Purification of Logic-Qubit Entanglement

Recently, the logic-qubit entanglement shows its potential application in future quantum communication and quantum network. However, the entanglement will suffer from the noise and decoherence. In this paper, we will investigate the first entanglement purification protocol for logic-qubit entanglement. We show that both the bit-flip error and phase-flip error in logic-qubit entanglement can be well purified. Moreover, the bit-flip error in physical-qubit entanglement can be completely corrected. The phase-flip in physical-qubit entanglement error equals to the bit-flip error in logic-qubit entanglement, which can also be purified. This entanglement purification protocol may provide some potential applications in future quantum communication and quantum network.

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  42 . In 2015, Sheng and Zhou described the first logic Bell-state analysis 43 . 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 entanglement [44][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 Φ + AB . 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 0 and 1 , 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 ρ 1 and ρ 2 , distributed from the entanglement source S. State ρ 1 is in the spatial modes a 1 , a 2 , b 1 and b 2 and state ρ 2 is in the spatial modes a 3 , a 4 , b 3 and b 4 , respectively. The whole system ρ 1 ⊗ ρ 2 can be described as follows. With the probability of F 2 , 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 ρ 1 and |Φ + 〉 A2B2 and |Ψ + 〉 A2B2 are the components in ρ 2 , respectively.
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 a 1 , a 2 will become State |φ − 〉 A1 in spatial modes a 1 , a 2 will become Here ± = ± ( 0 1 ) 1 2 . After passing through the CNOT gates and Hadamard gates, with the probability of F 2 , 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 3 3 and ψ + a b 3 3 are the physical Bell states described in Eq. (4) in spatial modes a 1 b 1 , a 3 b 3 , respectively. Interestingly, from Eq. (9) to Eq. (12), the qubits in spatial modes a 2 , b 2 , a 4 and b 4 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 a 1 , b 1 , a 3 and b 3 . Briefly speaking, as shown in Fig. 1, they let the qubits in a 1 , b 1 , a 3 and b 3 modes pass through the CNOT gates for a second time. The CNOT gates will make the state 13 Subsequently, Alice and Bob measure their qubits in spatial modes a 3 and b 3 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 φ + a b 1 1 , with the probability of F 2 , and ψ + a b 1 1 will the probability of (1 − F) 2 . In this way, they obtain a new mixed state a b a b a b If > F 1 2 , they can obtain F′ > F. State in Eq. (14) is the purified physical Bell state. The final step is to recover ρ a b 1 1 to logic Bell state. From Fig. 1, they perform the Hadamard operations on the qubits in spatial modes a 1 and b 1 and let four qubits in a 1 , b 1 , a 2 and b 2 pass through the CNOT gates, respectively.
Following the same principle, state ψ + a b 1 1 combined with 0 0 a b 2 2 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.
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 a 1 , a 2 , b 1 , b 2 , a 3 , a 4 , b 3 and b 4 passing through the CNOT gates and Hadamard gates, respectively. |Φ + 〉 A1B1 ⊗ |Φ + 〉 A2B2 will become φ φ , which is shown in Eq. (9). State |Φ + 〉 A1B1 ⊗ |Φ − 〉 A2B2 will become φ φ . 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 Ψ = + .
As shown in Fig. 2, they first choose two copies of the mixed states with the same form of ρ 3 . One mixed state ρ ab is in the spatial modes a 1 , b 1 , a 2 , b 2 , ···, a N , b N , and the other mixed state ρ cd is in the spatial modes c 1 , d 1 , c 2 , d 2 , ···, c N , d N , respectively. We first discuss the mixed state ρ ab . After passing through the CNOT gates and Hadamard gates, with the probability of F, state Φ +   Similar to Eqs (23) and (24), after passing through the CNOT gates and Hadamard gates, state ρ cd in the spatial modes c 1 , d 1 , c 2 , d 2 , ···, c N , d N can also evolve as 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 ρ ab and ρ cd , 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 a 1 mode and b 1 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 Φ + , which can be written as The mixed state can be written as Interestingly, after passing through the CNOT gates and Hadamard gates, 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 a 1 mode is the control qubit and the qubit in a 2 mode is the target qubit. State in Eq. (3) will become From Eqs (31) and (32), they only need to measure the physical qubit in a 2 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 a 2 mode and perform the CNOT operation with the physical qubit a 1 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 a 1 , a 2 , ···, a N modes pass through the N − 1 CNOT gates. In each CNOT gate, particle in a 1 mode is the control qubit and the other is the target qubit. It makes the state Φ ′ On the other hand, if a phase-flip error occurs on the logic qubit A, which makes |φ + 〉 ↔ |φ − 〉 . The state Φ + " AB 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 entanglement 13,14 , 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 experiment [49][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, respectively 49,50 .
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 microcavity 51 . The deterministic CNOT gate for spins 52 , electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities were also discussed 53 . Recently, the group of Du present a very important progress of experiment of fault-tolerant universal quantum gates with solid-state spins under ambient conditions 54 . 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.