a mixture of maximally hyperentangled pure states.In the polarization DOF, it is a mixture of the states

7

|φ+〉pAB ⊗ |φ+〉pCD, |φ+〉pAB ⊗ |ψ+〉pCD, |ψ+〉pAB ⊗ |φ+〉pCD,and |ψ+〉pAB ⊗ |ψ+〉pCD with the probabilities F 2

1 , (1 −F1)F1, (1 − F1)F1, and (1 − F1)

2, respectively. In thefirst-longitudinal-momentum DOF, it is a mixture of thestates |φ+〉FAB ⊗ |φ+〉FCD, |φ+〉FAB ⊗ |ψ+〉FCD, |ψ+〉FAB ⊗|φ+〉FCD, and |ψ+〉FAB ⊗ |ψ+〉FCD with the probabilitiesF 22 , (1 − F2)F2, (1 − F2)F2, and (1 − F2)

2, respectively.In the second-longitudinal-momentum DOF, it is a mix-ture of the states |φ+〉SAB ⊗ |φ+〉SCD, |φ+〉SAB ⊗ |ψ+〉SCD,|ψ+〉SAB ⊗ |φ+〉SCD, and |ψ+〉SAB ⊗ |ψ+〉SCD with the prob-abilities F 2

3 , (1 − F3)F3, (1 − F3)F3, and (1 − F3)2, re-

spectively.Our hyper-EPP for the nonlocal two-photon six-qubit

systems in hyperentangled Bell states with bit-flip er-rors in the polarization DOF and the two longitudinal-momentum DOFs can be achieved with two steps in eachround. The first step is completed with polarization andspatial-mode parity-check QNDs introduced in Sec. III.The second step of our hyper-EPP scheme is completedwith the SWAP gates introduced in Sec. IV. We discussthe principles of these two steps in detail as follows.

P-S-QND

2CPBS

1BS

3BS

4BS

2PH

rE

rI

rE

lI

rE

rI

lE

lI

rI

rE

lI

LrE

RlI

LrI

RrI

LlE

RlE

RrE

LlI

A

C

1PH

3PH

4PH

1CPBS

3CPBS

4CPBS

rE

2BS

LLL

FIG. 7: Schematic diagram of the first step of our hyper-EPPfor mixed hyperentangled Bell states with bit-flip errors inthe polarization DOF and the two longitudinal-momentumDOFs with P-S-QNDs. A beam splitter BSi (i = 1, 2) isused to perform the Hadamard operation [|E〉 → 1√

2(|E〉 +

|I〉), |I〉 → 1√2(|E〉−|I〉)] on the first-longitudinal-momentum

DOF of a photon and BSj (j = 3, 4) is used to perform theHadamard operation [|r〉 → 1√

2(|r〉+ |l〉), |l〉 → 1√

2(|r〉 − |l〉)]

on the second-longitudinal-momentum DOF of a photon. Dk

(k = LrE,RrE,LrI,RrI,LlE,RlE,LlI, orRlI) is a single-photon detector.

A. The first step of our hyper-EPP withpolarization and spatial-mode parity-check QND

The principle of the first step of our hyper-EPP isshown in Fig. 7. Alice performs the P-S-QNDs on pho-

tons AC and performs the Hadamard operations on thepolarization DOF and the spatial-mode DOFs of photonC. Bob performs the same operations on photons BD.

First, Alice and Bob perform the P-S-QNDs on thetwo two-photon systems AC and BD, respectively.Based on the results of the P-S-QNDs on the iden-tical two-photon systems, the states can be classifiedinto eight cases. In case (1), the two identical two-photon systems AC and BD are in the same parity-mode in all of the three DOFs, including the polariza-tion DOF, the first-longitudinal-momentum DOF, andthe second-longitudinal-momentum DOF. This case cor-responds to the states |φ+〉pAB |φ+〉

pCD or |ψ+〉pAB |ψ+〉pCD

in the polarization DOF, the states |φ+〉FAB|φ+〉FCD

or |ψ+〉FAB |ψ+〉FCD in the first-longitudinal-momentumDOF, and the states |φ+〉SAB|φ+〉SCD or |ψ+〉SAB |ψ+〉SCD

in the second-longitudinal-momentum DOF. The classi-fication of the eight cases is shown in Table I.

Now, let us discuss case (1), in which the paritymodes of AC and BD are same in the polarization DOF,the first-longitudinal-momentum DOF, and the second-longitudinal-momentum DOF, to bring to light of theprinciple of the first step in our hyper-EPP. Case (2), inwhich the parity modes of AC and BD are different inthe polarization DOF, the first-longitudinal-momentumDOF, and the second-longitudinal-momentum DOF, isdiscussed in the Appendix. The other cases are similarto these two cases with a little modification. When ACand BD are both in the even-parity modes in the po-larization DOF, the first-longitudinal-momentum DOF,and the second longitudinal momentum DOF, Alice andBob obtain the states

|Φ1〉p =1√2(|RRRR〉+ |LLLL〉)ABCD,

|Φ2〉p =1√2(|RLRL〉+ |LRLR〉)ABCD,

|Φ1〉F =1√2(|rrrr〉 + |llll〉)ABCD,

|Φ2〉F =1√2(|rlrl〉 + |lrlr〉)ABCD,

|Φ1〉S =1√2(|EEEE〉+ |IIII〉)ABCD,

|Φ2〉S =1√2(|EIEI〉+ |IEIE〉)ABCD.

(23)

On the contrary, when AC and BD are both inthe odd-parity modes in the polarization DOF, thefirst-longitudinal-momentum DOF, and the second-longitudinal-momentum DOF, Alice and Bob obtain the

9

TABLE II: The probabilities of different states corresponding to the eight cases, respectively.

CaseProbability

|φ+〉pAB |ψ+〉pAB |φ+〉FAB |ψ+〉FAB |φ+〉SAB |ψ+〉SAB

(1) F 21 (1− F1)

2 F 22 (1− F2)

2 F 23 (1− F3)

2

(2) F1(1− F1) F1(1− F1) F2(1− F2) F2(1− F2) F3(1− F3) F3(1− F3)

(3) F1(1− F1) F1(1− F1) F 22 (1− F2)

2 (1− F3)2 (1− F3)

2

(4) F 21 (1− F1)

2 F2(1− F2) F2(1− F2) F 23 (1− F3)

2

(5) F 21 (1− F1)

2 F 22 (1− F2)

2 F3(1− F3) F3(1− F3)

(6) F1(1− F1) F1(1− F1) F2(1− F2) F2(1− F2) F 23 (1− F3)

2

(7) F1(1− F1) F1(1− F1) F 22 (1− F2)

2 F3(1− F3) F3(1− F3)

(8) F 21 (1− F1)

2 F2(1− F2) F2(1− F2) F3(1− F3) F3(1− F3)

(2), Alice and Bob will discard the two photons. Other-wise, the second step of hyper-EPP with SWAP gates isrequired if the two two-photon systems ABCD are pro-jected into case (i) (i = 3 ∼ 8).

B. The second step of our hyper-EPP with SWAPgates

If the two two-photon systems AB and CD are pro-jected into the case (6), the fidelities of the Bell states ofthe photon pair AB in the polarization and the first-longitudinal-momentum DOFs are lower than the ini-tial ones and the fidelity in the second-longitudinal-momentum DOF is higher than the initial one. Al-ice and Bob seek another two two-photon systems A′B′

and C′D′ projected into the case (5), whose fidelity islower than the initial one in the second-longitudinal-momentum DOF and higher than the initial one in thepolarization and the first-longitudinal-momentum DOFs,respectively. The photons A, C, A′, and C′ belong to Al-ice and B, D, B′, and D′ belong to Bob. We discuss theresult of the second step when the states of the photonsAB and A′B′ are in |Φ〉AB0 = |ψ+〉p|φ+〉F |φ+〉S and|Φ〉A′B′0 = |φ+〉p|ψ+〉F |ψ+〉S , respectively, as an exam-ple.

First, Alice and Bob swap the polarization statesof the two-photon system AB and the states of thesystem A′B′ by using the P-P-SWAP gate shown inFig. 5. The state of the system composed of four-photon ABA′B′ and two NV centers are transformed

from |Φ〉0 = |Φ〉AB0|Φ〉A′B′0|φ+〉a|φ+〉b to the state

|Φ〉 =1

4[−(|RRRL〉+ |LLRL〉

+ |RRLR〉+ |LLLR〉)|+ 1〉a|+ 1〉b+ (|RRRL〉 − |LLRL〉− |RRLR〉+ |LLLR〉)|+ 1〉a| − 1〉b− (|RRRL〉 − |LLRL〉− |RRLR〉+ |LLLR〉)| − 1〉a|+ 1〉b+ (|RRRL〉+ |LLRL〉+ |RRLR〉+ |LLLR〉)| − 1〉a| − 1〉b]⊗ |φ+〉FAB|φ+〉SAB|ψ+〉FA′B′ |ψ+〉SA′B′ .

(26)

If the state of two NV centers a and b belonging to Aliceand Bob is |+1〉a|+1〉b or |−1〉a|−1〉b, Alice and Bob donothing. If the state of the NV centers is |+1〉a|−1〉b or |−1〉a|+1〉b, Alice and Bob perform a σP

z = |R〉〈R|−|L〉〈L|operation on photons A′ and B, respectively. Thus, Aliceand Bob obtain the states

|Φ〉AB1 = |φ+〉p|φ+〉F |φ+〉S ,|Φ〉A′B′1 = |ψ+〉p|ψ+〉F |ψ+〉S .

(27)

Second, Alice and Bob swap the polarization statesand the first-longitudinal-momentum states of photonsA, B, A′, and B′, respectively. The states of the twotwo-photon systems are transformed into the states


(28)

Third, Alice and Bob swap the polarization states ofthe two systems AB and A′B′ again and obtain the states

|Φ〉AB3 = |ψ+〉p|φ+〉F |φ+〉S ,|Φ〉A′B′3 = |φ+〉p|ψ+〉F |ψ+〉S .

(29)

Finally, Alice and Bob swap the polarization states andthe first-longitudinal-momentum states of photons A, B,A′, and B′, respectively. The states of the two systemsare changed into

|Φ〉AB4 = |φ+〉p|ψ+〉F |φ+〉S

|Φ〉A′B′4 = |ψ+〉p|φ+〉F |ψ+〉S .(30)

10

Here |Φ〉AB4 and |Φ〉A′B′4 are the final states of this casein the second step of hyper-EPP.If the two two-photon systems AB and CD are pro-

jected into the states in case (3), (4), (5), (7), or (8),the second step of the hyper-EPP can be completed, asdiscussed in the Appendix.

HaL

0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

F

Fid

elit

y

(b)0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

F

Eff

icie

ncy

FIG. 8: (a) The fidelities of our hyper-EPP for mixed hyper-entangled Bell states in three DOFs. The solid line, dash-dotted line, and dashed line represent the fidelities of the it-eration times n = 1, 2, and 3, respectively, for the cases withF1 = F2 = F3 = F . (b) The efficiencies of the of our hyper-EPP for mixed hyperentangled Bell states in three DOFs.The dashed line and the solid line represent the efficiencies ofthe first step and the two steps in the first round, respectivelyfor the cases with F1 = F2 = F3 = F .

After these two steps, the first round of our hyper-EPP process is completed and the state of the two-photonsystem AB is changed to be

ρ′AB =[F ′1|φ+〉pAB〈φ+|+ (1 − F ′

1)|ψ+〉pAB〈ψ+|]⊗ [F ′

2|φ+〉FAB〈φ+|+ (1− F ′2)|ψ+〉FAB〈ψ+|]

⊗ [F ′3|φ+〉SAB〈φ+|+ (1− F ′

3)|ψ+〉SAB〈ψ+|].(31)

Here F ′1 =

F 2

1

F 2

1+(1−F1)2

, F ′2 =

F 2

2

F 2

2+(1−F2)2

, and F ′3 =

F 2

3

F 2

3+(1−F3)2

. The fidelity of the finale state is F ′ =

F ′1F

′2F

′3 which is higher than the initial fidelity F =

F1F2F3 when Fi > 1/2 (i = 1, 2, 3). By iterating ourhyper-EPP process, the fidelity of the state of the two-photon system can be improved dramatically. The fideli-ties of the final states are shown in Fig. 8(a) with iteratingn = 1, 2, 3 times for the case with F1 = F2 = F3 = F .The efficiency of an EPP is defined as the probability of

obtaining a high-fidelity entangled photon system from apair of photon systems transmitted over a noisy channel

without photon loss. The efficiency of our hyper-EPPafter the first step in the first round is

Y1 = [F 21 + (1 − F1)

2][F 22 + (1− F2)

2][F 23 + (1− F3)

2].(32)

After the second step, the efficiency of the first roundof the hyper-EPP process is

Y2 =[F 21 + (1 − F1)

2][F 22 + (1− F2)

2][F 23 + (1− F3)

2]

+min{([2F1(1− F1)][F22 + (1 − F2)

2]

[F 23 + (1− F3)

2]), ([F 21 + (1− F1)

2]

[2F2(1− F2)][2F3(1− F3)])}+min{([F 2

1 + (1− F1)2][2F2(1− F2)]

[F 23 + (1− F3)

2]), ([2F1(1 − F1)]

[F 22 + (1− F2)

2][2F3(1− F3)])}+min{([F 2

1 + (1− F1)2][F 2

2 + (1− F2)2]

[2F3(1− F3)]), ([2F1(1− F1)]

[2F2(1− F2)][F23 + (1− F3)

2])}.(33)

The efficiencies Y1 and Y2 of our hyper-EPP for thecase with F1 = F2 = F3 = F are shown in Fig. 8(b),respectively. Obviously, the efficiency of our hyper-EPPis improved largely with the second step for purification.

VI. EXPANSION

We present a hyper-EPP for the nonlocal two-photon systems in the polarization and two longitudinal-momentum DOFs by two steps. The first step is com-pleted by the P-S-QNDs, Hadamard operations, andsome detections. After the first step, we can obtain theinformation of the probability of each Bell state in dif-ferent DOFs. If the probabilities of the three even-parityBell states in the three DOFs are higher than the ini-tial ones [corresponding to the case (1) in the first step],the second step is not need. If the probabilities of theeven-parity Bell states in the three DOFs are all lowerthan the initial ones [corresponding to the case (2) in thefirst step], the two-photon system is discarded. Other-wise [corresponding to the case (3), (4), (5), (6), (7) or(8)], the second step is required. The second step is com-pleted by using the combination of the P-P-SWAP gate,the P-F-SWAP gate, and the P-S-SWAP gate. The pur-pose of the second step is to improve the probability ofthe even-parity Bell state in one or two DOFs by costingsome other two-photon systems. For example, if the twotwo-photon systems ABCD are projected into the statesin case (4), the probability of the even-parity Bell statein the first-longitudinal-momentum DOF is lower thanthe initial one. We seek another two two-photon systemsA′B′C′D′ whose probability of even-parity Bell states inthe first-longitudinal-momentum DOF is higher than theinitial one and the fidelities of the Bell states in another

11

two DOFs are lower than the initial ones. Thus, with theP-P-SWAP gate and the P-F-SWAP gate, the fidelity ofthe two-photon system AB is improved. If we can con-struct the SWAP gate between the polarization state andarbitrary another longitudinal-momentum state withoutaffecting the states in other DOFs, the second step can beused to purify the two-photon systems in the polarizationDOF and multiple longitudinal-momentum DOFs. Thatis to say, using SWAP gates is a universal method for pu-rifying the two-photon systems in the polarization DOFand multiple longitudinal-momentum DOFs.

lIu

lId

lIu

lId

1XrEu

rIu

lEu

2CPBS

3CPBS

1CPBS

4CPBS

6CPBS

7CPBS

5CPBS

8CPBS

rEd

rId

lEd

rEu

rIu

lEu

rEd

rId

lEd

2X

3X

4X

FIG. 9: Schematic diagram of the SWAP gate between thepolarization state and the spatial-mode state in the thirdlongitudinal-momentum DOF of one photon.

We construct a SWAP gate between the polarizationstate and the state in the third-longitudinal-momentumDOF of a photon (P-T-SWAP gate) of being simulta-neously entangled in the polarization DOF and threelongitudinal-momentum DOFs as an example. The ini-tial state of a photon being simultaneously entangledwith other photons in the polarization DOF and threelongitudinal-momentum DOFs can be expressed as

|Φ〉 =(a1|R〉+ b1|L〉)⊗ (a2|r〉 + b2|l〉)⊗ (a3|E〉+ b3|I〉)⊗ (a4|u〉+ b4|d〉).

(34)

Here (a4|u〉 + b4|d〉) is the state of the photon A beingentangled with other photons in the third-longitudinal-momentum DOF (a4 and b4 include not only the pa-rameters for photon A but also the states for other pho-tons, so do ai and bi). u and d represent the upper andthe down spatial-mode states of the photon, respectively.The schematic diagram of the P-T-SWAP gate, which isconstructed with some linear optical elements, is shownin Fig. 9. After the photon A passes through the quan-tum circuit, its state is transformed into

|Φ′〉 =(a4|R〉+ b4|L〉)⊗ (a2|r〉+ b2|l〉)⊗ (a3|E〉+ b3|I〉) ⊗ (a1|u〉+ b1|d〉).

(35)

Here |Φ′〉 is the objective state of the P-T-SWAP gate.After the first step of the hyper-EPP, if the probabil-ity of the even-parity Bell state of the system ABCDin the third longitudinal-momentum DOF is lower thanthe initial one, we can seek another system A′B′C′D′

whose probability of the even-parity Bell state in this

DOF is higher. By using the P-P-SWAP gate and the P-T-SWAP gate, the second step of our hyper-EPP canbe completed to improve the efficiency of purificationlargely.

HaL

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

g� ΚΓ

FP-

QN

D

HbL

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

g� ΚΓ

PP-

QN

DFIG. 10: (a) The fidelity of the polarization-paritycheck QND. (b) The efficiency of the polarization-paritycheck QND. The solid line represents FP1 (ηP1) for|φ±〉p|φ±〉F |φ±〉S, |φ±〉p|φ±〉F |ψ±〉S, |φ±〉p|ψ±〉F |φ±〉S, and|φ±〉p|ψ±〉F |ψ±〉S. The dashed line represents FP2 (ηP2) for|ψ±〉p|φ±〉F |φ±〉S, |ψ±〉p|φ±〉F |ψ±〉S, |ψ±〉p|ψ±〉F |φ±〉S, and|ψ±〉p|ψ±〉F |ψ±〉S .

VII. DISCUSSION AND SUMMARY

With the P-S-QND and the SWAP gates, we constructour efficient hyper-EPP for the mixed hyperentangledBell states in the polarization DOF and two longitudinal-momentum DOFs with bit-flip errors. For constructingthe P-S-QND and the SWAP gates, we utilize the re-flection coefficient produced by the nonlinear interactionbetween the single photon and the NV-cavity system.Under the resonant condition ω0 = ωp = ωc, the re-flection coefficient of the input photon is affected by thecoupling strength g, the NV decay rate γ, and the cav-ity damping rate κ. If we neglect κ and γ, the fideli-ties and the efficiencies of the P-S-QND and the SWAPgates can reach 100%. However, in the realistic condi-tion, the fidelities and the efficiencies are suppressed byγ and κ. The fidelity is defined as F = |〈ϕr|ϕi〉|2. Here|ϕi〉 represents the final state in the ideal condition and|ϕr〉 represents the final state in the realistic condition.The efficiency is defined as η = nout/nin, where nout

represents the number of the output photons and nin

represents the number of the input photons. For the

12

HaL

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

g� ΚΓ

FS-QND

HbL

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

g� ΚΓ

PS-QND

FIG. 11: (a) The fidelity of the spatial-mode-parity checkQND. (b) The efficiency of the spatial-mode-parity checkQND. The solid line represents F1(η1) for |φ±〉p|φ±〉F |φ±〉F .The dash-dotted line represents FS2(ηS2) = FS4(ηS4) =FS6(ηS6) = FS7(ηS7) = FS8(ηS8) for |ψ±〉p|φ±〉F |φ±〉S,|ψ±〉p|φ±〉F |ψ±〉S , |ψ±〉p|ψ±〉F |φ±〉S, |φ±〉p|ψ±〉F |ψ±〉S , and|ψ±〉p|ψ±〉F |ψ±〉S . The dashed line represents F3(η3) =F5(η5) for |φ

±〉p|φ±〉F |ψ±〉S and |φ±〉p|ψ±〉F |φ±〉S .

polarization parity-check QND, there exist two fidelitiesFPi (i = 1, 2) and two efficiencies ηpi (i = 1, 2) corre-sponding to the two different polarization states |φ±〉pand |ψ±〉p. For the spatial-mode parity-check QND,there exist eight fidelities FSi (i = 1 ∼ 8) and eightefficiencies ηSi (i = 1 ∼ 8) corresponding to the states|φ±〉p|φ±〉F |φ±〉S , |ψ±〉p|φ±〉F |φ±〉S , |φ±〉p|φ±〉F |ψ±〉S ,|ψ±〉p|φ±〉F |ψ±〉S , |φ±〉p|ψ±〉F |φ±〉S , |ψ±〉p|ψ±〉F |φ±〉S ,|φ±〉p|ψ±〉F |ψ±〉S , and |ψ±〉p|ψ±〉F |ψ±〉S , respectively.As r0 is not affected by g, κ, nor γ, we have FS2(ηS2) =FS4(ηS4) = FS6(ηS6) = FS7(ηS7) = FS8(ηS8) andFS3(ηS3) = FS5(ηS5). The fidelities and the efficiencies(discussed the detail in the Appendix) of the polarizationparity-check QND, spatial parity-check QND, and the P-P-SWAP gate, which vary with the parameter g/

√κγ,

are shown in Fig. 10, Figs. 11, and 12, respectively.

By far, several groups have experimentally demon-strated the coupling between a diamond NV center and amicrocavity [49–52]. In 2009, Barclay et al. [49] reportedtheir experiment with the parameters [g, κ, γtot, γ]/2π =[0.30, 26, 0.013, 0.0004]GHz for coupling the diamond NVcenters to a chip-based microcavity. This experiment isimplemented in a weak coupling regime with a low-Q fac-tor. In their experiment, the NV center coherent couplingrate within the narrow-band zero phonon line (ZPL) isalmost one-third of the total coupling rate. We have r =0.94 at ω0 = ωp = ωc. Based on their experimental pa-

(a)

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

g� ΚΓ

FSWAP

(b)

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

g� ΚΓ

PSWAP

FIG. 12: (a) The fidelity of the P-P-SWAP gate. (b) Theefficiency of the P-P-SWAP gate.

rameters, the fidelities and the efficiencies of our polariza-tion parity-check QND can reach Fp1,2 = 99.76%, 99.91%and ηp1,2 = 94.84%, 94.54%, respectively. The fideli-ties and the efficiencies of our spatial-mode parity-checkQND can reach FS1,2,3 = 99.53%, 99.83%, 99.68% andηS1,2,3 = 89.95%, 89.38%, 89.66%, respectively. The fi-delity and the efficiency of the P-P-SWAP gate can reachFSWAP = 99.46% and ηSWAP = 90.08%, respectively.

In our proposal, the two cavity modes with right- andleft- circular polarizations, which couple to the two tran-sitions |+〉 ↔ |A2〉 and |−〉 ↔ |A2〉 respectively, are re-quired. Many good experiments that provide a cavitysupporting both of two circularly-polarized modes withthe same frequency have been realized [57–63]. For ex-ample, Luxmoore et al. [57] presented a technique forfine tuning of the energy split between the two circularly-polarized modes to just 0.15nm in 2012.

In summary, we have proposed a hyper-EPP forthe mixed hyperentangled Bell states with bit-flip er-rors in three DOFs, including the polarization and twolongitudinal-momentum DOFs. Our hyper-EPP is com-pleted with one or two steps. In the first step, Alice andBob perform the P-S-QNDs and Hadamard operationson their own photons. If the states of the two photonpairs are projected into the states in case (1), the hyper-entanglement purification process is accomplished and itcosts one photon pair to purify another one. If the statesof the two photon pairs are projected into cases (3), (4),(5), (6), (7) and (8), the second step of our hyper-EPPis needed. In the second step, Alice and Bob swap thestates between two-photon systems in the same DOF orbetween different DOFs of the one photon by the method

13

of combining the P-P-SWAP gate, the P-F-SWAP gate,and the P-S-SWAP gate. If the hyperentanglement pu-rification process is accomplished with two steps, it costsanother three pairs to purify one photon pair. We haveshowed the feasibility of our hyper-EPP as high fideli-ties and efficiencies of P-S-QND and SWAP gates can beobtained based on the existing experimental parametersand there exists the cavity that supports two circularly-polarized modes with the same frequency.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Sci-ence Foundation of China under Grant No. 11674033 andNo. 11474026, and the Fundamental Research Funds forthe Central Universities under Grant No. 2015KJJCA01.

APPENDIX

A. Case (2) in the first step of our hyper-EPP withSWAP gates

In case (2), the two identical two-photon systemsAC and BD are in different parity-mode in all ofthe three DOFs, including the polarization DOF,the first-longitudinal-momentum DOF, and the second-longitudinal- momentum DOF.

When the photons AC are in the even-parity modesand BD are in the odd-parity modes at the sametime in the polarization DOF, in the first-longitudinal-momentum DOF, and in the second-longitudinal-momentum DOF, Alice and Bob obtain the states

|Φ5〉p =1√2(|RRRL〉+ |LLLR〉)ABCD,

|Φ6〉p =1√2(|RLRR〉+ |LRLL〉)ABCD,

|Φ5〉F =1√2(|rrrl〉 + |lllr〉)ABCD,

|Φ6〉F =1√2(|rlrr〉 + |lrll〉)ABCD,

|Φ5〉S =1√2(|EEEI〉 + |IIIE〉)ABCD,

|Φ6〉S =1√2(|EIEE〉 + |IEII〉)ABCD.

(36)

On the contrary, when the photons AC are in theodd-parity modes and BD are in the even-parity modesat the same time in the polarization DOF, in thefirst-longitudinal-momentum DOF or in the second-longitudinal-momentum DOF, Alice and Bob obtain the

states

|Φ7〉p =1√2(|RRLR〉+ |LLRL〉)ABCD,

|Φ8〉p =1√2(|RLLL〉+ |LRRR〉)ABCD,

|Φ7〉F =1√2(|rrlr〉 + |llrl〉)ABCD,

|Φ8〉F =1√2(|rlll〉+ |lrrr〉)ABCD ,

|Φ7〉S =1√2(|EEIE〉+ |IIEI〉)ABCD,

|Φ8〉S =1√2(|EIII〉+ |IEEE〉)ABCD.

(37)

The states |Φ7〉p, |Φ8〉p, |Φ7〉F , |Φ8〉F , |Φ7〉S , and |Φ8〉Scan be transformed into states |Φ5〉p, |Φ6〉p, |Φ5〉F , |Φ6〉F ,|Φ5〉S , and |Φ6〉S , respectively.Next, Alice and Bob perform Hadamard operations

in the polarization and the two longitudinal-momentumDOFs on photons C and D, respectively. In the polar-ization DOF, the states |Φ5〉p and |Φ6〉p are transformedinto the states |Φ5′〉p and |Φ6′〉p, respectively. In thefirst-longitudinal-momentum DOF, the states |Φ5〉F and|Φ6〉F are transformed into |Φ′

5〉F and |Φ′6〉F , respectively.

In the second-longitudinal-momentum DOF, the states|Φ5〉S and |Φ6〉S are transformed into |Φ′

5〉S and |Φ′6〉S ,

respectively. Here

|Φ′5〉p =

1

2√2[(|RR〉+ |LL〉)AB(|RR〉 − |LL〉)CD

+ (|RR〉 − |LL〉)AB(−|RL〉+ |LR〉)CD],

|Φ′6〉p =

1

2√2[(|RL〉+ |LR〉)AB(|RR〉+ |LL〉)CD

+ (−|RL〉+ |LR〉)AB(−|RL〉 − |LR〉)CD],

|Φ′5〉F =

1

2√2[(|rr〉 + |ll〉)AB(|rr〉 − |ll〉)CD

+ (|rr〉 − |ll〉)AB(−|rl〉+ |lr〉)CD],

|Φ′6〉F =

1

2√2[(|rl〉+ |lr〉)AB(|rr〉 + |ll〉)CD

+ (−|rl〉+ |lr〉)AB(−|rl〉 − |lr〉)CD],

|Φ′5〉S =

1

2√2[(|EE〉+ |II〉)AB(|EE〉 − |II〉)CD

+ (|EE〉 − |II〉)AB(−|EI〉+ |IE〉)CD],

|Φ′6〉S =

1

2√2[(|EI〉+ |IE〉)AB(|EE〉+ |II〉)CD

+ (−|EI〉+ |IE〉)AB(−|EI〉 − |IE〉)CD].

(38)

Finally, the photons C and D are detected by single-photon detectors, respectively. If the photons CDare in the even-parity mode in the polarization DOF(the first-longitudinal-momentum DOF or the second-longitudinal-momentum DOF), nothing is needed to be

14

done. If the photons CD are in the odd-parity mode inthe polarization DOF (the first-longitudinal-momentumDOF or the second-longitudinal-momentum DOF), thephase-flip operation σp

z = |R〉〈R| − |L〉〈L| (σFz = |r〉〈r| −

|l〉〈l| or σSz = |E〉〈E| − |I〉〈I|) is operated on the photon

B. If AC and BD are projected into this case, Alice andBob will discard the two photon pairs.

B. The complete second step of our hyper-EPPwith SWAP gates

(a) If the two two-photon systems AB and CD are pro-jected into the states in case (3) in the first step, Aliceand Bob can seek another two two-photon systems A′B′

and C′D′ which are projected into the states in case (8).The P-P-SWAP gate shown in Fig. 5 can be used to com-plete the second step of our hyper-EPP. The states of thetwo photon pairs AB and A′B′ are transformed into thestates


(39)

Here |Φ〉AB5 and |Φ〉A′B′5 are the finale states of case (a)in the second step of our hyper-EPP.(b) If the two two-photon systems AB and CD are pro-

jected into the states in case (4) in the first step, Aliceand Bob can seek another two two-photon systems A′B′

and C′D′ which are projected into the states in case (7).The P-P-SWAP gate shown in Fig. 5 and the P-F-SWAPgate shown in Fig. 6(a) can be used to complete the sec-ond step of our hyper-EPP.First, Alice swaps the polarization states and the first-

longitudinal-momentum states of photons A and per-forms the same operations on photon A′. Bob performsthe same operations on photons B and B′, respectively.Second, Alice swaps the polarization states between thephotons A and A′. Bob swaps the polarization statesbetween the photons B and B′. Finally, Alice and Bobswap the polarization states and the first-longitudinal-momentum states on the photons A, B, A′ and B′ again.After these operations, the states of photon pairs become

|Φ〉AB6 = |ψ+〉p|ψ+〉F |φ+〉S ,|Φ〉A′B′6 = |φ+〉p|φ+〉F |ψ+〉S .

(40)

Here |Φ〉AB6 and |Φ〉A′B′6 are the final states of case (b)in the second step of our hyper-EPP.(c) If the two two-photon systems AB and CD are

projected into the states in case (5) in the first step, Aliceand Bob can seek another two two-photon systems A′B′

and C′D′ which are projected into the states in case (6).Alice and Bob perform the similar operations as whatthey do in case (b) in the second step. The differenceis that in this case, they use the P-S-SWAP gate shownin Fig. 6(b), instead of the P-F-SWAP gate. The finalstates of case (c) in the second step of our hyper-EPP

are

|Φ〉AB7 = |ψ+〉p|φ+〉F |ψ+〉S ,|Φ〉A′B′7 = |φ+〉p|ψ+〉F |φ+〉S .

(41)

(d) If the two two-photon systems AB and CD areprojected into the states in case (7) in the first step, Aliceand Bob can seek another two two-photon systems A′B′

and C′D′ which are projected into the states in case (4).The second step of our hyper-EPP is completed with theP-P-SWAP gates and the P-S-SWAP gates.First, Alice and Bob swap the polarization states of the

two two-photon system AB and the states of the systemA′B′. Second, Alice and Bob swap the polarization statesand the second longitudinal-momentum states of photonsA, B, A′, and B′, respectively. Third, Alice and Bobswap the polarization states of the two systems AB andA′B′ again. Finally, Alice and Bob swap the polariza-tion states of the second-longitudinal-momentum statesof photons A, B, A′, and B′, respectively. The states ofthe two systems are changed to be

|Φ〉AB8 = |φ+〉p|φ+〉F |ψ+〉S

|Φ〉A′B′8 = |ψ+〉p|ψ+〉F |φ+〉S .(42)

Here |Φ〉AB8 and |Φ〉A′B′8 are the final states of this casein the second step of our hyper-EPP.(e) If the two two-photon systems AB and CD are

projected into the states in case (8) in the first step, Al-ice and Bob can seek another two two-photon systemsA′B′ and C′D′ which are projected into the states in case(3). The second step of our hyper-EPP is completed withthe P-P-SWAP gates, P-F-SWAP gates, and P-S-SWAPgates.First, Alice and Bob swap the polarization states and

the first-longitudinal-momentum states of photons A, B,A′, and B′ by the P-F-SWAP gate, respectively. Sec-ond, Alice and Bob swap the polarization states betweensystem AB and system A′B′ with the P-P-SWAP gate.Third, Alice and Bob swap the polarization states andthe first-longitudinal-momentum states of photons A, B,A′, and B′ again with the P-F-SWAP gate, respectively.Fourth, Alice and Bob swap the polarization states andthe second-longitudinal-momentum states of photons A,B, A′, and B′ with the P-S-SWAP gate, respectively.Fifth, Alice and Bob swap the polarization states betweensystem AB and system A′B′ by the P-P-SWAP gate forthe third time. Finally, Alice and Bob swap the polariza-tion states and the second-longitudinal-momentum statesof photons A, B, A′, and B′ respectively again. Thus,the states of systems become

|Φ〉AB9 = |ψ+〉p|ψ+〉F |ψ+〉S

|Φ〉A′B′9 = |φ+〉p|φ+〉F |φ+〉S .(43)

Here |Φ〉AB9 and |Φ〉A′B′9 are the final states of this casein the second step of our hyper-EPP.

15

C. The fidelity and the efficiency

There exist two fidelities FP1,2 and efficiencies ηP1,2 of the polarization parity-check QND. The fidelity (efficiency)FP1(ηP1) corresponds to the states |φ±〉p|φ±〉F |φ±〉S , |φ±〉p|φ±〉F |ψ±〉S , |φ±〉p|ψ±〉F |φ±〉S , and |φ±〉p|ψ±〉F |ψ±〉S .The fidelity (efficiency) FP2(ηP2) corresponds to the states |ψ±〉p|φ±〉F |φ±〉S , |ψ±〉p|φ±〉F |ψ±〉S , |ψ±〉p|ψ±〉F |φ±〉S ,and |ψ±〉p|ψ±〉F |ψ±〉S . The fidelities of the polarization parity-check QND can be expressed as

FP1 =|2 + r2 + r20 |2

4[2 + |r2|2 + |r20 |2], FP2 =

|r − r0|22[|r|2 + |r0|2]

. (44)

The efficiencies of the polarization parity-check QND can be expressed as

ηp1 =1

4[2 + |r2|2 + |r20 |2], ηP2 =

1

2[|r0|2 + |r|2]. (45)

The fidelity and the efficiency of the polarization parity-check QND, which vary with the parameter g/√κγ, are shown

in Figs. 9(a) and 9(b), respectively.For the spatial-mode parity-check QND, there exist eight fidelities FSi(i = 1, 2, · · · , 8) and eight efficiencies ηSi(i =

1, 2, · · · , 8) corresponding to the states |φ±〉p|φ±〉F |φ±〉F , |ψ±〉p|φ±〉F |φ±〉F , |φ±〉p|φ±〉F |ψ±〉F , |ψ±〉p|φ±〉F |ψ±〉F ,|φ±〉p|ψ±〉F |φ±〉F , |ψ±〉p|ψ±〉F |φ±〉F , |φ±〉p|ψ±〉F |ψ±〉F , and |ψ±〉p|ψ±〉F |ψ±〉F , respectively. The fidelities of thespatial-mode parity-check QND can be expressed as

FS1 =|(r2 + r20)

2 + 4(r2 + r20) + 4|216[4(|r2|2 + |r20 |2) + (|r4|2 + 2|r2r20 |2 + |r40 |2) + 4]

,

FS2 =|r2r20 − 2rr0 + 1|2

4[|r2r20 |2 + 2|rr0|2 + 1],

FS3 =|(r2 + r20)(r − r0) + 2(r − r0)|2

8[(|r3|2 + |r30 |2 + |r0r2|2 + |r20r|2) + 2(|r|2 + |r0|2)],

FS4 =|(r − r0)(1 − rr0)|2

4[|rr20 |2 + |r0r2|2 + |r|2 + |r0|2],

FS5 =|(r2 + r20)(r − r0) + 2(r − r0)|2

8[(|r3|2 + |rr20 |2 + |r0r2|2 + |r30 |2) + 2(|r|2 + |r0|2)],

FS6 =|(r − r0)(1 − rr0)|2

4[|r|2 + |r0|2 + |rr20 |2 + |r0r2|2],

FS7 =|r2 + r20 − 2rr0|2

4[|r2|2 + 2|rr0|2 + |r20 |2],

FS8 =|r2 + r20 − 2rr0|2

4[|r2|2 + 2|rr0|2 + |r20 |2].

(46)

The efficiencies of the spatial-mode parity-check QND can be expressed as

ηS1 =1

16[4(|r2|2 + |r20 |2) + (|r4|2 + 2|r2r20 |2 + |r40 |2) + 4],

ηS2 =1

4[|r2r20 |2 + 2|rr0|2 + 1],

ηS3 =1

8[(|r3|2 + |r30 |2 + |r0r2|2 + |r20r|2) + 2(|r|2 + |r0|2)],

ηS4 =1

4[|rr20 |2 + |r0r2|2 + |r|2 + |r0|2],

ηS5 =1

8[(|r3|2 + |rr20 |2 + |r0r2|2 + |r30 |2) + 2(|r|2 + |r0|2)],

ηS6 =1

4[|r|2 + |r0|2 + |rr20 |2 + |r0r2|2],

ηS7 =1

4[|r2|2 + 2|rr0|2 + |r20 |2],

ηS8 =1

4[|r2|2 + 2|rr0|2 + |r20 |2].

(47)

16

As r0 = −1 is an exact value, we have FS2(ηS2) = FS4(ηS4) = FS6(ηS6) = FS7(ηS7) = FS8(ηS8) and FS3(ηS3) =FS5(ηS5). The fidelities and the efficiencies of the spatial-mode parity-check QND, which vary with the parameterg/

√κγ, are shown in Figs. 10(a) and 10(b), respectively.

The fidelity and the efficiency of the P-P-SWAP gate proposed in Sec.IV(A) can be expressed as

F ′SWAP =

|f ′1 + f ′

2 + f ′3 + f ′

4 + f ′4 + f ′

5 + f ′6 + f ′

7 + f ′8|2

η′SWAP

,

η′SWAP =1

32[|2α1α2r

2 + (α1β2 + β1α2)(r2r0 + r3 + r30 − r20r) + β1β2(r

4 + r40)|2

+ |2α1α2r2 + (α1β2 + β1α2)(r

2r0 + r3 − r30 + r20r) + β1β2(r4 − r40 + 2r2r20)|2

+ |2α1α2r − (α1β2 − β1α2)(r2 + r20)− β1β2(rr

20 + r3 + r30 − r0r

2)|2

+ |2α1α2r − (α1β2 − β1α2)(2rr0 + r2 − r20)− β1β2(rr20 + r3 − r30 + r0r

2)|2

+ |2α1α2r + (α1β2 − β1α2)(r2 + r20)− β1β2(rr

20 + r3 + r30 − r0r

2)|2

+ |2α1α2r + (α1β2 − β1α2)(2rr0 + r2 − r20)− β1β2(rr20 + r3 − r30 + r0r

2)|2

+ |2α1α2 − 2(α1β2 + β1α2)r0 + 2β1β2r20 |2

+ |2α1α2 − 2(α1β2 + β1α2)r + 2β1β2r2|2].

(48)

Here,

f ′1 =

1

16[2α1α2r

2 + (α1β2 + β1α2)(r2r0 + r3 + r30 − r20r) + β1β2(r

4 + r40)](α1 − β1)(α2 − β2),

f ′2 =

1

16[2α1α2r

2 + (α1β2 + β1α2)(r2r0 + r3 − r30 + r20r) + β1β2(r

4 − r40 + 2r2r20)](α1 + β1)(α2 + β2),

f ′3 =

1

16[2α1α2r − (α1β2 − β1α2)(r

2 + r20)− β1β2(rr20 + r3 + r30 − r0r

2)](α1 + β1)(α2 − β2),

f ′4 =

1

16[2α1α2r − (α1β2 − β1α2)(2rr0 + r2 − r20)− β1β2(rr

20 + r3 − r30 + r0r

2)](α1 − β1)(α2 + β2),

f ′5 =

1

16[2α1α2r + (α1β2 − β1α2)(r

2 + r20)− β1β2(rr20 + r3 + r30 − r0r

2)](α1 − β1)(α2 + β2),

f ′6 =

1

16[2α1α2r + (α1β2 − β1α2)(2rr0 + r2 − r20)− β1β2(rr

20 + r3 − r30 + r0r

2)](α1 + β1)(α2 − β2),

f ′7 =

1

16[2α1α2 − 2(α1β2 + β1α2)r0 + 2β1β2r

20 ](α1 + β1)(α2 + β2),

f ′8 =

1

16[2α1α2 − 2(α1β2 + β1α2)r + 2β1β2r

2](α1 − β1)(α2 − β2).

(49)

In our hyper-EPP, our P-P-SWAP gate is used to swap the maximally entangled states between two photon pairs,and α1 = β1 = α2 = β2. Thus, the fidelity and efficiency of the P-P-SWAP gate can be expressed as

FSWAP =|f1 + f2|2ηSWAP

,

ηSWAP =1

32[|r2 + r2r0 + r3 + r30 − r20r +

1

2(r4 + r40)|2

+ |r2 + r2r0 + r3 − r30 + r20r +1

2(r4 − r40 + 2r2r20)|2

+ 2|r − 1

2(rr20 + r3 + r30 − r0r

2)|2 + 2|r − 1

2(rr20 + r3 − r30 + r0r

2)|2

+ |1− 2r0 + r20 |2 + |1− 2r + r2|2].

(50)

Here f1 = 116 [2r

2 + 2(r2r0 + r3 − r30 + r20r) + r4 − r40 + 2r2r20 ] and f2 = 18 (r0 − 1)2. The fidelity and the efficiency of

our P-P-SWAP gate varying with the parameter g/√κγ are shown in Figs. 11(a) and 11(b), respectively.

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