Polarization entanglement purification for concatenated Greenberger–Horne–Zeilinger state

Accepted Manuscript Polarization entanglement purification for concatenated Greenberger-Horne-Zeilinger state Lan Zhou, Yu-Bo Sheng

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S0003-4916(17)30201-4 http://dx.doi.org/10.1016/j.aop.2017.07.012 YAPHY 67442

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Annals of Physics

Received date : 17 April 2017 Accepted date : 24 July 2017 Please cite this article as: L. Zhou, Y. Sheng, Polarization entanglement purification for concatenated Greenberger-Horne-Zeilinger state, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.07.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Polarization entanglement purification for concatenated Greenberger-Horne-Zeilinger state 1

Lan Zhou1,2 , Yu-Bo Sheng2,3∗

College of Mathematics & Physics, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China 2 Key Lab of Broadband Wireless Communication and Sensor Network Technology, Nanjing University of Posts and Telecommunications, Ministry of Education, Nanjing, 210003, China 3 Institute of Signal Processing Transmission, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China (Dated: June 27, 2017) Entanglement purification plays a fundamental role in long-distance quantum communication. In the paper, we put forward the first polarization entanglement purification protocol (EPP) for one type of nonlocal logic-quit entanglement, i. e., concatenated Greenberger-Horne-Zeilinger (C-GHZ) state, resorting to the photon-atom interaction in low-quality (Q) cavity. In contrast to existing EPPs, this protocol can purify the bit-flip error and phase-flip error in both physic and logic level. Instead of measuring the photons directly, this protocol only requires to measure the atom states to judge whether the protocol is successful. In this way, the purified logic entangled states can be preserved for further application. Moreover, it makes this EPP repeatable so as to obtain a higher fidelity of logic entangled states. As the logic-qubit entanglement utilizes the quantum error correction (QEC) codes, which has an inherent stability against noise and decoherence, this EPP combined with the QEC codes may provide a double protection for the entanglement from the channel noise and may have potential applications in long-distance quantum communication. PACS numbers: 03.67.Mn, 03.67.-a, 42.50.Dv Keywords: Entanglement purification, Concatenated Greenberger-Horne-Zeilinger state, Quantum communication

I.

INTRODUCTION

Entanglement is an indispensable resource which is widely applied in many aspects, such as quantum teleportation [1–3], quantum key distribution (QKD) [4], quantum secret sharing (QSS) [5], and quantum secure direct communication (QSDC) [6–8]. During the past two decades, many types of entanglement were investigated, such as polarization entanglement [9], hybrid entanglement [10], hyperentanglement [11, 12], and so on. Besides the entanglement described above, logic-qubit entanglement, which utilizes the quantum error correction (QEC) codes and encodes many physic qubits in a logic qubit, is also widely discussed, especially in quantum computation [13–15]. Interestingly, recent work showed that the logic-qubit entanglements also demonstrate the benefits in long-distance quantum communication [16–18]. In 2011, Fr¨owis and D¨ ur first investigated a new type of logic-qubit entanglement [19]. It is called concatenated Greenberger-Horne-Zeilinger (C-GHZ) state. The typical C-GHZ state can be written as [19–27] 1 + ⊗N − ⊗N |Φ± iN,M = √ (|GHZM i ± |GHZM i ), 2

(1)

where N and M are the number of logic qubits and the number of physic qubits in each logic qubit, respectively. In ± Eq. (1), the logic qubits |GHZM i are physic GHZ states of the form 1 ± |GHZM i = √ (|0i⊗M ± |1i⊗M ). 2

(2)

In 2014, Lu et al. first experimentally generated the polarization logic-qubit entangled states with M = 2 and N = 3 in linear optics [26]. They also showed that the C-GHZ states are useful for large-scale fibre-based quantum networks and multipartite QKD, QSS and third-man quantum cryptography.

∗

Email address: [email protected]

2 Unfortunately, the entanglement is generally fragile, where noise and decoherence can diminish or even destroy the desirable quantum features [28]. In the applications, a degraded quantum channel may degrade the fidelity of the teleportation, even more, it will make the quantum communication insecure. Prior to the application, we have to recover the degraded entangled states into the maximally entangled states. The key techniques in longdistance quantum communication, i. e., entanglement purification and concentration can distill the high quality entangled states from the low quality entangled states [29–55]. The logic-qubit entangled state may also suffer from the decoherence. It will make the maximally logic-qubit entangled state degrade to the mixed state. Existing EPPs cannot deal with the logic-qubit entanglement. Compared with the conventional physic-qubit entangled state, logic-qubit entangled state has more complex structure. Moreover, the mixed state of logic-qubit entangled state contains more errors than the mixed state of conventional physic-qubit entangled state. The mixed state of logicqubit entangled state contains not only the bit-flip error and phase-flip error in the logic qubits, but also the bit-flip error and phase-flip error in physic qubit. In this paper, we put forward the first EPP for the polarization C-GHZ state. Each C-GHZ state contains two logic qubits and each logic qubit contains M physic qubit and such C-GHZ state is a logic Bell state. We will show that the bit-flip error and phase-flip error in logic-qubit entanglement can be well purified. A bit-flip error in one physic qubit can be completely corrected. The phase-flip error in one physic qubit equals to the bit-flip error in the logic-qubit entanglement, which can also be well purified. For the logic-qubit entanglement usually contains the QEC codes, which can suppress some errors deterministically and has an inherent stability against noise and decoherence [19, 26], this EPP combined with the QEC codes may provide a double protection of the entanglement from the channel noise. This paper is organized as follows: In Sec. 2, we briefly introduce the basic principle of the photonic Faraday rotation. In Sec. 3, we explain the purification for the bit-flip error and phase-flip error in the logic-qubit entanglement. In Sec. 4, we describe the purification for the physic-qubit error. In Sec. 5, we present a discussion. In Sec. 6, we make a conclusion. II.

THE PHOTONIC FARADAY ROTATION

The quantum electrodynamics (QED) is a promising platform for performing the quantum information tasks due to the controllable interaction between atoms and photons. For a long time, with the atoms strongly interacting with local high-quality (Q) cavities, the spatially separated cavities could serve as quantum nodes, and construct a quantum network assisted by the photons acting as quantum buses [56–60]. However, the requirements for high-Q cavities and strong coupling to the confined atoms are stringent for current techniques. In 2009, An et al. successfully implemented the quantum information processing (QIP) tasks with the moderate cavity-atom coupling in the low-Q cavities [61]. It is so attractive and applicable to combine the input-output process with low-Q cavities, for if it is achieved, the high-quality QIP tasks can be accomplished with current available techniques. Following the scheme from An et al., various works based on the QED in low-Q cavity have been presented [62–66].

FIG. 1: The basic principle of the interaction between the photon pulse and the three-level atom in the low-Q cavity [66]. (a): A three-level atom is trapped in a low-Q cavity. (b): The three-level atom has an excited state |ei and two degenerate ground states |gL i and |gR i. The states |gL i and |gR i couple with a left (L) polarized and a right (R) polarized photon, respectively.

Here, we make a three-level atom trap in the low-Q cavity. The atomic structure and the interaction between the photon pulse and the three-level atom in the low-Q cavity are shown in Fig. 1 [67–70]. The atom has two degenerate ground states |gL i and |gR i and an excited state |ei. The transition between |gL i and |ei is assisted with a left-circularly polarized photon (|Li), while that between |gR i and |ei is assisted with a right-circularly polarized photon (|Ri). Suppose a single photon pulse with the frequency ωp enters the optical cavity. Using the adiabatic

3 approximation, we can solve the Langevin equations of motion for cavity and atomic lowering operators analytically. Then, we obtain the general expression of the reflection coefficient of the atom-cavity system in the form of [61, 71, 72] r(ωp ) ≡

[i(ωc − ωp ) − κ2 ][i(ω0 − ωp ) + γ2 ] + λ2 aout (t) . = ain (t) [i(ωc − ωp ) + κ2 ][i(ω0 − ωp ) + γ2 ] + λ2

(3)

Here, ain (t) and aout (t) are the cavity input operator and cavity output operator, respectively. κ and γ are the cavity damping rate and atomic decay rate. ωp , ωc , and ω0 are the frequency of the input photon, the cavity, and the atom, respectively. λ is the atom-cavity coupling strength. For r(ωp ), as the photon experiences an extremely weak absorption in the interaction process, we consider that the output photon only experiences a pure phase shift without any absorption as a good approximation. In this way, Eq. (3) can be rewritten as r(ωp ) ≃ eiθ . An et al. also numerically proved that the reflection coefficient r(ωp ) is insensitive to both the cavity decay and the atomic spontaneous emission [61]. In the case of the atom uncoupling to the cavity, which indicates λ = 0, the Eq. (3) can be simplified as r0 (ωp ) =

i(ωc − ωp ) − i(ωc − ωp ) +

κ 2 κ 2

.

(4)

r0 (ωp ) can be written as a pure phase shift as r0 (ωp ) = eiθ0 . Therefore, for an input single-photon state as |ϕin i = √12 (|Li + |Ri), if the initial atom state is in |gL i, the output photon state will convert to 1 |ϕout i− = √ (eiθ |Li + eiθ0 |Ri), 2

(5)

while if the initial atom state is in |gR i, the output photon state will convert to 1 |ϕout i+ = √ (eiθ0 |Li + eiθ |Ri). 2

(6)

It can be found that after passing through the low-Q cavity, the polarization direction of the output photon rotates θ0 −θ θ−θ0 an angle Θ− or Θ+ F = F = 2 2 , which is so called the photonic Faraday rotation. In the low-Q cavity (λ < κ), based on Eq. (3) and Eq. (4), under the special condition that ω0 = ωc , ωp = ωc − κ2 , γ = 0, if λ = κ2 , it can be obtained r(ωp ) = −1 (θ = π), while if λ = 0, it can be obtained r0 (ωp ) = i (θ0 = π2 ). Therefore, when the photon is reflected from the low-Q cavity, we can obtain the relationship between the input and output photon combined with the atomic state as [73] |Li|gL i → −|Li|gL i, |Ri|gL i → i|Ri|gL i, |Li|gR i → i|Li|gR i, |Ri|gR i → −|Ri|gR i. III.

(7)

THE PURIFICATION OF THE LOGIC-ENTANGLEMENT A.

The purification of the logic bit-flip error

We first introduce the purification for the logic C-GHZ state under the simplest case, that is, N = M = 2 in Eq. (1). We suppose two parties, say Alice and Bob share a maximally entangled logic Bell state |Φ+ iAB of the form 1 |Φ+ iAB = √ (|φ+ iA |φ+ iB + |φ− iA |φ− iB ). 2

(8)

Here, the subscripts A and B mean Alice and Bob, respectively. If a logic bit-flip error occurs with the probability of 1 − F , it will change |Φ+ iAB to |Ψ+ iAB as 1 |Ψ+ iAB = √ (|φ+ iA |φ− iB + |φ− iA |φ+ iB ). 2

(9)

In Eq. (8) and Eq. (9), |φ± i are two of the four polarization Bell states, which can be written as 1 |φ± i = √ (|Li|Li ± |Ri|Ri), 2 1 |ψ ± i = √ (|Li|Ri ± |Ri|Li). 2

(10)

4 Due to the logic bit-flip error, the initial photon state degrades to a mixed state as ρin = F |Φ+ iAB hΦ+ | + (1 − F )|Ψ+ iAB hΨ+ |. HWP

HWP a1 

S1



S2 

a2  a3  a4 



1

b1  b2  b3  b4

(11)

3

2

4





Delay

Delay

PBS

FIG. 2: The schematic drawing of the EPP for logic bit-flip error of the logic Bell state with N = M = 2. Two copies of same mixed photon states are generated from the photon sources S1 and S2, respectively. The parties need to prepare four three-level atoms with the form of √12 (|gL i + |gR i) trapped in four low-Q cavities, respectively. HWP represents the half-wave plate, and PBS means the polarization beam splitter. The ”Delay” setup is used to ensure each cavity only contains a photon at a time.

The schematic drawing of the entanglement purification process is shown in Fig. 2. The purification contains two steps. In the first step, Alice and Bob require to share two same copies of the mixed states in Eq. (11), here named ρin1 and ρin2 . ρin1 is in the spatial modes of a1 , a2 , b1 , and b2 , while ρin2 is in the spatial modes a3 , a4 , b3 , and b4 . ρin1 ⊗ ρin2 can be described as follows. It is in the state |Φ+ iA1B1 ⊗ |Φ+ iA2B2 with the probability of F 2 . It is in the state |Φ+ iA1B1 ⊗ |Ψ+ iA2B2 or |Ψ+ iA1B1 ⊗ |Φ+ iA2B2 with the equal probability of F (1 − F ). With the probability of (1 − F )2 , it is in the state |Ψ+ iA1B1 ⊗ |Ψ+ iA2B2 . Here, the subscripts A1 and A2 represent Alice, while B1 and B2 represent Bob, respectively. Before purification, Alice and Bob make each of the photons pass through a half-wave plate (HWP), which makes |Li → √12 (|Li + |Ri) and |Ri → √12 (|Li − |Ri). After the HWPs, |φ+ i will not change, but |φ− i will change to |ψ + i. Therefore, |Φ+ iAB and |Ψ+ iAB will evolve to |Φ′+ iAB and |Ψ′+ iAB , respectively, which can be written as 1 |Φ′+ iAB = √ (|φ+ iA |φ+ iB + |ψ + iA |ψ + iB ), 2 1 |Ψ′+ iAB = √ (|φ+ iA |ψ + iB + |ψ + iA |φ+ iB ). 2

(12)

√1 (|gL i + |gR i) (i = 1, 2, 3, 4). They make The parties prepare four three-level atoms in the same states of |Ω+ i i= 2 the four atoms here named atom ”1”, ”2”, ”3”, and ”4” trap in four low-Q cavities, respectively. Then, they make the photons in the a1 a2 and a3 a4 modes pass through two low-Q cavities and successively interact with atom ”1” and ”2”, the photons in the b1 b2 and b3 b4 modes successively enter two cavities and interact with atom ”3” and ”4”, respectively. It is noticed that the parties should ensure that each cavity only contains one photon at a time. In this way, in practical experiment, the time for the photon entering the cavity should be precisely controlled. In our protocol, the ”Delay” represents the time delay setup (optical fiber in a suitable length, for example). In practical experiment, the mean photon-atom interaction time (T ) is T ∼ 10−4 − 10−5 s, while the mean lifetimes of the atom (τat ) and the cavity (τcav ) are τat ∼ 10−5 − 10−2 s and τcav ∼ 10−4 − 10−1 s, respectively [74, 75]. Therefore, the parties should adjust the lengths of each path precisely at subwavelength. The time delay has also been used in previous quantum information processing protocol [73]. In our protocol, with the help of the time-delay setup, Alice first makes the photon in the a1 mode enter the cavity and interact with the atom ”1”. After the photon is reflected and exits the cavity, she lets the photon in the a2 mode enter the cavity. Based on the photon-atom interaction rules

5 in Eq. (7), we can obtain

→

→

→

→

1 1 |φ+ i|φ+ i ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) 2 2 1 1 |φ+ i|φ+ i ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i), 2 2 1 1 |φ+ i|ψ + i ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) 2 2 1 1 + + −|φ i|ψ i ⊗ √ (|gL i − |gR i) ⊗ √ (|gL i − |gR i), 2 2 1 1 |ψ + i|φ+ i ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) 2 2 1 1 + + −|ψ i|φ i ⊗ √ (|gL i − |gR i) ⊗ √ (|gL i − |gR i), 2 2 1 1 |ψ + i|ψ + i ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) 2 2 1 1 |ψ + i|ψ + i ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i). 2 2 (13)

Here, we define |Ω− i = √12 (|gL i − |gR i). In this way, after all the photons are reflected from the two cavities, if the initial state is |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 with the probability of F 2 , the photon-atom state will evolve to = + → + + +

+ + + |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 1 √ (|φ+ ia1 a2 |φ+ ib1 b2 + |ψ + ia1 a2 |ψ + ib1 b2 ) ⊗ √ (|φ+ ia3 a4 |φ+ ib3 b4 2 2 + + + + + + |ψ ia3 a4 |ψ ib3 b4 ) ⊗ |Ω1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 + + + + [|φ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 − − − |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i − − − |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i + + + |ψ + ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 ⊗ |Ω1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω+ 4 i].

(14)

If the initial state is |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 or |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 with the equal probability of F (1 − F ), the photon-atom state will evolve to

= + → + + +

+ + + |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 1 √ (|φ+ ia1 a2 |φ+ ib1 b2 + |ψ + ia1 a2 |ψ + ib1 b2 ) ⊗ √ (|φ+ ia3 a4 |ψ + ib3 b4 2 2 + + + + + + |ψ ia3 a4 |φ ib3 b4 ) ⊗ |Ω1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 + + − − [|φ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 + + − |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i + + − |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i + − − |ψ + ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i],

(15)

6 or

= + → + + +

+ + + |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 1 √ (|φ+ ia1 a2 |ψ + ib1 b2 + |ψ + ia1 a2 |φ+ ib1 b2 ) ⊗ √ (|φ+ ia3 a4 |φ+ ib3 b4 2 2 + + + |ψ + ia3 a4 |ψ + ib3 b4 ) ⊗ |Ω+ i ⊗ |Ω i ⊗ |Ω i ⊗ |Ω 1 2 3 4i 1 + + − − [|φ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 + + − |φ+ ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i + + − |ψ + ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i + + − + + + + |ψ ia1 a2 |ψ ia3 a4 |φ ib1 b2 |ψ ib3 b4 ⊗ |Ω1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω− 4 i].

(16)

For the initial state of |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 with the probability of (1 − F )2 , the whole photon-atom state will evolve to

= + → + + +

+ + + |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 1 √ (|φ+ ia1 a2 |ψ + ib1 b2 + |ψ + ia1 a2 |φ+ ib1 b2 ) ⊗ √ (|φ+ ia3 a4 |ψ + ib3 b4 2 2 + + + |ψ + ia3 a4 |φ+ ib3 b4 ) ⊗ |Ω+ i ⊗ |Ω i ⊗ |Ω i ⊗ |Ω 1 2 3 4i 1 + + + + [|φ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 − − − |φ+ ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i − − − |ψ + ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i + + + + + + + + |ψ ia1 a2 |ψ ia3 a4 |φ ib1 b2 |φ ib3 b4 ⊗ |Ω1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i].

(17)

It can be found that all the items from both the photon states |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 and |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 make all the four atoms have the same states, while all the items from both the photon states |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 and |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 make the states of atoms ”1” and ”2” be different with those of atoms ”3” and ”4”. Next, the parties perform the Hadamard (H) operation on the four atoms. The H operation on the atom can be performed by driving it with an external classical field (polarized lasers), which makes |gL i → √12 (|gL i + |gR i), and |gR i → √12 (|gL i − |gR i). It can be found that the H operation will make |Ω+ i → |gL i and |Ω− i → |gR i. Then, the parties measure the states of the four atoms in the basis of {|gL i, |gR i}. From Eq. (14) to Eq. (17), if the measurement result of the four atoms is |gL i1 |gL i2 |gL i3 |gL i4 or |gR i1 |gR i2 |gR i3 |gR i4 , the parties can pick up the items from |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 or |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 , and our purification protocol is successful. If the measurement result is |gL i1 |gL i2 |gR i3 |gR i4 or |gR i1 |gR i2 |gL i3 |gL i4 , the parties can only pick up the items from |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 or |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 , and the purification protocol fails. We take the measurement result of |gL i1 |gL i2 |gL i3 |gL i4 for example. Under this case, |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 will collapse to 1 √ (|φ+ ia1 a2 |φ+ ib1 b2 |φ+ ia3 a4 |φ+ ib3 b4 + |ψ + ia1 a2 |ψ + ib1 b2 |ψ + ia3 a4 |ψ + ib3 b4 ), 2

(18)

and |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 will collapse to 1 √ (|φ+ ia1 a2 |ψ + ib1 b2 |φ+ ia3 a4 |ψ + ib3 b4 + |ψ + ia1 a2 |φ+ ib1 b2 |ψ + ia3 a4 |φ+ ib3 b4 ). 2

(19)

So far, the first step is completed. In the second step, in order to obtain the output mixed state with the same form of Eq. (11), the parties first make all the photons pass through the HWPs, which can change |ψ + i to |φ− i, and keep |φ+ i constant. In this way, the state in Eq. (18) will change to 1 √ (|φ+ ia1 a2 |φ+ ib1 b2 |φ+ ia3 a4 |φ+ ib3 b4 + |φ− ia1 a2 |φ− ib1 b2 |φ− ia3 a4 |φ− ib3 b4 ) 2 1 1 = √ [|φ+ ia1 a2 |φ+ ib1 b2 ⊗ (|LLLLi + |LLRRi + |RRLLi + |RRRRi)a3 a4 b3 b4 2 2 1 + |φ− ia1 a2 |φ− ib1 b2 ⊗ (|LLLLi − |LLRRi − |RRLLi + |RRRRi)a3 a4 b3 b4 ]. 2

(20)

7 The state in Eq. (19) will change to 1 √ (|φ+ ia1 a2 |φ− ib1 b2 |φ+ ia3 a4 |φ− ib3 b4 + |φ− ia1 a2 |φ+ ib1 b2 |φ− ia3 a4 |φ+ ib3 b4 ) 2 1 1 = √ [|φ+ ia1 a2 |φ− ib1 b2 ⊗ (|LLLLi − |LLRRi + |RRLLi − |RRRRi)a3 a4 b3 b4 2 2 1 + − + |φ ia1 a2 |φ ib1 b2 ⊗ (|LLLLi + |LLRRi − |RRLLi − |RRRRi)a3 a4 b3 b4 ]. 2

(21)

Subsequently, the parties let the four photons in a3 , a4 , b3 and b4 modes pass through the polarization beam splitters (PBSs), which can transmit the |Li photon and reflect the |Ri photon, respectively. Finally, they measure the photons in the output modes. If the measurement results are |LLLLia3 a4 b3 b4 or |RRRRia3 a4 b3 b4 , the state in Eq. (20) will finally evolve to |Φ+ iA1B1 , while the state in Eq. (21) will finally evolve to |Ψ+ iA1B1 . In this way, the parties will finally obtain a new mixed state as ρout1 = F ′ |Φ+ iA1B1 hΦ+ | + (1 − F ′ )|Ψ+ iA1B1 hΨ+ |.

(22)

It can be found that ρout1 has the same form of ρin . Here, F′ =

F2 . F 2 + (1 − F )2

(23)

If the measurement results of the four photons are |LLRRia3 a4 b3 b4 or |RRLLia3 a4 b3 b4 , the state in Eq. (20) will finally evolve to |Φ− iA1B1 = √12 (|φ+ ia1 a2 |φ+ ib1 b2 − |φ− ia1 a2 |φ− ib1 b2 ), while the state in Eq. (21) will finally evolve to |Ψ− iA1B1 = √12 (|φ+ ia1 a2 |φ− ib1 b2 − |φ− ia1 a2 |φ+ ib1 b2 ). In this way, the parties will finally obtain another mixed state as ρ′out1 = F ′ |Φ− iA1B1 hΦ− | + (1 − F ′ )|Ψ− iA1B1 hΨ− |,

(24)

State ρ′out1 can be transformed to ρout1 by performing the bit-flip operations on all the physic qubits in one of the logic qubit. Similarly, if the measurement results in the first step are |gR i1 |gR i2 |gR i3 |gR i4 , the parties can also finally obtain the mixed state of ρout1 after above operations in the second step. Therefore, the success probability for our protocol is P0 = F 2 + (1 − F )2 .

(25)

So far, our EPP for bit-flip errors in the logic entanglement is completed. It can be calculated that F ′ > F under F > 12 . The higher F will lead to the higher F ′ . However, under F > 21 , the higher F would lead to the lower P0 . Interestingly, this purification protocol can be extended to the logic Bell states with each logic qubit being arbitrary GHZ state. Suppose Alice and Bob share the state as 1 + + − − |Φ+ M iAB = √ (|GHZM iA |GHZM iB + |GHZM iA |GHZM iB ), 2

(26)

where 1 ± |GHZM i = √ (|Li⊗M ± |Ri⊗M ). 2

(27)

+ If the bit-flip error in the logic entanglement occurs with the probability of (1 − F ), |Φ+ M iAB will convert to |ΨM iAB with the form of

1 + − − + |Ψ+ M iAB = √ (|GHZM iA |GHZM iB + |GHZM iA |GHZM iB ). 2

(28)

In this way, Alice and Bob share a mixed state as + + + ρinM = F |Φ+ M iAB hΦM | + (1 − F )|ΨM iAB hΨM |.

(29)

For completing the purification task, Alice and Bob also require to share two same copies of the mixed states as shown in Eq. (29). The first copy of mixed state is in the spatial modes a1 , b1 , a2 , b2 , · · · , am , bm , and the second

8 HWP  a1  a2  c1  c2 

HWP 1

a3 



2



4



6

3

c3  b1  b2  d1  d2 

5

3

b3 

7

d3 





8

Delay 

Delay

PBS

FIG. 3: The schematic drawing of the EPP for logic bit-flip error of the logic Bell state with M = 3. Two same copies of the mixed photon states are required. The parties need to prepare eight three-level atoms with the form of √12 (|gL i + |gR i) trapped in eight low-Q cavities, respectively. HWP  a1 a2 

HWP 1



2



4



6

c1 c2  a3  a4  c3  c4 

3

b1  b2  d1  d2  b3  b4  d3  d4 

5

3 7

Delay 





8

Delay

PBS

FIG. 4: The schematic drawing of the EPP for logic bit-flip error of the logic Bell state with M = 4. The parties also need to prepare eight three-level atoms with the form of √12 (|gL i + |gR i) trapped in eight low-Q cavities, respectively.

copy of mixed state is in the spatial modes c1 , d1 , c2 , d2 , · · · , cm , dm , respectively. Here, the photons in spatial modes ai and ci (i = 1, 2, 3, · · · ) belong to Alice, and the photons in spatial modes bi and di (i = 1, 2, 3, · · · ) belong to Bob, + respectively. Then, the whole photon state can be written as follows. It is in the state of |Φ+ M iAB ⊗ |ΦM iCD with + + + + 2 the probability of F . It is in the state of |ΦM iAB ⊗ |ΨM iCD or |ΨM iAB ⊗ |ΦM iCD with the equal probability of + F (1 − F ). With the probability of (1 − F )2 , it is in the state of |Ψ+ M iAB ⊗ |ΨM iCD . Here, the subscripts A and C belong to Alice and the subscripts B and D belong to Bob. For realizing the purification, the parties need to first ± perform the H operations on all the photons. We take the H operations on |GHZM iA for example. Suppose the

9 ± parties first make the photons in a1 a2 modes pass through the HWPs. After that, |GHZM iA will evolve to

1 √ [(|Lia1 + |Ria1 )(|Lia2 + |Ria2 )|Lia⊗M−2 ] + (|Lia1 − |Ria1 )(|Lia2 − |Ria2 )|Ria⊗M−2 3 ···am 3 ···am 2 2 1 − + = √ (|φ+ ia1 a2 |GHZM−2 ia3 ···am ), ia3 ···am + |ψ + ia1 a2 |GHZM−2 2 1 ] − (|Lia1 − |Ria1 )(|Lia2 − |Ria2 )|Ria⊗M−2 → √ [(|Lia1 + |Ria1 )(|Lia2 + |Ria2 )|Lia⊗M−2 3 ···am 3 ···am 2 2 1 + − = √ (|φ+ ia1 a2 |GHZM−2 ia3 ···am ). ia3 ···am + |ψ + ia1 a2 |GHZM−2 2

+ |GHZM iA →

− |GHZM iA

(30)

Then, the parties make the photons in the a3 a4 modes pass through the HWPs, which will make 1 1 − + + ia5 ···am ) ia5 ···am + |ψ + ia3 a4 |GHZM−4 |GHZM iA → √ [|φ+ ia1 a2 (|φ+ ia3 a4 |GHZM−4 2 2 1 + − + |ψ + ia1 a2 (|φ+ ia3 a4 |GHZM−4 ia5 ···am )] ia5 ···am + |ψ + ia3 a4 |GHZM−4 2 1 + ia5 ···am = [(|φ+ ia1 a2 |φ+ ia3 a4 + |ψ + ia1 a2 |ψ + ia3 a4 ) ⊗ |GHZM−4 2 − + + + + + (|φ ia1 a2 |ψ ia3 a4 + |ψ ia1 a2 |φ ia3 a4 ) ⊗ |GHZM−4 ia5 ···am ],

(31)

and 1 1 − + − |GHZM iA → √ [|φ+ ia1 a2 (|φ+ ia3 a4 |GHZM−4 ia5 ···am ) ia5 ···am + |ψ + ia3 a4 |GHZM−4 2 2 1 + − + |ψ + ia1 a2 (|φ+ ia3 a4 |GHZM−4 ia5 ···am )] ia5 ···am + |ψ + ia3 a4 |GHZM−4 2 1 − = [(|φ+ ia1 a2 |φ+ ia3 a4 + |ψ + ia1 a2 |ψ + ia3 a4 ) ⊗ |GHZM−4 ia5 ···am 2 + ia5 ···am ]. + (|φ+ ia1 a2 |ψ + ia3 a4 + |ψ + ia1 a2 |φ+ ia3 a4 ) ⊗ |GHZM−4

(32)

Similarly, after they make the remaining m − 4 photons pass through the HWPs two by two, they can obtain the ± ± factorization of |GHZM iA . When M is odd or even, the factorization of |GHZM i is slightly different. When M is ± ± odd, the last items of the iteration are |GHZ1 iam . After the HWPs, |GHZ1 iam will evolve to |GHZ1+ iam → |Liam ,

When M is even, the last items of the iteration are

|GHZ1− iam → |Riam .

|GHZ2± iam−1 am .

|GHZ2+ iam−1 am → |φ+ iam−1 am ,

(33)

After the HWPs, they can evolve to

|GHZ2− iam−1 am → |ψ + iam−1 am .

(34)

For explaining the purification process in detail, we take the cases with M = 3 and M = 4 for example. The purification processes can be straightly extended to the cases with arbitrary odd M and even M , respectively. Under M = 3, after the HWPs, |GHZ3± i will evolve to 1 |GHZ3+ i → √ (|φ+ i|Li + |ψ + i|Ri), 2

1 |GHZ3− i → √ (|φ+ i|Ri + |ψ + i|Li). 2

(35)

As shown in Fig. 3, the parties need to prepare eight three-level atoms with the form of √12 (|gL i + |gR i) trapped in eight low-Q cavities, respectively. They make the photons in the a1 a2 c1 c2 and b1 b2 d1 d2 modes pass through cavities and successively interact with atom ”1” ”2”, and ”5” ”6”, respectively, the photons in the a3 c3 and b3 d3 modes pass through cavities and successively interact with the atom ”3” ”4” and ”7” ”8”, respectively. According to the input-output relations in Eq. (7), we can obtain 1 1 1 1 |LLi ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) → |LLi ⊗ √ (|gL i − |gR i) ⊗ √ (|gL i − |gR i), 2 2 2 2 1 1 1 1 |RRi ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) → |RRi ⊗ √ (|gL i − |gR i) ⊗ √ (|gL i − |gR i), 2 2 2 2 1 1 1 1 |LRi ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) → −|LRi ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i), 2 2 2 2 1 1 1 1 |RLi ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i) → −|RLi ⊗ √ (|gL i + |gR i) ⊗ √ (|gL i + |gR i). 2 2 2 2

(36)

10 In this way, after all the photons exiting the cavity, the parties make a H operation on all the eight atoms and then measure the atoms in the basis of {|gL i, |gR i}. If the measurement result of the eight atoms is one of the following eight cases, say |gL i1 |gL i2 |gR i3 |gR i4 |gL i5 |gL i6 |gR i7 |gR i8 , |gL i1 |gL i2 |gR i3 |gR i4 |gR i5 |gR i6 |gL i7 |gL i8 , |gR i1 |gR i2 |gL i3 |gL i4 |gL i5 |gL i6 |gR i7 |gR i8 , |gR i1 |gR i2 |gL i3 |gL i4 |gR i5 |gR i6 |gL i7 |gL i8 , |gL i1 |gL i2 |gL i3 |gL i4 |gL i5 |gL i6 |gL i7 |gL i8 , |gL i1 |gL i2 |gL i3 |gL i4 |gR i5 |gR i6 |gR i7 |gR i8 , |gR i1 |gR i2 |gR i3 |gR i4 |gL i5 |gL i6 |gL i7 |gL i8 , and |gR i1 |gR i2 |gR i3 |gR i4 |gR i5 |gR i6 |gR i7 |gR i8 , our protocol will be successful. On the other hand, if the measurement result of the eight atoms is one of the following eight cases, |gL i1 |gL i2 |gR i3 |gR i4 |gL i5 |gL i6 |gL i7 |gL i8 , |gL i1 |gL i2 |gR i3 |gR i4 |gR i5 |gR i6 |gR i7 |gR i8 , |gR i1 |gR i2 |gL i3 |gL i4 |gL i5 |gL i6 |gL i7 |gL i8 , |gR i1 |gR i2 |gL i3 |gL i4 |gR i5 |gR i6 |gR i7 |gR i8 , |gL i1 |gL i2 |gL i3 |gL i4 |gL i5 |gL i6 |gR i7 |gR i8 , |gL i1 |gL i2 |gL i3 |gL i4 |gR i5 |gR i6 |gL i7 |gL i8 , |gR i1 |gR i2 |gR i3 |gR i4 |gL i5 |gL i6 |gR i7 |gR i8 , and |gR i1 |gR i2 |gR i3 |gR i4 |gR i5 |gR i6 |gL i7 |gL i8 , our protocol will fail. + We take the successful case |gL i1 |gL i2 |gR i3 |gR i4 |gL i5 |gL i6 |gR i7 |gR i8 for example. Under this case, |Φ+ 3 iAB ⊗|Φ3 iCD will collapse to 1 [(|φ+ ia1 a2 |Lia3 |φ+ ic1 c2 |Lic3 + |ψ + ia1 a2 |Ria3 |ψ + ic1 c2 |Ric3 ) 2 ⊗ (|φ+ ib1 b2 |Lib3 |φ+ id1 d2 |Lid3 + |ψ + ib1 b2 |Rib3 |ψ + id1 d2 |Rid3 ) + (|φ+ ia1 a2 |Ria3 |φ+ ic1 c2 |Ric3 + |ψ + ia1 a2 |Lia3 |ψ + ic1 c2 |Lic3 ) ⊗ (|φ+ ib1 b2 |Rib3 |φ+ id1 d2 |Rid3 + |ψ + ib1 b2 |Lib3 |ψ + id1 d2 |Lid3 )],

(37)

+ and |Ψ+ 3 iAB ⊗ |Ψ3 iCD will collapse to

1 [(|φ+ ia1a2 |Lia3 |φ+ ic1c2 |Lic3 + |ψ + ia1a2 |Ria3 |ψ + ic1c2 |Ric3 ) 2 ⊗ (|φ+ ib1b2 |Rib3 |φ+ id1d2 |Rid3 + |ψ + ib1b2 |Lib3 |ψ + id1d2 |Lid3 ) + (|φ+ ia1a2 |Ria3 |φ+ ic1c2 |Ric3 + |ψ + ia1a2 |Lia3 |ψ + ic1c2 |Lic3 ) ⊗ (|φ+ ib1b2 |Lib3 |φ+ id1d2 |Lid3 + |ψ + ib1b2 |Rib3 |ψ + id1d2 |Rid3 )].

(38)

In the second step, the parties make all the photons pass through the HWPs. Then, they make each of the photons in the c1 c2 c3 and d1 d2 d3 modes pass through a PBS, and detect the photons in the output modes. By detecting the photons in the output modes, the parties can finally obtain a mixed state as + + + ′ ρout3 = F3′ |Φ+ 3 iAB hΦ3 | + (1 − F3 )|Ψ3 iAB hΨ3 |

(39)

2

F ′ with F3′ = F 2 +(1−F )2 = F . Similarly, if the parties get other seven successful cases in the first step, they will finally obtain the same mixed states as ρout3 . So far, the purification process for the mixed state with M = 3 is completed. Under M = 4, after making all the photons pass through the HWPs, |GHZ4± i will evolve to

1 |GHZ4+ i → √ (|φ+ i|φ+ i + |ψ + i|ψ + i), 2

1 |GHZ4− i → √ (|φ+ i|ψ + i + |ψ + i|φ+ i). 2

(40)

As shown in Fig. 4, the parties also prepare eight three-level atoms with √12 (|gL i + |gR i) in eight low-Q cavities, respectively. The photons can be divided into four groups, that is, the photons in a1 a2 c1 c2 , a3 a4 c3 c4 , b1 b2 d1 d2 and b3 b4 d3 d4 spatial modes. The parties make the photons in each group pass through two cavities and interact with two atoms, successively. After all the photons exiting the cavities, they make a H operation on the eight atoms and measure them in the basis of {|gL i, |gR i}. Similar with the case of M = 3, if the parties obtain one of the following measurement results |gL i1 |gL i2 |gR i3 |gR i4 |gL i5 |gL i6 |gR i7 |gR i8 , |gL i1 |gL i2 |gR i3 |gR i4 |gR i5 |gR i6 |gL i7 |gL i8 , |gR i1 |gR i2 |gL i3 |gL i4 |gL i5 |gL i6 |gR i7 |gR i8 , |gR i1 |gR i2 |gL i3 |gL i4 |gR i5 |gR i6 |gL i7 |gL i8 , |gL i1 |gL i2 |gL i3 |gL i4 |gL i5 |gL i6 |gL i7 |gL i8 , |gL i1 |gL i2 |gL i3 |gL i4 |gR i5 |gR i6 |gR i7 |gR i8 , |gR i1 |gR i2 |gR i3 |gR i4 |gL i5 |gL i6 |gL i7 |gL i8 , and |gR i1 |gR i2 |gR i3 |gR i4 |gR i5 |gR i6 |gR i7 |gR i8 , our protocol is successful. In the second step, the parties also make all the photons pass through the HWPs, and make the photons in the c1 c2 c3 c4 and d1 d2 d3 d4 modes pass through PBSs. By detecting the photons in the output modes, they can finally obtain + + + ′ ρout4 = F4′ |Φ+ 4 iAB hΦ4 | + (1 − F4 )|Ψ4 iAB hΨ4 | 2

(41)

F ′ with F4′ = F 2 +(1−F )2 = F , and the purification task is completed. It can be found that the value of M would never influence the fidelity of the distilled new mixed state. Under F > 21 , the purification can be realized.

11 Ăϭ ĂϮ Ăϯ

^ϭ



ϭ

Ϯ

ϯ

ϰ

Ăϰ ďϭ

^Ϯ

ďϮ ďϯ ďϰ ĞůĂLJ

ĞůĂLJ

,tW

W^

FIG. 5: The schematic drawing of the EPP for logic phase-flip error of the logic Bell state with N = M = 2. The parties need to prepare four three-level atoms with the form of √12 (|gL i + |gR i) trapped in four low-Q cavities, respectively.

B.

The purification of the logic phase-flip error

Besides the logic bit-flip error, the logic phase-flip error is also unavoidable. We suppose a logic phase-flip error occurs with the probability of 1 − F , which will make |Φ+ iAB become |Φ− iAB . Under this case, the parties will obtain a mixed state as ρinp = F |Φ+ iAB hΦ+ | + (1 − F )|Φ− iAB hΦ− |.

(42)

The schematic drawing of the purification is shown in Fig. 5. The purification contains two steps. In the first step, Alice and Bob require to share two same copies of the mixed states in Eq. (42), here named ρinp1 and ρinp2 . ρinp1 is in the spatial modes of a1 , a2 , b1 , and b2 , while ρinp2 is in the spatial modes a3 , a4 , b3 , and b4 . In this way, the parties share the state |Φ+ iA1B1 ⊗ |Φ+ iA2B2 with the probability of F 2 . They share the state |Φ+ iA1B1 ⊗ |Φ− iA2B2 or |Φ− iA1B1 ⊗ |Φ+ iA2B2 with the equal probability of F (1 − F ). With the probability of (1 − F )2 , they share the state |Φ− iA1B1 ⊗ |Φ− iA2B2 . The parties prepare four three-level atoms named atom ”1”, ”2”, ”3”, and ”4” in the same states of |Ω+ i i (i = 1, 2, 3, 4) trapped in four low-Q cavities, respectively. Then, they make the photons in the a2 a3 modes pass through two low-Q cavities and successively interact with atom ”1” and ”2”, the photons in the b2 b3 modes successively enter two cavities and interact with atom ”3” and ”4”, respectively. Based on the input-output relation from Eq. (36), if the initial state is |Φ+ iA1B1 ⊗ |Φ+ iA2B2 , the photon-atom state will evolve to + + + |Φ+ iA1B1 ⊗ |Φ+ iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 − − − → [(|LLLLLLLLi + |RRRRRRRRi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 + + + − (|LLLLRRRRi + |RRRRLLLLi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i].

(43)

For the initial state is |Φ+ iA1B1 ⊗ |Φ− iA2B2 , the photon-atom state will evolve to + + + |Φ+ iA1B1 ⊗ |Φ− iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 − + + → [(|LLLLLLRRi + |RRRRRRLLi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 + − − − (|LLLLRRLLi + |RRRRLLRRi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i].

(44)

For the initial state is |Φ− iA1B1 ⊗ |Φ+ iA2B2 , the photon-atom state will evolve to + + + |Φ− iA1B1 ⊗ |Φ+ iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 − + + → [(|LLRRLLLLi + |RRLLRRRRi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 + − − − (|RRLLLLLLi + |LLRRRRRRi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i].

(45)

12 If the initial state is |Φ− iA1B1 ⊗ |Φ− iA2B2 , the photon-atom state will evolve to + + + |Φ− iA1B1 ⊗ |Φ− iA2B2 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 1 − − − → [(|LLRRLLRRi + |RRLLRRLLi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω− 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i 2 + + + − (|LLRRRRLLi + |RRLLLLRRi)a1a2 b1 b2 a3 a4 b3 b4 ⊗ |Ω+ 1 i ⊗ |Ω2 i ⊗ |Ω3 i ⊗ |Ω4 i].

(46)

After the photon-atom interaction, the parties perform the H operation on all the four atoms, which makes |Ω+ i → |gL i and |Ω− i → |gR i. Then, the parties measure the states of the four atoms in the basis of {|gL i, |gR i}. If the measurement results of the four atoms are |gL i1 |gL i2 |gL i3 |gL i4 or |gR i1 |gR i2 |gR i3 |gR i4 , our purification protocol is successful. If the measurement result is |gL i1 |gL i2 |gR i3 |gR i4 or |gR i1 |gR i2 |gL i3 |gL i4 , our purification protocol fails. Here, we take the measurement result of |gR i1 |gR i2 |gR i3 |gR i4 for example. Under this measurement result, Eq. (43) will collapse to 1 √ (|LLLLLLLLi + |RRRRRRRRi)a1 a2 b1 b2 a3 a4 b3 b4 , 2 (47) while Eq. (46) will collapse to 1 √ (|LLRRLLRRi + |RRLLRRLLi)a1a2 b1 b2 a3 a4 b3 b4 . 2 (48) In the second step, the parties make each of the photons in a3 a4 b3 b4 pass through a HWP and measure the polarization states of the output photons. In this way, if the measurement result is one of the eight following cases, |LLLLia3 a4 b3 b4 , |LLRRia3 a4 b3 b4 , |LRLRia3 a4 b3 b4 , |LRRLia3 a4 b3 b4 , |RLLRia3 a4 b3 b4 , |RLRLia3 a4 b3 b4 , |RRLLia3 a4 b3 b4 , and |RRRRia3 a4 b3 b4 , the state in Eq. (47) will finally evolve to |Φ+ iA1B1 , and the state in Eq. (48) will finally evolve to |Φ− iA1B1 . Therefore, the parties will obtain a new mixed state of ρoutp = Fp′ |Φ+ iAB hΦ+ | + (1 − Fp′ )|Φ− iAB hΦ− |.

(49)

On the other hand, if the measurement result is one of the eight following cases, |LRLLia3 a4 b3 b4 , |LLLRia3 a4 b3 b4 , |LLRLia3 a4 b3 b4 , |RLLLia3 a4 b3 b4 , |LRRRia3 a4 b3 b4 , |RLRRia3 a4 b3 b4 , |RRLRia3 a4 b3 b4 , and |RRRLia3 a4 b3 b4 , the state in Eq. (47) will finally evolve to |Φ′′+ iA1B1 = √12 (|LLLLia1 a2 b1 b2 − |RRRRia1 a2 b1 b2 ), and the state in Eq. (48) will finally evolve to |Φ′′− iA1B1 = √12 (|LLRRia1 a2 b1 b2 − |RRLLia1 a2 b1 b2 ). Therefore, the parties will obtain another new mixed state of ρ′′outp = Fp′ |Φ′′+ iAB hΦ′′+ | + (1 − Fp′ )|Φ′′− iAB hΦ′′− |,

(50)

which can be transformed to ρoutp by performing the bit-flip operations on all the physic qubits in one of the logic qubits. Similarly, if parties obtain the measurement result of |gL i1 |gL i2 |gL i3 |gL i4 in the first step, they can also finally obtain the same states after the second step. F2 So far, our purification for the logic phase-flip error is completed. The fidelity of ρoutp is Fp′ = F 2 +(1−F )2 , and the 1 success probability of our protocol is the same as P0 in Eq. (25). Therefore, under the case of F > 2 , we can obtain Fp′ > F . We can also extend the protocol to the logic Bell states with each logic qubit being the arbitrary GHZ state. − Under this case, if the logic phase-flip error occurs with the possibility of (1 − F ), |Φ+ M iAB will convert to |ΦM iAB = + + − − 1 √ (|GHZ iA |GHZ iB − |GHZ iA |GHZ iB ). Therefore, the parties will obtain a mixed state as M M M M 2 + − − ρinpM = F |Φ+ M ihΦM | + (1 − F )|ΦM ihΦM |.

(51)

The purification process for ρinpM is quite similar with that for ρinp . Alice and Bob share two same copies of the mixed state in Eq. (51). The first copy of mixed state is in the a1 , b1 , a2 , b2 , · · · , am , bm , and the second copy of mixed state is in the spatial modes c1 , d1 , c2 , d2 , · · · , cm , dm , respectively. The photons in the spatial modes ai (i = 1, 2 · · · m) and ci (i = 1, 2 · · · m) belong to Alice, while the photons in the spatial modes bi (i = 1, 2 · · · m) and di (i = 1, 2 · · · m) belong to Bob. The parties prepare four three-level atoms with the form of √12 (|gL i + |gR i) trapped in four low-Q cavities, respectively, and make the photons in the am c1 modes pass through two cavities and interact with the two atoms ”1” and ”2”, successively, and the photons in bm d1 pass through two cavities and interact with

13 the two atoms ”3” and ”4”, successively. Then, they perform the H operation on the four atoms and measure the atom states. If the atomic state measurement result is |gL i1 |gL i2 |gL i3 |gL i4 or |gR i1 |gR i2 |gR i3 |gR i4 , our purification protocol is successful. In the second step, by making each of the photons in the c1 , d1 , c2 , d2 , · · · , cm , dm pass through a HWP and measuring the polarization state of the output photons, the parties can finally distill a new mixed state with the same form of ρinpM in Eq. (51). The fidelity of the new mixed state and the success probability of our protocol are the same as those under M = 2. IV.

THE PURIFICATION OF THE PHYSIC-QUBIT ERROR

In above section, we have successfully purified the bit-flip error and phase-flip error in the logic qubit. On the other hand, in the practical applications, the single physic qubit can also suffer from the bit-flip error or phase-flip error. 

Ăϭ ĂϮ Ăϯ

ϭ Ϯ ϯ

Ăϰ ĞůĂLJ

FIG. 6: The physic bit-error can be completely selected with the help of some three-level atoms with the form of trapped in the low-Q cavities.

√1 (|gL i+|gR i) 2

For |Φ+ M iAB in Eq. (26), we suppose a bit-flip error occurs in one of the physic qubits of the logic-qubit A. It makes ± ± one of the physic qubits in |GHZM i become |Li ↔ |Ri. In this way, |GHZM i will change to √12 (|LL · · · LRL · · · Li ± |RR · · · RLR · · · Ri). As the error occurs locally, the parties can completely select the physic qubit with the bit-flip error by the local operations as shown in Fig. 6. ± We suppose the bit-flip error occurs in one of the physic qubits of |GHZM iA . Alice should prepare some three-level 1 atoms with the form of √2 (|gL i + |gR i) trapped in the low-Q cavities, respectively. She first makes the photons in the spatial modes a1 and a2 pass through the cavity and interact with atom ”1”, successively. After the photon in a2 exiting the cavity, she makes the H operation on atom ”1” and then measures the state of it. If the measurement result is |gL i, the polarization states of the photons in a1 and a2 modes must be different. In this way, a bit-flip error occurs on the photon in a1 or a2 mode. In order to ensure which photon has the bit-flip error, Alice makes the photons in a2 and a3 modes enter the cavity and interact with the atom ”2”, successively. After the photon-atom interaction, she makes the H operation on atom ”2” and then measures the state of it. If it is also in |gL i, it means the polarization state of the photons in a2 and a3 modes are different. Under this case, the bit-flip error must occur on the photon in a2 mode. If the measurement result of atom ”2” is |gR i, it means the polarization states of the photons in a2 and a3 modes are the same. She can confirm the bit-flip error occurs on the photon in a1 mode. On the other hand, if the measurement result of atom ”1” is |gR i, it means the polarization states of the photons in a1 and a2 modes are the same, that is, no bit-flip error occurs on the photons of a1 and a2 modes. Next, she makes the photons in a2 and a3 modes interact with atom ”2”, successively. After the interaction, if the measurement result of atom ”2” is |gL i, the bit-flip error occurs on the photon in a3 mode. If the measurement of atom ”2” is |gR i, it means no bit-flip error occurs on the photons in a1 a2 a3 modes. Next, Alice continues to make the photons in a3 and a4 modes interact with atom ”3”, successively, and so forth. Once the measurement result of atom ”n” is |gL i (n ≥ 2), the bit-flip error occurs on the photon in an+1 mode. In the whole process, Alice requires two atoms at least corresponding to the error in a1 or a2 modes and M − 1 atoms at most corresponding to the error in aM (M ≥ 3) mode. After selecting the error photon, she can correct the error with a bit-flip operation. Similarly, if a bit-flip error occurs on the second logic qubit B, Bob can also completely correct it with the same principle. As the parties only require to measure the atom state, the corrected photon state can be remained perfectly for other applications. + − On the other hand, we consider a phase-flip error occurs in the logic-qubit A, which makes |GHZM i ↔ |GHZM i. + − + + − 1 + Under this case, the |ΦM iAB will change to √2 (|GHZM iA |GHZM iB + |GHZM iA |GHZM iB ) = |Ψ iAB . Obviously,the phase-flip error in logic-qubit A equals to the bit-flip error in the logic entanglement. Therefore, the parties can complete the purification based on the EPP described above.

14 V.

DISCUSSION

In common physic-qubit entanglement, there are only two kinds of errors, say bit-flip error and phase-flip error. The traditional EPPs for the physic-qubit entanglement can directly purify the bit-flip error. For the phase-flip error, they need to convert it to the bit-flip error first, and purify it in the next step. If both errors occur, the EPPs can also work [29]. For the logic-qubit entanglement, the error modes are more complicated. There are four different kinds of errors, say the bit-flip error and phase-flip error in the logic-qubit entanglement, and the bit-flip error and phase-flip error in the physic-qubit entanglement, respectively. As the logic-qubit is encoded by the several physic qubits, the logic error comes from the physic error. For example, as shown in Eq. (8) and Eq. (9), the logic bit-flip error essentially is the phase-flip error in the logic qubit B. On the other hand, as shown in Eqs. (42)-(46), the logic phase-flip error essentially is the physic bit-flip error occurring on both physic qubits in one logic qubit. Entanglement purification model does not consider the specific noise model, for it is a general model. The advantage of entanglement purification is that it considers all the possible errors in entanglement, and discuss how to purify or correct the errors. Certainly, there are other specific models to deal with the errors, such as quantum entanglement distribution over an arbitrary collective-noise channel [76], which focuses on the pure state, not the mixed state. In a practical complex noisy environment, we usually cannot predict in advance which error will occur, and all possible errors may occur in principle. In this way, we can apply our different purification protocols sequentially to obtain a high quality entanglement. For some special noisy environment, where one of errors may occur with higher probability, we can repeat our corresponding protocol to further purify this error. This recyclable EPP can work based on the fact that the purified entanglement can be remained for next purification step or further application. On the other hand, if multiple errors occur on several qubits, our entanglement purification can also work. The reason is that any error in logic-qubit entanglement must be one of the four errors, and one can purify these errors in prober order. In the protocol, both the logic bit-flip error and logic phase-flip error can be purified directly. For the logic bit-flip error, the parties require two same copies of the initial mixed photon states. For completing the purification task, they need to prepare 2M (M is even) or 2(M + 1) (M is odd) three-level atoms in the form of √12 (|gL i + |gR i) trapped in the low-Q cavities, respectively. For the logic phase-flip error, the parties only require four three-level atoms in the form of √12 (|gL i + |gR i) trapped in the low-Q cavities. They make the photons enter the cavities and interact with the atoms successively. After the photon-atom interaction, the parties measure the states of all the atoms and the second copy of photon. Based on the atom and photon measurement results, the parties can finally distill the new mixed states with the same form of the initial mixed state. The fidelity of the new mixed state (F ′ ) is higher than the fidelity (F ) of the initial mixed state, under the case that F > 21 . We also prove that the physic bit-flip error in one of the physic qubits of a logic qubit can be selected and completely corrected with the help of the photon-atom interaction in low-Q cavities and the bit-flip operation. The phase-flip error in one physic qubit of a logic qubit can be transformed to the logic bit-flip error, which can be also well purified by our protocol. In this way, our protocol can completely deal with all the four kinds of errors of arbitrary logic Bell state. Moreover, as the parties only measure the atom states to judge whether the protocols are successful, all the distilled photon states can be well remained for other applications. The key element of our protocol is the low-Q cavity. Recently, some attractive experiment results about the low-Q cavity have been reported. For example, in 2005, Nuβmann et al. showed that they can precisely control and adjust the individual ultracold 85 Rb atoms coupled to a high-finesse optical cavity [77]. The states of |F i = 2. mF = ±1 of the 5S1/2 are chosen to be the two ground states |gL i and |gR i, respectively. The transition frequency between the 15 ground states and the excited state at λ = 780nm is ω0 = 2πc λ ≈ 2.42 × 10 Hz. The cavity length, cavity rate and the finesse are L = 38.6µm, K = 2π × 53M Hz and F = 37000, respectively. In 2007, the group of Fortier experimentally realized the deterministic loading of single 87 Rb atoms into the cavity by incorporating a deterministic loaded atom conveyor [78]. In the same year, Colombe et al. also reported their experiment on realizing the strong atom-field coupling for Bose-Einstein condensates (BEC) in a fiber-based F-P cavity on a chip [79]. They showed that the 87 Rb BEC can be positioned deterministically anywhere within the cavity and localized entirely within a single antinode of the standing-wave cavity field. Current research showed that the single-electron spin in a single quantum dot inside a micro-cavity and the nitrogen-vacancy (N-V) defect center in diamond can induce the giant optical Faraday rotation [80–82]. In this way, our EPP can also be suitable for entangled electrons using a quantum dot and microcavity coupled system. As shown in Sec. 2, in order to obtain the input-output relations as Eq. (7), we must precisely control ω0 = ωc , ωp = ωc − κ2 , γ = 0, λ = κ2 (λ = 0) to obtain θ = π (θ0 = π2 ). From Eq.(4), it is easy to control θ0 = π2 , for the photon couples with the empty cavity and the frequencies ω0 , ωc and ωp usually can be well controlled. However, from Eq. (3), θ is decided by six coefficients as κ, γ and λ, ω0 , ωc and ωp . In this way, it is hard to precisely control θ = π, even under the condition that we can well control the three kinds of frequencies. The small fluctuation of κ, γ or λ will contribute to the change of θ. In our error model, we suppose θ = π + σ, and the addition of σ to θ is actually the contributions from several parameters.

15 Under this case, the input-output relations in Eq. (7) should be rewritten as |Li|gL i → −eiσ |Li|gL i, |Ri|gL i → i|Ri|gL i, |Li|gR i → i|Li|gR i, |Ri|gR i → −eiσ |Ri|gR i.

(52)

In this imperfect condition, the input-output relations in Eq. (13) will also be changed. For example, for the photon state of |φ+ i|φ+ i, the item |LLLLi will make the atom state be √12 (e4iσ |gL i+|gR i), |RRRRi will make the atom state be √12 (|gL i + e4iσ |gR i), while other two items |LLRRi and |RRLLi will not change the atom state. All the four items of |ψ + i|ψ + i will not change the atom state. For |φ+ i|ψ + i and |ψ + i|φ+ i, the items |LLLRi, |LLRLi, |LRLLi, |RLLLi will change the atom state to √12 (−e3iσ |gL i + eiσ |gR i), while the items |RRLRi, |RRRLi, |LRRRi, and |RLRRi will change it to √12 (eiσ |gL i − e3iσ |gR i). These imperfect input-output relations will influence the measurement results of the atom states. Therefore, in the first step of our protocol, the item selections will be influenced. Then, we will analysis our EPP for the logic bit-flip error with M = N = 2 under the imperfect condition in detail. As shown in Sec. 3.1, in the first step, after making all the photons pass through the HWPs, the photon state may be in the state |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 with the probability of F 2 . It is in the state |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 or |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 with the equal probability of F (1 − F ). With the probability of (1 − F )2 , it is in the state |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 . We will prove that the imperfect input-output condition will cause some errors in our protocol. The errors can be divided into two groups. The first kind of errors corresponds to the initial photon states of |Φ′+ iA1B1 ⊗|Ψ′+ iA2B2 and |Ψ′+ iA1B1 ⊗|Φ′+ iA2B2 . Under the imperfect input-output condition in Eq. (52), some items would cause the measurement results of all the four atoms to be the same in some probability. This will make the parties distill some unwanted items from |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 and |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 . In detail, the photon state of |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 can be written as 1 + [|φ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 + |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 2 + |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 + |ψ + ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 ].

|Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 =

(53)

For |φ+ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 1 = (|LLLLia1 a2 a3 a4 + |LLRRia1 a2 a3 a4 + |RRLLia1 a2 a3 a4 + |RRRRia1 a2 a3 a4 ) 2 1 ⊗ (|LLLRib1 b2 b3 b4 + |LLRLib1 b2 b3 b4 + |RRLRib1 b2 b3 b4 + |RRRLib1 b2 b3 b4 ), 2

(54)

all the four items |LLLLLLLRi, |LLLLLLRLi, |RRRRRRLRi, and |RRRRRRRLi in a1 a2 a3 a4 b1 b2 b3 b4 modes may make the measurement results of atoms ”1” and ”2” be the same as the atoms ”3” and ”4” with the probabil2iσ 2 ity of | √12 (e4iσ |gL i + |gR i) √12 (−e3iσ |gL i + eiσ |gR i)|2 = |1−e4 | . All the four items |LLLLRRLRi, |LLLLRRRLi, |RRRRLLLRi, and |RRRRLLRLi in a1 a2 a3 a4 b1 b2 b3 b4 modes may make the measurement results of atoms ”1” and ”2” be the same as the atoms ”3” and ”4” with the probability of | √12 (e4iσ |gL i + |gR i) √12 (eiσ |gL i − e3iσ |gR i)|2 =

|1−e6iσ |2 . The left eight items may cause the same measurement results 4 2iσ 2 ity of | √12 (|gL i + |gR i) √12 (−e3iσ |gL i + eiσ |gR i)|2 = |1−e4 | . Therefore, |φ+ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 can be written as

Perror1 =

of all the four atoms with the probabilthe error probability of our protocol for

1 |1 − e2iσ |2 1 |1 − e6iσ |2 1 |1 − e2iσ |2 3|1 − e2iσ |2 |1 − e6iσ |2 × + × + × = + . 4 4 4 4 2 4 16 16

(55)

It is obvious that the error probability of our protocol for |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 is the same as Perror1 . Next, for |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 1 = (|LRLLia1 a2 a3 a4 + |LRRRia1 a2 a3 a4 + |RLLLia1 a2 a3 a4 + |RLRRia1 a2 a3 a4 ) 2 1 ⊗ (|LRLRib1 b2 b3 b4 + |LRRLib1 b2 b3 b4 + |RLLRib1 b2 b3 b4 + |RLRLib1 b2 b3 b4 ), 2

(56)

all the sixteen items may make the measurement results of atoms ”1” and ”2” be the same as the atoms ”3” and 2iσ 2 ”4” with the probability of |1−e4 | . In this way, the error probability for |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 can be

16 written as Perror2 =

|1 − e2iσ |2 . 4

(57)

Similarly, for |ψ + ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 , the error probability of our protocol is also the same as Perror2 . Therefore, if the initial photon state is |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 , the error probability of our protocol is Perror =

1 1 7|1 − e2iσ |2 |1 − e6iσ |2 Perror1 + Perror2 = + . 2 2 32 32

(58)

Similarly, for the initial state of |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 , we can also calculate the error probability, which equals to Perror of |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 . In the second step, the parties make all the photons pass through the HWPs and make a BSM for the photons in a3 a4 b3 b4 modes. Based on the above calculations, after the BSM measurements, the above distilled items would finally convert to the following states. In detail, under the initial photon state of |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 , if the measurement result in the second step is |LLLLia3 a4 b3 b4 or |RRRRia3 a4 b3 b4 , |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 will finally collapse to √ (59) |Θ1 i = |ϕ1 ia1 a2 b1 b2 + |ϕ2 ia1 a2 b1 b2 + 2a|φ− ia1a2 |φ− ib1b2 . If the measurement result is |LLRRia3 a4 b3 b4 , |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 will finally collapse to |Θ2 i = |ϕ1 ia1 a2 b1 b2 + a|φ− ia1a2 |φ− ib1b2 ,

(60)

while if the measurement result is |RRLLia3 a4 b3 b4 , |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 will finally collapse to |Θ3 i = |ϕ2 ia1 a2 b1 b2 + a|φ− ia1a2 |φ− ib1b2 .

(61)

Here, |ϕ1 ia1 a2 b1 b2 and |ϕ2 ia1 a2 b1 b2 can be written as |ϕ1 ia1 a2 b1 b2 = a(|LLLLi + |RRRRi + |RRLLi)a1 a2 b1 b2 + b|LLRRia1a2 b1 b2 , |ϕ2 ia1 a2 b1 b2 = a(|LLLLi + |RRRRi + |LLRRi)a1 a2 b1 b2 + b|RRLLia1a2 b1 b2 , 2iσ

(62)

6iσ

where a = |1−e2 | , and b = |1−e2 | , respectively. Under the initial state of |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 , if the measurement result in the second step is |LLLLia3a4 b3 b4 , or |RRRRia3 a4 b3 b4 , |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 will finally collapse to √ √ |Θ4 i = ( 2a|φ+ ia1a2 |φ− ib1b2 + 2a|φ− ia1a2 |φ+ ib1b2 ) = 2a|Ψ+ iA1B1 . (63) If the measurement result is |LLRRia3 a4 b3 b4 , |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 will finally collapse to √ |Θ5 i = |ϕ3 ia1 a2 b1 b2 + |ϕ5 ia1 a2 b1 b2 + 2a|Ψ+ iA1B1 , while if the measurement result is |RRLLia3 a4 b3 b4 , |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 will finally collapse to √ |Θ6 i = |ϕ4 ia1 a2 b1 b2 + |ϕ6 ia1 a2 b1 b2 ) + 2a|Ψ+ iA1B1 .

(64)

(65)

Here √ 2(b|LLi + a|RRi)a1 a2 |φ− ib1b2 , √ |ϕ4 ia1 a2 b1 b2 = 2(a|LLi + b|RRi)a1 a2 |φ− ib1b2 . √ |ϕ5 ia1 a2 b1 b2 = 2|φ− ia1a2 (a|LLi + b|RRi)b1 b2 , √ |ϕ6 ia1 a2 b1 b2 = 2|φ− ia1a2 (b|LLi + a|RRi)b1 b2 .

|ϕ3 ia1 a2 b1 b2 =

(66)

It can be found that the above distilled photon states caused by the first kind of errors would increase the success probability of our protocol, but they can not contribute to the wanted |Φ+ iA1B1 except σ = π2 . It is noteworthy that when σ = π2 , we can obtain a = b. Under this case, after two purification steps, the distilled photon states from |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 and |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 can finally convert to |Φ+ iA1B1 and |Ψ+ iA1B1 , respectively.

17 The second kind of errors corresponds to the initial photon state of |Φ′+ iA1B1 ⊗|Φ′+ iA2B2 or |Ψ′+ iA1B1 ⊗|Ψ′+ iA2B2 . Under the imperfect input-output condition, some items will cause the measurement results of all the four atoms to be the same with some probability, not for 100%. In this way, this kind of error would reduce the probability for the parties to select the items. For example, the photon state of |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 can be written as 1 + [|φ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 + |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 2 + |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 + |ψ + ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 ].

|Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 =

(67)

For |φ+ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 1 = (|LLLLia1 a2 a3 a4 + |LLRRia1 a2 a3 a4 + |RRLLia1 a2 a3 a4 + |RRRRia1 a2 a3 a4 ) 2 1 ⊗ (|LLLLib1 b2 b3 b4 + |LLRRib1 b2 b3 b4 + |RRLLib1 b2 b3 b4 + |RRRRib1 b2 b3 b4 ), 2

(68)

both the two items |LLLLRRRRi and |RRRRLLLLi in a1 a2 a3 a4 b1 b2 b3 b4 modes make the same measure8iσ 2 ment results of all the four atoms with the probability of |1+e4 | . Each of the eight items |LLLLLLRRi, |LLLLRRLLi, |RRRRLLRRi, |RRRRRRLLi, |LLRRLLLLi, |LLRRRRRRi, |RRLLLLLLi, and |RRLLRRRRi in the a1 a2 a3 a4 b1 b2 b3 b4 modes causes the same measurement results of all the four atoms with the probability of |1+e4iσ |2 . The left six items would make the same measurement results of all the atoms with the probability of 100%. 4 In this way, for |φ+ ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 , the error probability can be written as ′ Perror1 =

1 |1 + e8iσ |2 1 |1 + e4iσ |2 5 |1 + e8iσ |2 |1 + e4iσ |2 (1 − ) + (1 − )= − − . 8 4 2 4 8 32 8

(69)

For |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 1 = (|LLLRia1 a2 a3 a4 + |LLRLia1 a2 a3 a4 + |RRLRia1 a2 a3 a4 + |RRRLia1 a2 a3 a4 ) 2 1 ⊗ (|LLLRib1 b2 b3 b4 + |LLRLib1 b2 b3 b4 + |RRLRib1 b2 b3 b4 + |RRRLib1 b2 b3 b4 ), 2

(70)

all the eight items |LLLRRRLRi, |LLLRRRRLi, |LLRLRRLRi, |LLRLRRRLi, |RRLRLLLRi, |RRLRLLRLi, |RRRLLLLRi, and |RRRLLLRLi in a1 a2 a3 a4 b1 b2 b3 b4 modes may make the same measurement results of the four 4iσ 2 atoms with the equal probability of |1+e4 | . The left eight items would make the same measurement results of all the atoms with the probability of 100%. In this way, for |φ+ ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 , we can obtain the error probability as ′ Perror2 =

1 |1 + e4iσ |2 1 |1 + e4iσ |2 (1 − )= − . 2 4 2 8

(71)

′ . Similarly, for |ψ + ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 , the error probability is also Perror2 For

|ψ + ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 1 = (|LRLRia1 a2 a3 a4 + |LRRLia1 a2 a3 a4 + |RLLRia1 a2 a3 a4 + |RLRLia1 a2 a3 a4 ) 2 1 ⊗ (|LRLRib1 b2 b3 b4 + |LRRLib1 b2 b3 b4 + |RLLRib1 b2 b3 b4 + |RLRLib1 b2 b3 b4 ), 2

(72)

All the sixteen items will make the states of all the four atoms be the same. Therefore, the error probability is zero. According to the calculations above, for |Φ′+ iA1B1 ⊗ |Φ′+ iA2B2 , the total error probability is ′ Perror =

1 ′ 13 |1 + e8iσ |2 3|1 + e4iσ |2 1 ′ Perror1 + Perror2 = − − . 4 2 32 128 32

(73)

18 We can also calculate the error probability of our protocol for the initial photon state of |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 with the form of 1 + [|φ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 + |φ+ ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 2 + |ψ + ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 + |ψ + ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 ].

|Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 =

(74)

It is obvious that for |φ+ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 and |ψ + ia1 a2 |ψ + ia3 a4 |φ+ ib1 b2 |φ+ ib3 b4 , our protocol has the same error probabilities. In this way, we take |φ+ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 for example, which can be written as |φ+ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 1 = (|LLLLia1 a2 a3 a4 + |LLRRia1 a2 a3 a4 + |RRLLia1 a2 a3 a4 + |RRRRia1 a2 a3 a4 ) 2 1 ⊗ (|LRLRib1 b2 b3 b4 + |LRRLib1 b2 b3 b4 + |RLLRib1 b2 b3 b4 + |RLRLib1 b2 b3 b4 ). 2

(75)

The eight items |LLLLLRLRi, |LLLLRLRLi, |LLLLLRRLi, |LLLLRLLRi, |RRRRLRLRi, |RRRRRLRLi, |RRRRLRRLi, and |RRRRRLLRi in a1 a2 a3 a4 b1 b2 b3 b4 modes cause the same measurement results of the four 4iσ 2 atoms with the probability of |1+e4 | , while other eight items would lead to the same measurement results of all the four atoms definitely. Therefore, for |φ+ ia1 a2 |φ+ ia3 a4 |ψ + ib1 b2 |ψ + ib3 b4 , the error probability can be written as ′′ Perror1 =

1 |1 + e4iσ |2 1 |1 + e4iσ |2 (1 − )= − . 2 4 2 8

(76)

Similar, for |φ+ ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 and |ψ + ia1 a2 |φ+ ia3 a4 |φ+ ib1 b2 |ψ + ib3 b4 , our protocol has the same error probability. For example, for |φ+ ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 1 = (|LLLRia1 a2 a3 a4 + |LLRLia1 a2 a3 a4 + |RRLRia1 a2 a3 a4 + |RRRLia1 a2 a3 a4 ) 2 1 ⊗ (|LLLRib1 b2 b3 b4 + |LLRLib1 b2 b3 b4 + |RRLRib1 b2 b3 b4 + |RRRLib1 b2 b3 b4 ), 2

(77)

the eight items |LLLRRRLRi, |LLLRRRRLi, |LLRLRRLRi, |LLRLRRRLi, |RRLRLLLRi, |RRLRLLRLi, |RRRLLLLRi, and |RRRLLLRLi in a1 a2 a3 a4 b1 b2 b3 b4 modes may make the same measurement results of all the 4iσ 2 four atoms with the probability of |1+e4 | , the other eight items would lead to the same measurement results of four atoms definitely. In this way, for |φ+ ia1 a2 |ψ + ia3 a4 |ψ + ib1 b2 |φ+ ib3 b4 , we can calculate the error probability as ′′ Perror2 =

1 |1 + e4iσ |2 1 |1 + e4iσ |2 (1 − )= − . 2 4 2 8

(78)

Therefore, if the initial state is |Ψ′+ iA1B1 ⊗ |Ψ′+ iA2B2 , the total error probability of our protocol is ′′ Perror =

1 ′′ 1 ′′ 1 |1 + e4iσ |2 Perror1 + Perror2 = − . 2 2 2 8

(79)

So far, we have calculated both two kinds of possible errors. Based on above calculations, under the imperfect input-output condition, the total success probability of our protocol can be written as ′ ′′ P = F 2 (1 − Perror ) + (1 − F )2 (1 − Perror ) + 2F (1 − F )Perror .

(80)

When our protocol is successful, the parties can finally obtain the mixed state as ′ ′′ ρ′′out1 = F 2 (1 − Perror )|Φ+ iA1B1 hΦ+ | + (1 − F )2 (1 − Perror )|Ψ+ iA1B1 hΨ+ | + F (1 − F )|error1 iA1B1 herror1 | + F (1 − F )|error2 iA1B1 herror2 |,

(81)

where |error1 i and |error2 i represent the distilled states from |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 and |Ψ′+ iA1B1 ⊗ |Φ′+ iA2B2 , respectively. From Eqs. (59)-(66), the |error1 i and |error2 i can be written as 1 |error1 iA1B1 = √ (|Θ1 i + |Θ2 i + |Θ3 i), 4 2

1 |error2 iA1B1 = √ (|Θ4 i + |Θ5 i + |Θ6 i). 4 2

(82)

19 It can be proved that the probability for obtaining |error1 iA1B1 or |error2 iA2B2 equals to Perror . Especially, when σ = π2 , the |error1 i from |Φ′+ iA1B1 ⊗ |Ψ′+ iA2B2 will finally evolve to |Φ+ iA1B1 . In this way, under the case that σ = π2 , the fidelity (F ′′ ) of the distilled mixed state can be calculated as ′ F ′′ = [F 2 (1 − Perror ) + F (1 − F )Perror ]/P = F.

When σ 6=

π 2,

(83)

the F ′′ of the distilled mixed state can be written as ′ F ′′ = F 2 (1 − Perror )/P.

(84)





FIG. 7: The total success probability (P ) for our protocol as a function of σ. Here, we make F = 0.6 and F = 0.8, respectively, and σ ∈ [0, π]. σ = 0 means under the perfect input-output condition, the value of P equals to P0 = F 2 + (1 − F )2 . At the point σ = π/2, P = 1.

 ′′

FIG. 8: The fidelity of the new mixed state (F ) a function of σ. Here, we make F = 0.6 and F = 0.8, respectively, and F2 σ ∈ [0, π]. Under the perfect input-output condition (σ = 0), the value of F ′′ equals to F ′ = F 2 +(1−F . Under the imperfect )2 ′′ ′ ′′ input-output condition (σ > 0), F < F . When σ = π/2, F = F .

Here, we calculate the values of P and F ′′ as a function of σ, which are shown in Fig. 7 and Fig. 8, respectively. Here, we suppose the initial fidelity is 0.6 and 0.8, respectively. Here, for ensuring the completeness of Fig. 7 and Fig. 8, we make σ take value up to π. It can be found that both P and F ′′ are on axis symmetry of σ = π2 . There are two special points in both two figures. When σ = 0, which means the perfect input-output condition in Eq. (7)

20

FIG. 9: The fidelity (FN′′ ) of the new mixed state after the protocol has been performed for N times. Here, we make F = 0.54 and σ = 0, 0.05, 0.1, and 0.15, respectively. We change the recycle number N from 0 to 5. N = 0 presents that the purification does not start. σ = 0 means the perfect input-output condition, while σ = 0.05, 0.1, and 0.15 mean the imperfect input-output conditions.

2

F ′ can be realized, the values of P and F ′′ can reach P = F 2 + (1 − F )2 = P0 and F ′′ = F 2 +(1−F )2 = F . On the other π π ′′ hand, when σ = 2 , we can obtain P = 1 and F = F . Actually, under the case that σ = 2 , the Eq. (7) will evolve to

|Li|gL i → −i|Li|gL i, |Ri|gL i → i|Ri|gL i, |Li|gR i → i|Li|gR i, |Ri|gR i → −i|Ri|gR i.

(85)

Under above input-output relation, after the photon-atom interaction, all the possible photon states |φ+ i|φ+ i, |φ+ i|ψ + i, |ψ + i|φ+ i, and |ψ + i|ψ + i would cause the atom states to be √12 (|gL i + |gR i). In this way, under σ = π2 , the distilled new mixed state is absolutely the same as the input mixed state, and our protocol does not work. Therefore, under this case, the total success probability is 100%, and F ′′ mutates to F . As shown in Fig. 7, under σ ∈ (0, π2 ), increasing the σ can increase the success probability of our protocol, so that it looks like it is better to add more noise. However, the fact is not. From Eq. (82), it can be found the existence of σ would make the parties distill some unwanted error states with some probability. As shown in Eq. (80), the probability to obtain the unwanted error state also contributes to the total success probability (the third items of P in Eq. (80)). Under the scale of σ ∈ (0, π2 ), the probability to obtain the unwanted error state will increase with the growth of σ, which lead to the increase of the total success probability. However, the growth of σ will largely reduce the fidelity of the distilled state. It can be calculated that when F = 0.6, the parties can obtain F ′′ < F under the scale of σ > 0.293, while when F = 0.8, they can obtain F ′′ < F under the scale of σ > 0.448. F ′′ < F will make our protocol completely fail. In this way, in practical experiment, the parties should try to diminish the σ. Based on the experimental results from Ref. [77–79], by manipulating the position of a single 87 Rb atom, precisely tuning the atom-cavity coupling strength, and controlling the reflectivity of the input coherent state, the parties can make the value of σ small. In this way, they could make the P and F ′′ quite close to those in the ideal condition. As all the distilled photon states can be well remained, our protocol can be repeated to further purify the distilled photon states. We also take the purification for the logic bit-flip error under N = M = 2 for example. In Sec. 3.1, after the first purification round, the parties take two copies of ρout1 in Eq. (22) as the new input mixed states and can repeat the protocol with the help of four new atoms in the form of √12 (|gL i + |gR i) trapped in the cavities, respectively. By repeating the protocol, the fidelity of the distilled new mixed state can be further increased. In Fig. 9, we calculate the fidelity of the new mixed state FN′′ after the protocol has been performed for N times as a function of N . Here, we suppose the initial fidelity F = 0.54. When F = 0.54, the parties can obtain FN′′ > F under the scale of σ < 0.193. In this way, we select σ = 0, 0.05, 0.1, 0.15, and N = 0, 1, 2, 3, 4 and 5, respectively. It can be found that by increasing the recycle number N , the fidelity of the distilled new mixed state can be effectively increased. For example, under the perfect experimental condition (σ = 0), we can obtain F1′′ = 0.57949, while F5′′ = 0.99412. According to Eq. (23) and Eq. (25), under the scale of F > 12 , relatively low initial fidelity F would lead to relatively high success probability but low fidelity of the distilled new mixed state. However, by repeating our protocol, the fidelity of the distilled mixed states can be largely increased with the growth of recycle number N . In this way, the relatively high success probability and fidelity can be realized simultaneously by repeating the protocol. On the other hand, under the imperfect experimental condition, the exist of σ would diminish the FN′′ . When σ is as low as 0.05,

21 FN′′ is quite closed to the ideal value with σ = 0. With the growth of σ, it is obvious that FN′′ is lower than the ideal value. Especially, when the recycle number N is large, the difference between the two values becomes bigger. VI.

CONCLUSION

In conclusion, in practical applications, four kinds of errors may occur in the logic Bell state, that is, the bit-flip error and phase-flip error in the logic qubits, and the bit-flip error and phase-flip error in the physic qubits. In the paper, we put forward an EPP to deal with the four kinds of errors in the logic Bell state, where each logic qubit is arbitrary M -particle GHZ state. For the logic bit-flip error and logic phase-flip error, the parties require two copies of initial mixed photon states and some auxiliary single three-level atoms trapped in the low-Q cavities. With the help of the photonic Faraday rotation effect and the atomic and photon states measurement, they can finally distill new mixed photon state. The phase-flip error in the physic qubit equals to the bit-flip error in the logic qubits, which can also be purified with above EPP. On the other hand, we prove that a bit-flip error in one physic qubit can be selected with the help of some auxiliary three-level atoms in the low-Q cavities. Then, the parties can completely correct it with the physic bit-flip operation. In our protocol, all the distilled new photon states can be well remained, so that the parties can repeat this EPP to obtain a higher fidelity of the entangled states. As the logic-qubit entanglement utilizes the QEC codes, and it has an inherent stability against noise and decoherence, this EPP may provide a double protection for the entanglement from the channel noise. According to the above advantages, our EPP may be useful in the future long-distance quantum communication based on logic-qubit entanglement. VII.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11474168 and 61401222), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20151502), and a Project Funded by the Priority Academic Development Program of Jiangsu Higher Education Institutions, China.

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Highlights of the manuscript “Polarization entanglement purification for concatenated Greenberger-Horne-Zeilinger state”(YAPHY_67442) 1

This protocol is the first polarization entanglement purification protocol (EPP) for concatenated Greenberger-Horne-Zeilinger (C-GHZ) state in optical system, resorting to the photon-atom interaction in low-quality (Q) cavity. 2 In contrast to existing optical EPPs, this protocol can purify the bit-flip error and phase-flip error in both physic and logic level. 3 With the help of photonic Faraday rotation, the purified entangled state can be remained for future application and this protocol can also be repeated to obtain a higher fidelity of the mixed state.