Introduction

Entanglement plays an important role in quantum information areas. Quantum teleportation1, quantum key distribution (QKD)2, quantum secure direct communication (QSDC)3,4,5, quantum repeaters6,7 and other important quantum information protocols8,9,10,11,12, all need entanglement. Before starting the quantum communication protocol, the parties should set up the maximally entanglement channel first. Usually, they create the entanglement locally and distribute the entangled state to distant locations in fiber or free space. Noise is one of the main obstacles in entanglement distribution. It will degrade the entanglement. The degraded entanglement will decrease the efficiency of the communication and even make the quantum communication insecure.

Entanglement purification is to distill the high quality entangled states from the low quality entangled states with local operation and classical communications (LOCC). In 1996, Bennett et al. proposed the concept of entanglement purification13. Subsequently, there are many efficient entanglement purification protocols (EPPs) proposed14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36. For example, in 2001, Pan et al. described a feasible EPP with linear optics16. In 2008, Sheng et al. described an EPP which can be repeated to obtain a higher fidelity18. In 2010, the first deterministic EPP was proposed19. The deterministic EPPs19,20,21,22 are quite different from the previous EPPs13,14,15,16,17,18. The deterministic EPP can obtain the maximally entangled state in one or two steps with the 100% success probability, in principle, while the previous EPPs are to increase the fidelity of mixed states by repeating the EPPs. In 2014, the EPP for hyperentanglement was presented27. Recent researches showed that the entanglement purification can be used to benefit the blind quantum computation29. There are also some important EPPs for solid systems, such as the EPP for spins30, short chains of atoms32,33, ionic states36, and so on.

The EPPs described above all focus on the entanglement encoded in the physical qubit directly, for existing quantum communication protocols are usually based on the physical-qubit entanglement. Recently, Froẅis and Dür investigated a new type of entanglement, named concatenated Greenberger-Horne-Zeilinger (C-GHZ) state37. The C-GHZ state can be written as37,38,39,40,41,42,43,44,45,46,47,48

Here M is the number of the logic qubit and N is the number of the physical qubit in each logic qubit. Each logic qubit is a physical GHZ state of the form

In 2014, Lu et al. realized the first experiment of logic-qubit entanglement in linear optics42. In 2015, Sheng and Zhou described the first logic Bell-state analysis43. They showed that we can perform the logic-qubit entanglement swapping and it is possible to perform the long-distance quantum communication based on logic-qubit entanglement44,45,46. These theory and experiment researches may provide an important avenue that the large-scale quantum networks and the quantum communication may be based on logic-qubit entanglement in future.

Though many EPPs were proposed and discussed, none protocol discusses the purification of logic-qubit entanglement. In this paper, we will investigate the first model of entanglement purification for logic-qubit entanglement. We show that both the bit-flip error and phase-flip error in logic-qubit entanglement can be well purified. With the help of controlled-not (CNOT) gate, the EPP of logic-qubit entanglement can be simplified to the EPP of physical-qubit entanglement, which can be easily purified in the next step. Moreover, we also show that if a bit-flip error occurs in one of a physical-qubit entanglement locally, it can be well corrected. Moreover, the phase-flip error in one of a physical-qubit entanglement equals to the bit-flip error in the logic qubit entanglement, which can be well purified.

This paper is organized as follows: In Sec. II, we explain the purification for logic qubit error. In Sec. III, we describe the purification for physical qubit error. In Sec. IV, we present a discussion and conclusion.

Results

Suppose that Alice and Bob share the maximally logic Bell state . The four logic Bell states can be described as

Here |ϕ±〉 and |ψ±〉 are four physical Bell states of the form

with |0〉 and |1〉 being the physical qubits, respectively. From Eq. (3), the Bell states |ϕ+〉 and |ϕ−〉 can be regarded as the logic qubit and , respectively. |Φ+〉AB essentially is the state with M = N = 2 in Eq. (1). If a bit-flip error occurs on the logic qubit with the probability of 1 − F, |Φ+〉AB will become |Ψ+〉AB. The whole mixed state can be described as

As shown in Fig. 1, Alice and Bob share two copies of mixed states, named ρ1 and ρ2, distributed from the entanglement source S. State ρ1 is in the spatial modes a1, a2, b1 and b2 and state ρ2 is in the spatial modes a3, a4, b3 and b4, respectively. The whole system ρ1 ⊗ ρ2 can be described as follows. With the probability of F2, it is in the state |Φ+〉A1B1 ⊗ |Φ+〉A2B2. With the equal probability of F(1 − F), it is in the states |Φ+〉A1B1 ⊗ |Ψ+〉A2B2 or |Ψ+〉A1B1 ⊗ |Φ+〉A2B2. With the probability of (1 − F)2, it is in the state |Ψ+〉A1B1 ⊗ |Ψ+〉A2B2. Here states |Φ+〉A1B1 and |Ψ+〉A1B1 are the components in ρ1 and |Φ+〉A2B2 and |Ψ+〉A2B2 are the components in ρ2, respectively.

Figure 1: Schematic diagram of the purification of logic Bell-state analysis.
figure 1

H represents the Hadamard operation and M represents the measurement in the basis {|0〉, |1〉}.

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We first discuss the item |Φ+〉A1B1 ⊗ |Φ+〉A2B2. It can be written as

From Fig. 1, they let all qubits pass through the controlled-not (CNOT) gate. State |ϕ+〉A1 in spatial modes a1, a2 will become

State |ϕ−〉A1 in spatial modes a1, a2 will become

Here . After passing through the CNOT gates and Hadamard gates, with the probability of F2, state in Eq. (6) can be evolved as

Following the same principle, with the probability of F(1 − F), state |Φ+〉A1B1 ⊗ |Ψ+〉A2B2 can be evolved as

and state |Ψ+〉A1B1 ⊗ |Φ+〉A2B2 can be evolved as

With the probability of (1 − F)2, state |Ψ+〉A1B1 ⊗ |Ψ+〉A2B2 can be evolved as

Here , , and are the physical Bell states described in Eq. (4) in spatial modes a1b1, a3b3, respectively. Interestingly, from Eq. (9) to Eq. (12), the qubits in spatial modes a2, b2, a4 and b4 disentangle with the other qubits. The purification of logic Bell state can be transformed to the purification of the physical Bell state in spatial modes a1, b1, a3 and b3. Briefly speaking, as shown in Fig. 1, they let the qubits in a1, b1, a3 and b3 modes pass through the CNOT gates for a second time. The CNOT gates will make the state13

Subsequently, Alice and Bob measure their qubits in spatial modes a3 and b3 in {0, 1} basis, respectively. With classical communication, if the measurement results are the same, the purification is successful. Otherwise, if the measurement results are different, the purification is a failure. From Eq. (13), if it is successful, they will obtain , with the probability of F2, and will the probability of (1 − F)2. In this way, they obtain a new mixed state

Here

If , they can obtain F′ > F. State in Eq. (14) is the purified physical Bell state. The final step is to recover to logic Bell state. From Fig. 1, they perform the Hadamard operations on the qubits in spatial modes a1 and b1 and let four qubits in a1, b1, a2 and b2 pass through the CNOT gates, respectively. State combined with evolves as

Following the same principle, state combined with evolve to |Ψ+〉A1B1. Finally, they will obtain a new mixed state

In this way, they have completed the purification.

On the other hand, if a phase-flip error occurs, it will make the state in Eq. (3) become

The whole mixed state can be written as

The mixed state in Eq. (19) can also be purified with the same principle. Briefly speaking, as shown in Fig. 1, they first choose two copies of the states in Eq. (19). After the qubits in spatial modes a1, a2, b1, b2, a3, a4, b3 and b4 passing through the CNOT gates and Hadamard gates, respectively. |Φ+〉A1B1 ⊗ |Φ+〉A2B2 will become , which is shown in Eq. (9). State |Φ+〉A1B1 ⊗ |Φ−〉A2B2 will become . State |Φ−〉A1B1 ⊗ |Φ+〉A2B2 will become , and state |Φ−〉A1B1 ⊗ |Φ−〉A2B2 will become . They only need to add the Hadamard operations on each qubit, which make |ϕ+〉 not change, and |ϕ−〉 become |ψ+〉. They essentially transform the phase-flip error to bit-flip error, which has the same form as described above. In this way, the phase-flip error in logic-qubit entanglement can also be purified.

So far, we have described the EPP for logic-qubit entanglement. Each logic qubit is encoded in a physical Bell state. It is straightforward to extend this approach to the logic-qubit entanglement with arbitrary physical GHZ state encoded in a logic qubit. Suppose that Alice and Bob share the state

The noise makes the state become

Here

As shown in Fig. 2, they first choose two copies of the mixed states with the same form of ρ3. One mixed state ρab is in the spatial modes a1, b1, a2, b2, ···, aN, bN, and the other mixed state ρcd is in the spatial modes c1, d1, c2, d2, ···, cN, dN, respectively. We first discuss the mixed state ρab. After passing through the CNOT gates and Hadamard gates, with the probability of F, state becomes

Figure 2: Schematic diagram of the EPP with each logic qubit being arbitrary GHZ state.
figure 2

On pair of mixed state ρab is in the spatial modes a1, b1, a2, b2, ···, aN and bN. The other copy of mixed state ρcd is in the spatial modes c1, d1, c2, d2, ···, cN and dN.

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With the same principle, with the probability of 1 − F, state becomes

Similar to Eqs (23) and (24), after passing through the CNOT gates and Hadamard gates, state ρcd in the spatial modes c1, d1, c2, d2, ···, cN, dN can also evolve as

and

Here the subscripts a, b, c and d are the spatial modes as shown in Fig. 2. From Eqs (23) to (26), by choosing two copies of mixed states ρab and ρcd, they can be simplified to the purification of the physical Bell state, which can be easily performed, similar to Eqs (9) to (12). After they obtaining the purified high fidelity physical Bell state, the last step is also to recover the physical Bell state to arbitrary logic Bell state. They first perform the Hadamard operation on the qubits in a1 mode and b1 mode, respectively. Subsequently, they both let the N qubits pass through N − 1 CNOT gates, respectively. Finally, they can obtain a high fidelity of arbitrary logic-qubit entangled state.

On the other hand, if a phase-flip error occurs, it makes the state become , which can be written as

The mixed state can be written as

Interestingly, after passing through the CNOT gates and Hadamard gates, state will become

From Eqs (23), (28) and (29), we can find that the phase-flip error in the logic-qubit entanglement can be simplified into the phase-flip error of the physical-qubit Bell entanglement, which can be transformed to the bit-flip error and purified in the next step. In this way, they can purify arbitrary logic-qubit entanglement.

In above section, we showed that the bit-flip error and phase-flip error in the logic-qubit entanglement can be simplified into the bit-flip error and phase-flip error in the physical-qubit entanglement, respectively. Subsequently, the errors in the physical-qubit entanglement can be well purified with the similar approach as refs 13 and 14. Besides the errors in the logic-qubit entanglement, the single physical qubit can also suffer from the error. For example, as shown in Eq. (3), if a bit-flip error occurs in one of the physical qubits in the logic-qubit A, it makes |ϕ+〉A become |ψ+〉A and |ϕ−〉A become |ψ−〉A, respectively. Therefore, it makes the state |Φ+〉AB become

Compared with Eq. (3) and Eq. (30), we find that the error occurs locally. In this way, they only require to choose one copy of the mixed state to perform the error correction. They let the logic-qubit A pass through the CNOT gate. The qubit in a1 mode is the control qubit and the qubit in a2 mode is the target qubit. State in Eq. (3) will become

and state in Eq. (30) will become

From Eqs (31) and (32), they only need to measure the physical qubit in a2 mode in the basis {0, 1}. If it becomes |1〉, it means that a bit-flip error occurs. If Alice and Bob exploit the quantum nondemolition (QND) measurement, which does not destroy the physical qubit, they are only required to perform a bit-flip operation to correct the bit-flip error. On the other hand, if the measurement is destructive, they can prepare another physical qubit |0〉 in a2 mode and perform the CNOT operation with the physical qubit a1 mode in logic qubit A to recover the whole state to |Φ+〉AB. If the bit-flip error occurs on the second logic qubit B, they can also completely correct it with the same principle.

If the logic qubit is N-particle GHZ state, a bit-flip error on the logic-qubit A will make the state become

They let the particles in a1, a2, ···, aN modes pass through the N − 1 CNOT gates. In each CNOT gate, particle in a1 mode is the control qubit and the other is the target qubit. It makes the state become

From Eq. (34), by measuring the physical qubit in aN mode in the basis {0, 1}, if it becomes |1〉, it means that a bit-flip error occurs. Following the same principle, it can be completely corrected.

On the other hand, if a phase-flip error occurs on the logic qubit A, which makes |ϕ+〉 ↔ |ϕ−〉. The state with a phase-flip error in logic qubit A can be written as

Interestingly, from Eq. (35), we find that the phase-flip error in the two physical qubits essentially equals to the logic bit-flip error as shown in Eq. (3). In this way, we have completely explained our EPP.

Discussion

In traditional EPPs for physical-qubit entanglement13,14, they should purify two kinds of errors. The one is the bit-flip error and the other is the phase-flip error. Using the CNOT gate, the bit-flip error can be purified directly. The phase-flip error can be transformed to the bit-flip error and be purified in the next step. In our EPP, we show that the logic-qubit entanglement may contain four kinds of errors. The bit-flip error and phase-flip error occur in the logic-qubit entanglement and physical-qubit entanglement, respectively. From our description, the bit-flip error and phase-flip error in logic-qubit entanglement can be simplified to the bit-flip error and phase-flip error in physical-qubit entanglement, which can be purified with the previous approach in the next step. On the other hand, if a bit-flip error occurs in one logic qubit, the error can be completely corrected locally. Moreover, if a phase-flip error occurs in one logic qubit, we find that it equals to the bit-flip error in the logic-qubit entanglement. In this way, all errors can be purified. In our EPP, the key element to realize the protocol is the CNOT gate. There are some important progresses in construction of the CNOT gate, which shows that it is possible to realize the deterministic CNOT gate in future experiment49,50,51,52,53. For example, with the help cross-Kerr nonlinearity, Nemoto et al. and Lin et al. described a near deterministic CNOT gate for polarization photons, respectively49,50. Recently, Wei and Deng designed a deterministic CNOT gate on two photonic qubits by two single-photon input-output processes and the readout on an electron-medium spin confined in an optical resonant microcavity51. The deterministic CNOT gate for spins52, electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities were also discussed53. Recently, the group of Du present a very important progress of experiment of fault-tolerant universal quantum gates with solid-state spins under ambient conditions54. They realized a universal set of logic gates in a nitrogen-vacancy center in diamond with an average single-qubit gate fidelity of 0.999952 and two-qubit (CNOT) gate fidelity of 0.992. Such high fidelity CNOT gate shows that it is feasible to realize this EPP in future.

In conclusion, we have described the first EPP for logic-qubit entanglement. We first described the purification for both the bit-flip error and phase-flip error in the logic qubit. The entanglement purification for logic-qubit entanglement can be simplified to the purification of the physical-qubit entanglement, which can be performed in the next step. On the other hand, we also discussed the purification of the errors occurring in the physical-qubit entanglement. The bit-flip error on the physical qubit can be completely corrected locally. The phase-flip error occurs on the physical qubit equals to the bit-flip error on the logic qubit, which can also be well purified. Our EPP is suitable for the case that each logic qubit being arbitrary N-particle GHZ state. Our EPP may be useful for future long-distance quantum communication based on logic-qubit entanglement.

Additional Information

How to cite this article: Zhou, L. and Sheng, Y.-B. Purification of Logic-Qubit Entanglement. Sci. Rep. 6, 28813; doi: 10.1038/srep28813 (2016).