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Invited Paper

Generation of hybrid four-qubit entangled decoherence-free states assisted by the cavity-QED system You-Sheng Zhou a,b, Xian Li c, Yun Deng d,n, Hui-Ran Li e, Ming-Xing Luo e a

School of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China School of Electronic Engineering, Dublin City University, Dublin 9, Ireland c School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450046, China d School of Computer Science, Sichuan University of Science & Engineering, Zigong 64300, China e Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu 610031, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 18 May 2015 Received in revised form 29 November 2015 Accepted 24 December 2015

We propose three effective protocols to generate four-qubit entangled decoherence-free states assisted by the cavity-QED system. These schemes are based on optical selection rules realized with a single electron charged self-assembled GaAs/InAs quantum dot in a micropillar resonator. Compared with previous photonic protocols, the ﬁrst scheme is to replace the entangled-state resources with much simpler single-photon resources and has a deterministic success probability. Moreover, the cavity-QED system may be used to generate four-spin entangled decoherence-free states and hybrid four-qubit of spin-photon entangled decoherence-free states. These states may be applied up to different requirements because of different superiorities of photons and spins. All schemes may be implemented with current physical technologies. Crown Copyright & 2015 Published by Elsevier B.V. All rights reserved.

Keywords: Quantum optics Quantum communications Decoherence-free state Cavity-QED system

1. Introduction Quantum entanglement occurs when different particles are generated or interact in ways such that each particle cannot be described independently. Bipartite or multipartite entanglements as essential ingredients for testing local hidden variable have become most important resources for quantum teleportation [1,2], quantum computation [3–5], quantum key distribution [6–8], quantum dense coding [9–11], etc. Most of these protocols require maximal entanglements or noiseless quantum channels [1–11]. However, in practice, they are easily degraded because of the coupling between the quantum system and the environment or equipments [12], which may greatly reduce the application ﬁdelity. Different ways have been explored to deal with these quantum decoherences. One way is to concentrate the decoherenced quantum entanglements. Entanglement concentration is used to get the maximal entanglement from partially entangled pure states. Bennett et al. [13] introduced the ﬁrst entanglement concentration protocol (ECP) using the Schmidt projection method and collective measurements for two-photon systems. After that, many interesting ECPs have been proposed for photon systems [14–23]. The other typical scheme uses the quantum errorn Corresponding author at: School of Computer Science, Sichuan University of Science & Engineering, Zigong 64300, China. E-mail address: [email protected] (Y. Deng).

correction and dynamical decoupling techniques. Another useful way is to encode information into the symmetry state in the decoherence-free subspace to avoid the system–environment interaction. Thus the decoherence-free subspace is inherently immune to quantum decoherence and robust to perturbing error processes. Therefore, the decoherence-free states are very useful for longdistance quantum communication and quantum computation and applied in quantum error correction codes [12]. The N-qubit decoherence-free states were originally proposed by Kempe et al. [24], and are invariant under any identical unitary transformation on each of the qubits. One decoherence-free singlet state is special EPR state

|ψ −〉 =

1 (|01〉 − |10〉)12 2

(1)

Another nontrivial example is the four-qubit entangled decoherence-free state

|Φ〉 = α|Ψ0〉 + β|Ψ1〉

(2)

with

|Ψ0〉 = |ψ −〉12 |ψ −〉34 ,

|Ψ1〉 =

1 1 1 + |0011〉 + |1100〉 − |ψ 〉12 |ψ +〉34 , 3 3 3

http://dx.doi.org/10.1016/j.optcom.2015.12.065 0030-4018/Crown Copyright & 2015 Published by Elsevier B.V. All rights reserved.

Please cite this article as: Y.-S. Zhou, et al., Optics Communications (2016), http://dx.doi.org/10.1016/j.optcom.2015.12.065i

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2

and |ψ +〉 =

1 2

(|01〉 + |10〉). This four-qubit state is sufﬁcient to fully

protect an arbitrary logical qubit against collective decoherence in contrast to the two-qubit state. With its interesting applications, Bourennane et al. [25] have generated a four-photon polarizationentangled decoherence-free states via a spontaneous parametric down-conversion source. They coherently overlapped two fourphoton sources |Ψ1〉 and |Ψ2〉. Recently, Zou et al. [26] and Gong et al. [27] proposed schemes to generate four-photon polarization-entangled decoherence-free states based on linear optical elements and post-selection strategy. However, those schemes mentioned in Refs. [25-27] work in the destructive way because the generated four-photon polarization-entangled decoherence-free states cannot be used for further quantum information processing and quantum computation when a four-photon coincidence measurement was made on the photonic states. Wang et al. [28] propose a probabilistic linear-optics-based scheme for local conversion of four Einstein–Podolsky–Rosen photon pairs into four-photon polarization-entangled decoherence-free states. In this paper, we propose deterministic schemes to generate the four-photon entangled decoherence-free states, four-spin entangled decoherence-free states, hybrid four-qubit entangled decoherencefree states assisted by the cavity-QED system. Hybrid systems (photon-spin) [29,30] have been explored to effectively enable strong nonlinear interactions between single photons [31] in the weakcoupling regime. The optical selection rules realized with a single electron charged self-assembled GaAs/InAs quantum dot in a micropillar resonator [32,33] may be applied to construct qubit gates on photon systems [31,34–38]. We ﬁrst present a theoretical preparation scheme of four-qubit entangled decoherence-free states with CNOT gates and one-qubit rotations. And then, we generate fourphoton entangled decoherence-free states by constructing the CNOT gate on a two-photon system with the help of the cavity-QED system. Moreover, by constructing the CNOT gate on a two-spin system, we can generate four-spin entangled decoherence-free states with the help of the cavity-QED system. Furthermore, with the hybrid CNOT gate on a photon-spin or spin-photon system, hybrid four-qubit entangled decoherence-free states may be generated. These schemes may be experimentally realized with present technology.

2. Generations of four-qubit entangled decoherence-free states In order to generate four-qubit entangled decoherence-free states deterministically, we consider its theoretical decomposition circuits using the elementary gates of the CNOT gate and singlequbit rotations. Notice that |Φ〉 may be rewritten as |Φ〉= 1 1 1 β|00〉|11〉 − β|01〉|01〉 + α|10〉|10〉 + β|11〉|00〉 with |01〉= |ψ +〉 3

3

3

and |10〉 = |ψ −〉. Using the two-qubit logic gate T11·T12·T13, the initial state |0011〉 is changed into

⎛ 1 ⎞ 1 1 β|00〉 − β|01〉 − α|10〉 + β|11〉⎟ |11〉34 ⎜ ⎝ 3 ⎠12 3 3

(4)

where ⎛ ⎜ ⎜ ⎜ T11 = ⎜ ⎜ ⎜ ⎜⎜ ⎝

1 2 0 0 1 2

0 0 − 1 0 0 1 0 0

⎛ ⎜ ⎜ β 0 ⎜ 0 β T13 = ⎜ ⎜ −α 0 ⎜ 0 −α ⎜⎜ ⎝

1 ⎞ ⎟ 2⎟ ⎟ 0 ⎟ , 0 ⎟ 1 ⎟ ⎟⎟ 2 ⎠

⎞ ⎟ α 0⎟ 0 α⎟ ⎟ β 0⎟ 0 β⎟ ⎟⎟ ⎠

⎛ ⎜ ⎜ ⎜ T12 = ⎜ − ⎜ ⎜ ⎜⎜ ⎝

2 3

1

1

2 3

3 0 0

3

0 0

⎞ 0 0⎟ ⎟ ⎟ 0 0 ⎟, ⎟ ⎟ 1 0 ⎟⎟ 0 1⎠

And then using two CNOT gates on qubits (1,3) and qubits (2,4), we can get

⎛ 1 ⎞ 1 1 β|0011〉 − β|0110〉 − α|1001〉 + β|1100〉⎟ ⎜ ⎝ 3 ⎠1234 3 3

which may be changed into |Φ〉 using two two-qubit logic gates −1 ·CNOT 1 on the ﬁrst twoT2 = CNOT 1·T14·CNOT 1 and T3 = CNOT 1·T14 qubit (1,2) and the last two-qubit (3,4) respectively. Here,

⎛1 ⎜ ⎜0 ⎜ T14 = ⎜ ⎜0 ⎜ ⎜0 ⎝

0 1 2 0 1 2

0 ⎞ ⎟ 1 ⎟ 0 − 2⎟ ⎟ 1 0 ⎟ 1 ⎟ 0 ⎟ 2 ⎠ 0

(7)

Now, using the CNOT gate and qubit rotations [39,40], each controlled-U can be realized with (I2 ⊗ A)·CNOT1· (I2 ⊗ B )·CNOT 1·(I2 ⊗ C ) , where U = Rz (α )R y(θ )Rz (β ), A = Rz (α )R y(θ /2), B = R y( − θ /2)Rz ( − (α + β ) /2), and C = Rz ((β − α ) /2). Thus we can get the following decompositions: T11 = (X ⊗ I2)· CNOT 1·(A1−1 ⊗ I2)·CNOT 2·(A1 ⊗ I2)·CNOT 2·CNOT 1·(X ⊗ I2), T12 = (X ⊗ A2−1)·CNOT 1·(I2 ⊗ A2 )·CNOT 1·(X ⊗ I2), T13 = A3 ⊗ I2, T14 = (A1 ⊗ I2)·CNOT 2·(A1−1 ⊗ I2)·CNOT 2

(8)

where X denote the Pauli ﬂip,

⎛π⎞ A1 = R y⎜ ⎟, ⎝ 4⎠

⎛θ ⎞ A2 = R y⎜ 1 ⎟, ⎝ 2⎠

A3 = R y( − θ2)

(9)

with θ1 = arctan(1/ 2 ) , θ2 = 2arctan(α /β ), and CNOT1 denotes the controlled NOT gate with the ﬁrst qubit is the controlling qubit while CNOT2 denotes the controlled NOT gate with the second qubit as the controlling qubit. R y(θ ) = cos(θ /2)|0〉 and 〈0| − sin(θ /2)|0〉〈1| + sin(θ /2)|1〉〈0| + cos(θ /2)|1〉〈1| Rz (φ)= denote the qubit rotations exp(iφ /2)|0〉〈0| + exp( − iφ /2)|1〉〈1| along the y-axis or z-axis in Pauli sphere respectively. X denotes the Pauli ﬂip. 2.1. A singely charged quantum dot in an one-side optical microcavity The cavity-QED system used in our proposal is constructed by a singly charged In(Ga)As quantum dot located in the center of a one-side optical cavity [41–43], as shown in Fig. 1. The single electron states have Jz = ± 1/2 spin ( | ↑ 〉, | ↓ 〉) and the holes have Jz = ± 3/2 ( |⇑〉, |⇓〉). The two electrons form a singlet state and therefore have total spin zero, which prevents electron spin interactions with the hole spin [41]. Photon polarization is commonly deﬁned with respect to the direction of propagation, i.e. zaxis, where the absolute rotation direction of its electro-magnetic ﬁelds does not change. Label the optical states by their circular polarization ( |L〉 and |R〉 for left and right circular polarization respectively). A negatively charged exciton | ↑ ↓ ⇑〉 or | ↓ ↑ ⇓〉 may be created by resonantly absorbing |L〉 or |R〉, respectively. The input–output relation of this one-side cavity system can be calculated from the Heisenberg equation [41] of motions for the cavity ﬁeld operator and dipole operator

⎛ κ ⎞ da^ κ = − ⎜⎜ iΔωc + + s ⎟⎟a^ − gσ^− − 2 2⎠ dt ⎝ (5)

(6)

κ a^in,

⎛ dσ^− ς⎞ = − ⎜⎜ iΔωx + ⎟⎟σ^− − gσ^za^, dt 2⎠ ⎝

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Fig. 1. (a) A charged QD inside a one-side micropillar microcavity interacting with circularly polarized photons. (b) The optical selection rules due to the Pauli exclusion principle. | ↑ 〉 → | ↑ ↓ ⇑〉 is driven by the left-circularly polarized photon ( |L〉) and | ↓ 〉 → | ↓ ↑ ⇓〉 is driven by the right-circularly polarized photon ( |R〉).

where Δωc = ωc − ω , Δωx = ωx − ω . ωc , ω and ωx are the frequencies of the cavity mode, the input probe light, and the dipole transition, respectively. g is the coupling strength between the cavity and dipole. ς/2, κ/2, and κs/2 are the decay rates of the dipole, the cavity ﬁeld, and the cavity side leakage mode, respectively. a^in and a^out satisfy a^out = a^in + κ a^ . If the dipole stays in the ground state most of the time [42], a cavity with a dipole behaves like a linear beamsplitter whose reﬂection coefﬁcient r is given by

r (ω) =

i2Δωc + κs + κ + g^ i2Δω + κ − κ + g^ c

s

(11)

g^ = g /(i2Δωx + ς ). In the following we consider the case of a dipole tuned into resonance with the cavity mode ( Δωx = 0), probed with resonant light ( g = 0, ζ → 0 ). If the radiation is not coupled to the dipole transition ( g = 0, ζ → ∞), the reﬂection coefﬁcient in Eq. (9) becomes

Fig. 2. (a) The photonic CNOT gate assisted by a one-side micropillar microcavity. (b) The generation circuit. cPBSi represent polarizing beam splitter in the circular basis, which transmits |R〉 and reﬂects |L〉. Aj represent special wave-plates to perform rotation operations along the y-axis in the Pauli sphere with the phases deﬁned in Eq. (9). A−j 1 denote the inverse of Aj. Cj represent special wave-plates to perform rotation operations along the z-axis in the Pauli sphere with the phases deﬁned in Eq. (9). C −j 1 denote the inverse of Cj. X denotes the wave-plate to perform the ﬂip |R〉〈L| + |L〉〈R|. Hj denote the half wave-plate to perform the Hadamard operation on the photon. X denotes the wave-plate to perform the ﬂip |R〉〈L| + |L〉〈R|. Wj represent the Hadamard operation on the spin e in the Cavity. All photons evolve from left to right.

2

r0(ω) =

i2Δω + κ + κs i2Δω + κs − κ

(12)

The reﬂection coefﬁcients can reach |r0(ω)| ≈ 1 and |rh(ω)| ≈ 1 when the cavity side leakage is negligible [42,43]. If the probe beam is in the state α|R〉 + β|L〉 and an one-side cavity-QED system has a superposition spin (| ↑ 〉 + | ↓ 〉)/ 2 , the state of the system consisting of the light and the spin after the reﬂection is

eiθ 0 [| ↑ 〉(α|R〉 + βeiΔθ |L〉) + | ↓ 〉(αeiΔθ |R〉 + β|L〉)]

(13)

where Δθ = θh − θ0 with θ0 = arg[r0(ω)] and θh = arg[rh(ω)]. By adjusting the frequencies of the light and the cavity mode, the phase difference Δθ for the left and the right circularly polarized photons may be reached to π [34,38]. Thus the optical selection rules may be described as

cPBS1, Cy 1

|ϕ1〉A |ϕ2〉B | + 〉e ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ cPBS2

2

[| ↑ 〉e (a1|R〉 + b1|L〉)A + | ↓ 〉e (a1|R〉 − b1|L〉)A]|ϕ2〉B

W1

⟶(a1| ↑ 〉e |R〉A + b1| ↓ 〉e |L〉A )|ϕ2〉B H1

⟶(a1| ↑ 〉e |R〉A + b1| ↓ 〉e |L〉A )(a2′ |R〉 + b 2′ |L〉)B cPBS3, Cy

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→[a1| ↑ 〉e |R〉A (a2′ |R〉 + b 2′ |L〉)B + b1| ↓ 〉e |L〉A (a2′ |R〉 − b 2′ |L〉)B ] cPBS 4 W2, H2

⎯⎯⎯⎯⎯⎯⎯⎯⎯→| ↑ 〉e [a1|R〉A (a2|R〉 + b 2|L〉)B + α 2|L〉a (b 2|R〉 + a2|L〉)B ] + | ↓ 〉e [a1|R〉A (a2|R〉 + b 2|L〉)B − b1|L〉A (b 2|R〉 + a2|L〉)b ]

|R〉| ↑ 〉↦ − |R〉| ↑ 〉, |R〉| ↓ 〉↦|R〉| ↓ 〉, |L〉| ↑ 〉↦|L〉| ↑ 〉, |L〉| ↓ 〉↦ − |L〉| ↓ 〉.

for a deterministic quantum computation with the present experimental techniques, an appealing platform for quantum information processing is the cavity-QED system [42,43,48]. The deterministic CNOT gate on a two-photon system [34–37] is ﬁrstly implemented. The initial states of the spin e and the photons A and B are | + 〉e , |ϕ1〉A = a1|R〉 + b1|L〉 and |ϕ2〉B = a2|R〉 + b2|L〉, respectively, as shown in Fig. 2(a). The joint system of the photons A and B and the spin e is changed as follows:

Me

⟶a1|R〉A (a2|R〉 + b 2|L〉)B + α 2|L〉a (b 2|R〉 + a2|L〉)B

(14)

2.2. Photonic four-qubit entangled decoherence-free states From the theoretical preparation shown in Eqs. (4)–(8), the CNOT gate is the key in the experimental realization. Knill et al. [44] ﬁrstly proposed a probabilistic CNOT gate on two photonic qubits by using linear optical elements and postselection. The cross-Kerr nonlinearity or charge detection has also been used to implement the CNOT gate [45–47]. Of course, a strong cross-Kerr nonlinearity is still a big challenge in experiment at present. To achieve a nontrivial nonlinearity between two individual qubits

(15)

Here, a2′ = (a2 + b2) / 2 and b2′ = (a2 − b2) / 2 . This is the controlled-NOT gate (CNOT) on two photons. Me denotes the measurement of the spin e under the basis {| ↑ 〉, | ↓ 〉} and one phase ﬂip Z is performed on the photon A for the measurement outcome | ↓ 〉. Moreover, the SWAP of two photons A and B may be realized with three CNOT gates as CNOT 1·CNOT 2·CNOT 1 shown in Fig. 2(b). With the theoretical preparation shown in Section 2.1, and the photonic CNOT circuit in Fig. 2(a), photonic four-qubit entangled decoherence-free states may be generated in Fig. 2(c). The detailed evolution of four photons A , B , C , D in the state |RRLL〉ABCD is deﬁned as

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4

T13

|RRLL〉ABCD ⟶ T12

1

T11

6 1

⟶ ⟶

6

1

CNOT (B, D)

1

T3

3 1

⟶

3

6

This is the controlled-NOT gate (CNOT) on two spins, i.e.,

(β|RR〉 + α|LR〉)AB |LL〉CD

(β|RR〉 − β|RL〉 +

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ T2

2

| ↑ 〉〈 ↑ | ⊗ (| ↑ 〉〈 ↑ | + | ↓ 〉〈 ↓ |) + | ↓ 〉〈 ↓

( 2 β|RR〉 − β|RL〉 +

CNOT (A, C )

⟶

1

3 α|LR〉)AB |LL〉CD

| ⊗ (| ↑ 〉〈 ↓ | + | ↓ 〉〈 ↑ |)

3 α|LR〉 + β|LL〉)AB |LL〉CD

(β|RRLL〉 − β|RLLR〉 +

3 α|LRRL〉 + β|LLRR〉)ABCD

(β|RRLL〉ABCD − β|RL〉AB |LR〉CD +

3 α|LR〉AB |RL〉CD + β|LLRR〉ABCD )

(β|RRLL〉ABCD − β|RL〉AB |RL〉CD +

3 α|LR〉AB |LR〉CD + β|LLRR〉ABCD )

(16)

where |RL〉 = (|RL〉 + |LR〉) / 2 and |LR〉 = (|RL〉 − |LR〉) / 2 .

The schematic deterministic CNOT gate on a two-spin system is shown in Fig. 3(a). Here, the spin e1 is the control qubit while the spin e2 is the target qubit. The initial states of the spins e1 and e2, and an auxiliary photon A are |ϕ1〉e1 = a1| ↑ 〉 + b1| ↓ 〉 and

|ϕ2〉e2 = a2| ↑ 〉 + b2| ↓ 〉, (|R〉 + |L〉) / 2 , respectively, shown in Fig. 3 (a). The joint system of the spins e1 and e2 and the photon A is changed as follows: 1 2

|ϕ1〉e1|ϕ2〉e2 (|R〉 + |L〉)A

cPBS1, Cy1

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ cPBS2

1 2

Here, a2′ = (a2 + b2) / 2 and b2′ = (a2 − b2) / 2 . MA denotes the measurement of the photon A under the basis {(|R〉 ± |L〉) / 2 } and one phase ﬂip Z is performed on the spin e1 for the measurement outcome (|R〉 − |L〉) / 2 . The SWAP of two spins e1 and e2 may be realized with three CNOT gates on two-spin system as CNOT 1·CNOT 2·CNOT 1. With the theoretical preparation shown in Section 2.1, and the CNOT circuit in Fig. 3(a), from four spins e1, e2, e3, e4 in the state | ↑ ↑ ↓ ↓ 〉e1e2e3e4 , four-spin entangled decoherence-free states may be generated as

1 1 1 β| ↑ ↑ 〉| ↓ ↓ 〉 − β|↑↓〉|↑↓〉 + α|↓↑〉|↓↑〉 + β| ↓ ↓ 〉| ↑ ↑ 〉 3 3 3

2.3. Four-spin entangled decoherence-free states

[|R〉e (a1| ↑ 〉 + b1| ↓ 〉)e1 + |L〉e (a1| ↑ 〉 − b1| ↓ 〉)e1]|ϕ2〉e2

(18)

(19)

with |↑↓〉 = (| ↑ ↓ 〉 + | ↓ ↑ 〉)/ 2 and |↓↑〉 = (| ↑ ↓ 〉 − | ↓ ↑ 〉)/ 2 . 2.4. Hybrid four-qubit entangled decoherence-free states The schematic deterministic CNOT gate on a spin-photon system is shown in Fig. 3(b). Here, the spin e is the control qubit, while the photon A is the target qubit. The initial states of the spin e and the photon A are |ϕ1〉e = a1| ↑ 〉 + b1| ↓ 〉 and |ϕ2〉A = a2|R〉 + b2|L〉 respectively. The photon A passes through the H1, cPBS, Cy, cPBS, H2, sequentially. The joint system of the spin e and the photon A is changed from |ϕ1〉e |ϕ2〉A into

H1

a1| ↑ 〉(a2|R〉A + b2|L〉A ) + b1| ↓ 〉e (a2|L〉A + b2|R〉A )

W1

This is the controlled-NOT gate (CNOT) on a spin-photon system, i.e.,

⟶(a1| ↑ 〉e1|R〉A + b1| ↓ 〉e2 |L〉A )|ϕ2〉e2 ⟶(a1| ↑ 〉e1|R〉A + b1| ↓ 〉e2 |L〉A )(a2′| ↑ 〉 + b 2′ | ↓ 〉)B cPBS3, Cy2

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→[a1| ↑ 〉e1|R〉A (a2′| ↑ 〉 + b 2′ | ↓ 〉)e2 + b1| ↓ 〉e |L〉A (a2′| ↑ 〉 − b 2′ | ↓ 〉)e2] cPBS4

| ↑ 〉〈 ↑ | ⊗ (|R〉〈R| + |L〉〈L|) + | ↓ 〉〈 ↓ | ⊗ (|R〉〈L| + |L〉〈R|)

W2

⟶[a1| ↑ 〉e1|R〉A (a2| ↑ 〉 + b 2| ↓ 〉)e2 + b1| ↓ 〉e |L〉A (b 2| ↑ 〉 + a2| ↓ 〉)e2] MA

⟶a1| ↑ 〉e1(a2| ↑ 〉 + b 2| ↓ 〉)e2 + b1| ↓ 〉e (b 2| ↑ 〉 + a2| ↓ 〉)e2

(17)

(20)

(21)

The SWAP of two spins e1 and e2 may be realized with three CNOT gates on two-spin system as CNOT 1·CNOT 2·CNOT 1. With the theoretical preparation shown in Section 2.1, and the CNOT circuits in Figs. 2(a), 3(a) and 3(b), from two spins e1 and e2 and two photons A, B in the state | ↑ ↑ LL〉e1e2AB , hybrid four-qubit entangled decoherence-free states may be generated as

1 1 1 β| ↑ ↑ 〉|RR〉 − β|↑↓〉|RL〉 + α|↓↑〉|LR〉 + β| ↓ ↓ 〉|LL〉 3 3 3

(22)

with |RL〉 = (|RL〉 + |LR〉) / 2 , |LR〉 = (|RL〉 − |LR〉) / 2 . Moreover, if the input joint system is in the state | ↑ 〉e1|L〉A | ↑ 〉e2 |L〉B , different hybrid four-qubit entangled decoherence-free states may be generated as

1 1 1 β| ↑ R〉| ↑ R〉 − β|↑L〉|↑L〉 + α|↓R〉〉|↓R〉 + β| ↓ L〉| ↓ L〉 3 3 3

(23)

with |↑L〉 = (| ↑ L〉 + | ↓ R〉) / 2 and |↓R〉 = (| ↑ L〉 − | ↓ R〉) / 2 . The schematic deterministic CNOT gate on a photon-spin system is shown in Fig. 3(c). Here, the photon A is the control qubit, while the spin e is the target qubit. The initial states of the photon A and the spin e are |ϕ1〉A = a1|R〉 + b1|L〉 and |ϕ2〉e = a2| ↑ 〉 + b2| ↓ 〉 respectively. The photon A passes through the cPBS7, W1, Cy, W2, cPBS8, sequentially. The joint system of the photon A and the spin e is changed from |ϕ1〉A |ϕ2〉e into

a1|R〉A (a2| ↑ 〉 + b2| ↓ 〉)e + b1|L〉A (b2| ↑ 〉 + a2| ↓ 〉)e

(24)

This is the controlled-NOT gate (CNOT) on a photon-spin system, i.e.,

|R〉〈R| ⊗ (| ↑ 〉〈 ↑ | + | ↓ 〉〈 ↓ |) + |L〉〈L| ⊗ (| ↑ 〉〈 ↓ | + | ↓ 〉〈 ↑ |) Fig. 3. (a) The CNOT gate on the spins in one-side micropillar microcavity. (b) The CNOT gate on a spin-photon system. (c) The CNOT gate on a photon-spin system. cPBSi, Wi, Hi are the same to those deﬁned in Fig. 2.

(25)

With the theoretical preparation shown in Section 2.1, and the CNOT circuits in Figs. 2(a), 3(a) and 3(c), from two photons A, B and

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two spins e1, e2 in the state |RR ↓ ↓ 〉 ABe1e2 , hybrid four-qubit entangled decoherence-free states may be generated as

1 1 1 β|R ↑ 〉|R ↑ 〉 − β|L↑〉|L↑〉 + α|R↓〉|R↓〉 + β|L ↓ 〉|L ↓ 〉 3 3 3

4. Discussion and conclusion

3. The feasibilities and efﬁciencies of our schemes All the schemes for the generations of four-qubit entangled decoherence-free states are based on the negligible side leakage rate κs, i.e., in ideal conditions, the reﬂection coefﬁcients are |r0(ω)| ≈ 1 and |rh(ω)| ≈ 1. Our schemes have 100% ﬁdelity and success probability. Now consider the κs ≠ 0. In the resonant condition with ωc = ωx = ω , the optical selection rules in Eq. (13) should be changed into

|R〉| ↑ 〉↦r0|R〉| ↑ 〉, |R〉| ↓ 〉↦r|R〉| ↓ 〉, (27) 2

the cavity photon lifetime [58]. The heavy-light hole mixing could be reduced by engineering the shape, size and type of charged exciton [49].

(26)

with |L↑〉 = (|L ↑ 〉 + |R ↓ 〉) / 2 and |R↓〉 = (|L ↑ 〉 − |R ↓ 〉) / 2 .

|L〉| ↑ 〉↦r|L〉| ↑ 〉, |L〉| ↓ 〉↦r0|L〉| ↓ 〉.

5

Deﬁne the ﬁdelity F = ∫ |〈Φ|Φf 〉| , where |Φ〉 and |Φf 〉 are the ﬁnal states of an ideal condition and a real situation with side leakage, respectively. The efﬁciency is deﬁned P as the ratio of the numbers of the input photons and the output photons. Using similar procedures shown in Section 2, we evaluate the ﬁdelity and the efﬁciency as shown in Figs. 4 and 5 respectively. Generally, the cooperativity C ≔2g 2/(κγ ) of cavity QED and relative cavity loss ratio κs/κ are all required for the high ﬁdelity and efﬁciency. Experimentally, the side leakage and cavity loss rate have been reduced to κs/κ ≈ 0.7 with C ≈ 7 [50,51,33]. In this case, the ﬁdelities and efﬁciencies are greater than 82.5% and 80.6% respectively. The cooperativity C has been raised to 13.5 by improving the sample designs, growth, and fabrication [52]. For our generations, if C ≈ 13.5 and κs/κ ≈ 0.3, the ﬁdelities are greater than 94.6% while efﬁciencies are greater than 91.5%. Recently, a quantum gate between the spin state of a single trapped atom and the polarization state of an optical photon contained in a faint laser pulse has been experimentally achieved [53]. We believe that their hybrid gates may soon be extended to our general generations of four-qubit entangled decoherence-free states. The preparation and the qubit operation of a spin can be completed using nanosecond spin resonance microwave pulses [54,55]. In experiment, the spin coherence time can be extended to μs using spin echo techniques [49,56,57] for photonic applications. The optical coherence time of an exciton is ten times longer than

If choose the 87Rb as an experimental atom as a singly charged quantum dot in an one-side optical microcavity [36,51], the second four-qubit entangled decoherence-free states are atomic states which are more stable for storing information in fault-tolerant applications. The auxiliary input photon pulse is very convenience to transmit over large distances using existing telecommunication ﬁber technology [12–22] and quantum controlling switch [59]. Thus, the second scheme may be completed with remote atoms of multipartite if some photonic channels are permitted [27]. Moreover, our third scheme is to generate a hybrid four-qubit entangled decoherence-free states of photon and spins. Similarly, if the 87Rb is used, the photon-atom four-qubit entangled decoherence-free states may be constructed locally. These entanglements may be easily applied to complete quantum dense coding [59-62], where the photon-subsystem is used to encode information and be transmitted over photonic channel. When the input photons are allowed to transmit over photonic channel, these entanglements can be generated with bipartite. These hybrid bipartite entanglements are applicable for large-scale application such as Shor's algorithms, where different systems may be involved. Of course, if they are used as hybrid quantum channels, different physical systems may be teleported, i.e., two-photon system may be teleported into two-photon system [63,64], two-atom system may be teleported into two-atom system [65,66], two-photon system may be teleported into two-atom system while two-atom system may be teleported into two-photon system. Finally, our hybrid generation schemes in this paper are elements for general quantum state generations, the optical selection rules in Eq. (12) are useful for generations of other quantum states. In conclusion, we ﬁrstly proposed the detailed generation circuit of the four-qubit entangled decoherence-free states. And then, based on a single electron spin charged self-assembled GaAs/InAs quantum dot in a micropillar resonator, we generated four-photon polarization-entangled decoherence-free states, four-spin entangled decoherence-free states, and hybrid four-qubit polarization-entangled decoherence-free states. Compared with the fourphoton polarization-entangled decoherence-free states [25,26], our photonic scheme is not destructive way and may be used for further quantum information processing and quantum

Fig. 4. (a) The average ﬁdelity of four-photon entangled decoherence-free states. (b) The average ﬁdelity of the four-spin entangled decoherence-free states. The average ﬁdelity is computed as the expectation of random α, β.

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6

Fig. 5. (a) The efﬁciency of the preparation of four-photon entangled decoherence-free states. (b) The efﬁciency of the preparation of the four-spin entangled decoherencefree states. The efﬁciency is computed as the expectation of random α, β.

computation. Different from the probabilistic linear-optics-based scheme [27], our success probabilities of all generation schemes are 100% in principle. Moreover, with current technology, our schemes are feasible. Our schemes are applicable since the spin and photon may be arbitrarily combined to satisfy speciﬁc requirement. For example, the photon is convenient for transmission in quantum communication while the spin is stable for storage in quantum computation. We hope that our work will be useful for future quantum computation and communication networks.

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

Acknowledgements

[26] [27]

This work is supported by the National Natural Science Foundation of China (Nos. 61303039 and 14CTQ026), the Fundamental Research Funds for the Central Universities (No. 2682014CX095), the Chongqing Research Program of Application Foundation and Advanced Technology (No. cstc2014jcyjA-40028), and the State Scholarship Fund of China (No. 201408505070).

[28] [29] [30]

[31] [32] [33]

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