aa r X i v : . [ qu a n t - ph ] N ov Purification of Logic-Qubit Entanglement
Lan Zhou , and Yu-Bo Sheng ∗ College of Mathematics & Physics,Nanjing University of Posts and Telecommunications,Nanjing, 210003, China Key Lab of Broadband Wireless Communication and Sensor Network Technology,Nanjing University of Posts and Telecommunications,Ministry of Education, Nanjing, 210003, China (Dated: January 23, 2018)Recently, the theoretical work of Fr¨owis and W. D¨ur (Phys. Rev. Lett. , 110402 (2011))and the experiment of Lu et al. (Nat. Photon. , 364 (2014)) both showed that the logic-qubitentanglement has its potential application in future quantum communication and quantum network.However, the entanglement will suffer from the noise and decoherence. In this paper, we willinvestigate the 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. Moreover, the bit-fliperror and in physical-qubit entanglement can be completely corrected. The phase-flip error equalsto the bit-flip error in logic-qubit entanglement which can also be purified. This EPP may providesome potential applications in future quantum communication and quantum network. PACS numbers: 03.67.Pp, 03.67.Mn, 03.67.Hk, 42.50.-p
I. INTRODUCTION
Entanglement plays an important role in quantum in-formation areas. quantum teleportation [1], quantum keydistribution (QKD)[2], quantum secret sharing (QSS) [3],quantum secure direct communication (QSDC)[4, 5], andquantum repeaters [6, 7], all need entanglement. Beforestarting the quantum communication protocol, the par-ties should set up the maximally entanglement channelfirst. Usually, they create the entanglement locally anddistribute the entangled state to the distant locations infiber or free space. Noise is one of the main obstaclein entanglement distribution. It will degrade the entan-glement. The degraded entanglement will decrease theefficiency of the communication and also make the quan-tum communication insecure.Entanglement purification is to distill the high qual-ity entangled stats from the low quality of entangledstates [8–31]. In 1996, Bennett et al. proposed the con-cept of entanglement purification [8]. Subsequently, thereare many efficient entanglement purification protocols(EPPs) proposed. For example, in 2001, Pan et al. de-scribed the feasible EPP with linear optics [11]. In 2008,Sheng et al. described an EPP which can be repeatedto obtain a higher fidelity [14]. In 2010, the determin-istic EPP was also proposed [15]. In 2014, the EPP forhyperentanglement was presented [23]. Recent researchesshowed that the entanglement purification can be used tobenefit the blind quantum computation [24, 26]. Thereare also some important EPPs for solid systems, such asthe EPP for spins [27], short chains of atoms [30, 31], and ∗ [email protected] so on.The EPPs described above all focus on the entan-glement encoded in the physical qubit directly, for ex-isting quantum communication protocols are usuallybased on the physical-qubit entanglement. Recently,Fro ¨wis and W. D¨ur investigated a new type of entangle-ment, named concatenated Greenberger-Horne-Zeilinger(C-GHZ) state [32]. The C-GHZ state can be written as[32–41] | Φ ± i N,M = 1 √ | GHZ + N i ⊗ M ± | GHZ − N i ⊗ M ) . (1)Here M is the number of the logic qubit and N is thenumber of physical qubit in each logic qubit. Each logicqubit is a physical GHZ state of the form | GHZ ± N i = 1 √ | i ⊗ N ± | i ⊗ N ) . (2)In 2014, Lu et al. realized the first experiment of logic-qubit entanglement in linear optics [37]. In 2015, Shengand Zhou described the first logic Bell-state analysis [38].They showed that we can perform the logic-qubit en-tanglement swapping and it is possible to perform thelong-distance quantum communication based on logic-qubit entanglement [39, 40]. These theory and experi-ment researches may provide an important avenue thatthe large-scale quantum networks and the quantum com-munication may be based on logic-qubit entanglement infuture.Though many EPPs were proposed and discussed,none protocol discusses the purification of logic-qubit en-tanglement. In this paper, we will investigate the firstmodel of entanglement purification for logic-qubit entan-glement. 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 oflogic-qubit entanglement can be simplified to the EPP ofphysical-qubit entanglement, which can be easily purifiedin the next step. Moreover, we also show that if a bit-flip error occurs in one of a physical-qubit entanglementlocally, it can be well corrected. Moreover, the phase-fliperror in one of a physical-qubit entanglement equals tothe bit-flip error in the logic qubit entanglement, whichcan be well purified.This paper is organized as follows: In Sec. II, we ex-plain the purification for logic qubit error. In Sec. III,we describe the purification for physical qubit error. InSec. IV, we present a discussion and conclusion. II. PURIFICATION OF LOGIC-QUBIT ERROR
Suppose that Alice and Bob share the maximally en-tangled state | Φ + i AB of the formThe four logic Bell states can be described as | Φ + i AB = 1 √ | φ + i A | φ + i B + | φ − i A | φ − i B ) . (3)From Eq. (3), the Bell states | φ + i and | φ − i can be re-garded as the logic qubit | i and | i , respectively. If abit-flip error occurs on the logic qubit with the probabil-ity of 1 − F , | Φ + i AB will become | Ψ + i AB of the form | Ψ + i AB = 1 √ | φ + i A | φ − i B + | φ − i A | φ + i B ) . (4)Here | φ ± i and | ψ ± i are four physical Bell states of theform | φ ± i = 1 √ | i| i ± | i| i ) , | ψ ± i = 1 √ | i| i ± | i| i ) , (5)with | i and | i are the physical qubit, respectively. | Φ + i AB essentially is the state with m = N = 2 in Eq.(1). The whole mixed state can be described as ρ = F | Φ + i AB h Φ + | + (1 − F ) | Ψ + i AB h Ψ + | . (6)As shown in Fig. 1, Alice and Bob share two copiesof mixed states, named ρ and ρ , distributed from theentanglement source S . State ρ is in the spatial modes a , a , b and b and state ρ is in the spatial modes a , a , b and b , respectively. The whole system ρ ⊗ ρ can be described as follows. With the probability of F ,it is in the state | Φ + i A B ⊗ | Φ + i A B . With the equalprobability of F (1 − F ), they are in the states | Φ + i A B ⊗| Ψ + i A B and | Ψ + i A B ⊗ | Φ + i A B , respectively. Withthe probability of (1 − F ) , it is in the state | Φ + i A B ⊗| Φ + i A B . Here states | Φ + i A B and | Ψ + i A B are thecomponents in ρ and | Φ + i A B and | Ψ + i A B are thecomponents in ρ , respectively. FIG. 1: Schematic diagram of the purification of logic Bell-state analysis. H represents the Hadamard operation and M represents the measurement in the basis {| i , | i} . We first discuss the item | Φ + i A B ⊗ | Φ + i A B . It canbe written as | Φ + i A B ⊗ | Φ + i A B = 1 √ | φ + i A | φ + i B + | φ − i A | φ − i B ) ⊗ √ | φ + i A | φ + i B + | φ − i A | φ − i B )= 12 ( | φ + i A | φ + i A | φ + i B | φ + i B + | φ + i A | φ − i A | φ + i B | φ − i B + | φ − i A | φ + i A | φ − i B | φ + i B + | φ − i A | φ − i A | φ − i B | φ − i B ) . (7)From Fig. 1, they let all qubits pass through thecontrolled-not (CNOT) gate. State | φ + i A in spatialmodes a , a will become | φ + i A = 1 √ | i a | i a + | i a | i a ) → √ | i a | i a + | i a | i a )= | + i a | i a . (8)State | ψ + i A in spatial modes a , a will become | φ − i A = 1 √ | i a | i a − | i a | i a ) → √ | i a | i a − | i a | i a )= |−i a | i a . (9)Here |±i = √ ( | i ± | i ). After passing through theCNOT gates and Hadamard gates, with the probabilityof F , state in Eq. (7) can be evolved as | Φ + i A B ⊗ | Φ + i A B →
12 ( | + i a | i a | + i a | i a | + i b | i b | + i b | i b + | + i a | i a |−i a | i a | + i b | i b |−i b | i b + |−i a | i a | + i a | i a |−i b | i b | + i b | i b + |−i a | i a |−i a | i a |−i b | i b |−i b | i b ) → | φ + i a b | φ + i a b | i a | i b | i a | i b . (10)Following the same principle, with the probability of F (1 − F ), state | Φ + i A B ⊗ | Ψ + i A B can be evolvedas | Φ + i A B ⊗ | Ψ + i A B → | φ + i a b | ψ + i a b | i a | i b | i a | i b , (11)and state | Ψ + i A B ⊗ | Φ + i A B can be evolved as | Ψ + i A B ⊗ | Φ + i A B → | ψ + i a b | φ + i a b | i a | i b | i a | i b . (12)With the probability of (1 − F ) , state | Ψ + i A B ⊗| Ψ + i A B can be evolved as | Ψ + i A B ⊗ | Ψ + i A B → | ψ + i a b | ψ + i a b | i a | i b | i a | i b . (13)Here | φ + i a b , | ψ + i a b , | φ + i a b and | ψ + i a b are thephysical Bell states as described in Eq. (5) in spatialmodes a b , a b , respectively. Interestingly, from Eq.(10) to Eq. (13), the qubits in spatial modes a , b , a and b disentangle with the other qubits. The purifica-tion of logic Bell state can be transformed to the purifi-cation of the physical Bell state in spatial modes a , b , a and b . Briefly speaking, as shown in Fig. 1, they letthe qubits in a , b , a and b pass through the CNOTgate in a second time. The CNOT gate will make thestate [8] | φ + i a b | φ + i a b → | φ + i a b | φ + i a b , | φ + i a b | ψ + i a b → | φ + i a b | ψ + i a b , | ψ + i a b | φ + i a b → | ψ + i a b | ψ + i a b , | ψ + i a b | ψ + i a b → | ψ + i a b | φ + i a b . (14)Subsequently, Alice and Bob measure their qubits in spa-tial modes a and b in { , } basis, respectively. Withclassical communication, if the measurement results arethe same, both 0, or 1, the purification is successful. Oth-erwise, if the measurement results are different, the pu-rification is a failure. From Eq. (14), if it is successful,they will obtain | φ + i a b , with the probability of F , and | ψ + i a b will the probability of (1 − F ) . In this way, theyobtain a high fidelity of mixed state ρ a b = F ′ | φ + i a b h φ + | + (1 − F ′ ) | ψ + i a b h ψ + | . (15)Here F ′ = F F + (1 − F ) . (16)If F > , they can obtain F ′ > F . State in Eq.(15) is thepurified physical Bell state. The final step is to recover ρ a b to logic Bell state. From Fig. 1, they perform theHadamard operations on the qubits in spatial modes a and b and let four qubits in a , b , a and b pass through the CNOT gates, respectively. State | φ + i a b combinedwith | i a | i b evolve as | φ + i a b | i a | i b = 1 √ | i a | i b + | i a | i b ) | i a | i b → √ √ | i a + | i a ) | i a ⊗ √ | i b + | i b ) | i b + 1 √ | i a − | i a ) | i a ⊗ √ | i b − | i b ) | i b ] → √ √ | i a | i a + | i a | i a ) ⊗ √ | i b | i b + | i b | i b )+ 1 √ | i a | i a − | i a | i a ) ⊗ √ | i b | i b − | i b | i b )]= | Φ + i A B . (17)Following the same principle, state | ψ + i a b combinedwith | i a | i b evolve to | Ψ + i A B . Finally, they willobtain a new mixed state ρ ′ = F ′ | Φ + i A B h Φ + | + (1 − F ′ ) | Ψ + i A B h Ψ + | . (18)In this way, they have completed the purification.On the other hand, if a phase-flip error occurs, it willmake the state in Eq. (3) become | Φ − i AB = 1 √ | φ + i A | φ + i B − | φ − i A | φ − i B ) . (19)The whole mixed state can be written as ρ ′ = F | Φ + i AB h Φ + | + (1 − F ) | Φ − i AB h Φ − | . (20)The mixed state in Eq. (20) can also be purified withthe same principle. Briefly speaking, as shown in Fig.1, they first choose two copies of the states in Eq. (20).After the qubits in spatial modes a , a , b , b , a , a , b and b passing through the CNOT gates andHadamard gates, respectively, | Φ + i A B ⊗ | Φ + i A B will become | φ + i a b | φ + i a b | i a | i b | i a | i b , whichis shown in Eq. (10). State | Φ + i A B ⊗ | Φ − i A B will become | φ + i a b | φ − i a b | i a | i b | i a | i b .State | Φ − i A B ⊗ | Φ + i A B will become | φ − i a b | φ + i a b | i a | i b | i a | i b , andstate | Φ − i A B ⊗ | Φ − i A B will become | φ − i a b | φ − i a b | i a | i b | i a | i b . They only needto add the Hadamard operations on each qubit, whichmake | φ + i do not change, and | φ − i become | ψ + i . Theyessentially transform the phase-flip error to bit-flip error,which has the same form described above. In this way,the phase-flip error in logic-qubit entanglement can alsobe purified.So far, we have described the EPP for logic-qubit en-tanglement. Each logic qubit is encoded in a physical FIG. 2: Schematic diagram of the EPP with each logic qubitbeing arbitrary GHZ state. On pair of mixed state ρ ab is inthe spatial modes a , b , a , b , · · · , a N and b N . The othercopy of mixed state ρ cd is in the spatial modes c , d , c , d , · · · , c N and d N . Bell state. It is straightforward to extend this approachto the logic-qubit entanglement with arbitrary physicalGHZ state encoded in a logic qubit. Suppose that Aliceand Bob share the state | Φ +1 i AB = 1 √ | GHZ + N i A | GHZ + N i B + | GHZ − N i A | GHZ − N i B ) . (21)The noise makes the state become ρ = F | Φ +1 i AB h Φ +1 | + (1 − F ) | Ψ +1 i AB h Ψ +1 | . (22)Here | Ψ +1 i AB = 1 √ | GHZ + N i A | GHZ − N i B + | GHZ − N i A | GHZ + N i B ) . (23)As shown in Fig. 2, they first choose two copies of themixed states with the same form of ρ . One mixed state ρ ab is in the spatial modes a , b , a , b , · · · , a N , b N , andthe other mixed state ρ cd is in the mixed state c , d , c , d , · · · , c N , d N , respectively. We first discuss the mixedstate ρ ab . After passing through the CNOT gates andHadamard gates, with the probability of F , state | Φ +1 i ab becomes | Φ +1 i ab = 1 √ | GHZ + N i a | GHZ + N i b + | GHZ − N i a | GHZ − N i b )= 1 √ √ | i a | i a · · · | i a N + | i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N + | i b | i b · · · | i b N )+ 1 √ | i a | i a · · · | i a N − | i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N − | i b | i b · · · | i b N )] → √ √ | i a | i a · · · | i a N + | i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N + | i b | i b · · · | i b N )+ 1 √ | i a | i a · · · | i a N − | i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N − | i b | i b · · · | i b N )] → √ | i a | i b + | i a | i b ) ⊗ | i a · · · | i a N | i b · · · | i b N . (24)With the same principle, with the probability of 1 − F ,state | Ψ +1 i ab becomes | Ψ +1 i ab → √ | i a | i b + | i a | i b ) ⊗ | i a · · · | i a N | i b · · · | i b N . (25)Similar to Eqs. (24) and (25), after passing throughthe CNOT gates and Hadamard gates, state ρ cd in thespatial modes c , d , c , d , · · · , c N , d N can also evolveas | Φ +1 i cd → √ | i c | i d + | i c | i d ) ⊗ | i c · · · | i c N | i d · · · | i d N , (26)and | Ψ +1 i cd → √ | i c | i d + | i c | i d ) ⊗ | i c · · · | i c N | i d · · · | i d N . (27)Here the subscripts a , b , c and d are the spatial modes asshown in Fig. 2. From Eqs. (24) to (27), by choosing twocopies of mixed states ρ ab and ρ cd , they can be simplifiedto the purification of the physical Bell state, which can beeasily performed, similar to Eqs. (10) to (13). After theyobtaining the purified high fidelity physical Bell state, thelast step is also to recover Bell state to arbitrary logic Bellstate. They first perform the Hadamard operation onthe qubit in a and b , respectively. Subsequently, theyboth let the N qubits pass through N − | Φ +1 i AB become | Φ − i AB , which can be writtenas | Φ − i AB = 1 √ | GHZ + N i A | GHZ + N i B − | GHZ − N i A | GHZ − N i B ) . (28)The mixed state can be written as ρ = F | Φ +1 i AB h Φ +1 | + (1 − F ) | Φ − i AB h Φ − | . (29)Interestingly, after passing through the CNOT gates andHadamard gates, state | Φ − i ab will become | Φ − i cd → √ | i a | i b − | i a | i b ) ⊗ | i a · · · | i a N | i b · · · | i b N . (30)From Eqs. (24), (29) and (30), we can find that thephase-flip error in the logic-qubit entanglement can besimplified into the phase-flip error of the physical-qubitBell entanglement, which can be transformed to the bit-flip error and be purified in the next step. In this way,they can purify arbitrary logic-qubit entanglement. III. PURIFICATION OF PHYSICAL-QUBITERROR
In above section, we showed that the bit-flip error andphase-flip error in the logic-qubit entanglement can besimplified into the bit-flip error and phase-flip error in thephysical-qubit entanglement, respectively. Subsequently,the errors in the physical-qubit entanglement can be wellpurified with the similar approach as Refs.[8, 9]. Be-sides the errors in the logic-qubit entanglement, the sin-gle physical qubit can also suffer from the error. Forexample, as shown in Eq. (3), if a bit-flip error occursin one of the physical qubit in the logic-qubit A , whichmakes | φ + i A become | ψ + i A and | φ − i A become | ψ − i A ,respectively. Therefore, if the error occurs, it makes thestate | Φ + i AB become | Φ + i ′ AB = 1 √ | ψ + i A | φ + i B + | ψ − i A | φ − i B ) . (31)Compared with Eq. (3) and Eq. (31), we find that theerror occurs locally. In this way, they only require tochoose one copy of the mixed state to perform the errorcorrection. They let the logic-qubit A pass through theCNOT gate. The qubit in a is the control qubit andthe qubit in a is the target qubit. State in Eq. (3) willbecome | Φ + i AB → √ | + i a | i a | φ + i B + |−i a | i a | φ − i B ) , (32)and state in Eq. (31) will become | Φ + i ′ AB → √ | + i a | i a | φ + i B + |−i a | i a | φ − i B ) . (33)From Eqs. (32) and (33), they only need to measure thephysical qubit in a in the basis { , } . If it becomes | i ,it means that a bit-flip error occurs. If Alice and Bob ex-ploit the quantum nondemolition (QND) measurement,which do not destroy the physical qubit, they are onlyrequired to perform a bit-flip operation to correct the bit-flip error. On the other hand, if the measurement isdestructive, they can prepare another physical qubit | i in a and perform the CNOT operation with the physicalqubit a in logic qubit A to recover the whole state to | Φ + i AB . If the bit-flip error occurs on the second logicqubit B , they can also completely correct it with thesame principle.If the logic qubit is N -particle GHZ state, a bit-fliperror on the logic-qubit A will make the state become | Φ +1 i ′ AB = 1 √ √ | i a | i a · · · | i a N + | i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N + | i b | i b · · · | i b N )+ 1 √ | i a | i a · · · | i a N − | i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N − | i b | i b · · · | i b N )] . (34)They let the particles a , a , · · · , a N pass through the N − a mode is the control qubit and the other is the targetqubit. It makes the state | Φ +1 i ′ AB become | Φ +1 i ′ AB → [( | + i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N + | i b | i b · · · | i b N )+ ( |−i a | i a · · · | i a N ) ⊗ √ | i b | i b · · · | i b N − | i b | i b · · · | i b N )] . (35)From Eq. (35), by measuring the physical qubit a N inthe basis { , } , if it becomes | i , it means that a bit-flip error occurs. Following the same principle, it can becompletely corrected.On the other hand, if a phase-flip error occurs on thelogic qubit A , which makes | φ + i ↔ | φ − i . The state | Φ + i ′′ AB with a phase-flip error in logic qubit A can bewritten as | Φ + i ′′ AB = 1 √ | φ − i A | φ + i B + | φ + i A | φ − i B ) . (36)Interestingly, from Eq. (36), we find that the phase-fliperror in the two physical qubits essentially equals to thelogic bit-flip error as shown in Eq. (4). In this way, wehave completely explained our EPP. IV. DISCUSSION AND CONCLUSION
In traditional EPPs for physical-qubit entanglement[8, 9], they should purify two kinds of errors. The oneis the bit-flip error and the other is the phase-flip error.Using the CNOT gate, the bit-flip error can be purifieddirectly. The phase-flip error can be transformed to thebit-flip error and be purified in the next step. In our EPP,we show that the logic-qubit entanglement may containfour kinds of errors. The bit-flip error and phase-flip erroroccur in the logic-qubit entanglement and physical-qubitentanglement, respectively. From our description, thebit-flip error and phase-flip error in logic-qubit entangle-ment can be simplified to the bit-flip error and phase-fliperror in physical-qubit entanglement, which can be puri-fied with the previous approach in the next step. On theother hand, if a bit-flip error occurs in one of logic qubit,the error can be completely corrected locally. Moreover,if a phase-flip error occurs in one of the logic qubit, wefind that it equals to the bit-flip error in the logic-qubitentanglement. In this way, all errors can be purified.In our EPP, the key element to realize the protocol isthe CNOT gate. There are some important progressesin construction of the CNOT gate, which shows that itis possible to realize the deterministic CNOT gate infuture experiment [42–46]. For example, with the helpcross-Kerr nonlinearity, Nemoto et al. and Lin et al. de-scribed a near deterministic CNOT gate for polarizationphotons, respectively [42, 43]. Recently, Wei and Dengdesigned a deterministic controlled-not gate on two pho-tonic qubits by two single-photon input-output processesand the readout on an electron-medium spin confined inan optical resonant microcavity [44]. The deterministicCNOT for spins [45], electron-spin qubits assisted by di-amond nitrogen-vacancy centers inside cavities were also discussed [46].In conclusion, we have described the first EPP forlogic-qubit entanglement. We first described the purifica-tion for both the bit-flip error and phase-flip error in thelogic qubit. The entanglement purification for logic-qubitentanglement can be simplified to the purification of thephysical-qubit entanglement, which can be performed inthe next step. On the other hand, we also discussed thepurification of the errors occurring in the physical-qubitentanglement. The bit-flip error on the physical qubitcan be completely corrected locally. The phase-flip erroroccurs on the physical qubit equals to the bit-flip erroron the logic qubit, which can also be well purified. OurEPP is suitable for the case that each logic qubit beingarbitrary N -particle GHZ state. Our EPP may be usefulfor future long-distance quantum communication basedon logic-qubit entanglement. ACKNOWLEDGEMENTS
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