Greenberger-Horne-Zeilinger States-Based Blind Quantum Computation with Entanglement Concentration

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OPEN Greenberger-Horne-Zeilinger states-based blind quantum computation with entanglement Received: 24 March 2017 Accepted: 16 June 2017 concentration Published: xx xx xxxx Xiaoqian Zhang1, Jian Weng1, Wei Lu2, Xiaochun Li3, Weiqi Luo1 & Xiaoqing Tan3

In blind quantum computation (BQC) protocol, the quantum computability of servers are complicated and powerful, while the clients are not. It is still a challenge for clients to delegate quantum computation to servers and keep the clients’ inputs, outputs and algorithms private. Unfortunately, quantum channel noise is unavoidable in the practical transmission. In this paper, a novel BQC protocol based on maximally entangled Greenberger-Horne-Zeilinger (GHZ) states is proposed which doesn’t need a trusted center. The protocol includes a client and two servers, where the client only needs to own quantum channels with two servers who have full-advantage quantum computers. Two servers perform entanglement concentration used to remove the noise, where the success probability can almost reach 100% in theory. But they learn nothing in the process of concentration because of the no-signaling principle, so this BQC protocol is secure and feasible.

Blind quantum computation (i.e. BQC)1–7 is still a challenging research feld, where a client has not enough quantum computability, and delegates her quantum computing to the servers who have full-advanced quantum computers. In long-distance BQC, quantum entanglement plays an important role and three mainly blind entangled states have already been studied which are blind brickwork state1, blind topological state2 and Afeck-Kennedy-LiebTasaki (i.e. AKLT) state3. Some BQC protocols1, 4–6 are based on the blind brickwork state which is proposed by Broadbent et al.1. Later, Barz et al.7 demonstrated the blindness of the brickwork state. Broadbent et al.1 in 2009 proposed a single-server BQC protocol based on single-qubit states and double-server BQC protocol based on the entanglement swapping of Bell states. However, the quantum entanglement of Bell states in double-server BQC protocol1 will sufer quantum channel loss due to the infuence of noisy channel. To solve this problem, Morimae and Fujii4 proposed a method of entanglement distillation to extract high-fdelity Bell states, meanwhile its security can also be guaranteed. Li et al.5 proposed a triple-server BQC protocol based on Bell states. Sheng and Zhou6 proposed a double-server BQC protocol based on Bell states, where the deterministic entanglement distillation can remove the noise that transforms pure entangled states into mixed entangled states. As we can see that the aims of BQC proto- 1, 4–6 ππ2 7π cols are all to obtain the single-qubit states ±θ with θ ∈…0, ,,, to create the blind brickwork i i {}4 4 4 states1. The other two blind graph states2, 3 can also be used to perform BQC successfully. The Raussendorf-Harrington-Goyal (i.e. RHG) lattice2, which the blindness is guaranteed in a topological manner, is used to perform four quantum measurements {X, Y, Z, T} only known by clients. Compared with the cluster states, AKLT states can be prepared efciently and simply in linear optics with biphotons8. Recently, more and more inter- esting BQC protocols are proposed9–18. In BQC, the quantum channel noise is still an urgent problem. Previous works4, 6, 14 have studied quantum channel noises in BQC protocols. For example, Takeuchi et al.14 proposed three BQC protocols based on decoherence-free subspace (i.e. DFS) to resist the collective noise of quantum channel. Te new BQC protocol is based on maximally GHZ entangled states, where there are three participants (a client Alice, two servers Bob and Charlie). Te BQC protocol is divided into four steps. First, Bob prepares initial GHZ states, remains one photon and sends other two photons to Alice. Alice disturbs the orders of two photons and sends to Charlie. Second, Bob and Charlie perform entanglement concentration to get ideal maximally entangled states, where two identical less-entangled states can be used to concentrate a maximally entangled state by two-step

1Department of Computer Science, Jinan University, Guangzhou, 510632, China. 2School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510006, China. 3Department of Mathematics, Jinan University, Guangzhou, 510632, China. Correspondence and requests for materials should be addressed to J.W. (email: [email protected])

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Figure 1. Schematic diagram of BQC protocol is based on maximally GHZ entangled states with z-basis ({|0〉, −iθi |1〉}) measurement M1, basis 01± e measurement M2, Pauli operations U, Bell measurement BM, Charlie’s measurement outcome ci. Te order of photons sequences Aj and Cj (j ∈…{1,2,3,,n}) are disturbed and rewritten as A ′ and C ′ (t , t ∈…{1,2,3,,n}). Photons A ′ and C ′ belong to Charlie, and B belongs to t2 t1 1 2 t2 t1 j Bob, where A ′ , B and C ′ (t ≠ t ≠ j) belong to a GHZ state. t2 j t1 1 2

parity check and project measurements. Tird, Bob performs Pauli operations on his photons under Alice’s instruc- tion. Then Charlie performs measurement on one photon with the basis {|0〉, |1〉}. Alice randomly chooses ππ7 θi ∈…0, ,, and sends to Charlie. Charlie performs measurement on the other photon and Bob gets the (){}4 4 single-qubit. Finally, Alice and Bob perform single-server BQC protocol. Tis BQC protocol has four contributions. First, two servers can communication with each other without degrading the security. Second, it does not need a trusted center. Te task of preparing initial entangled states can be assigned to Bob. Tird, Bob and Charlie don’t need to exchange their classical information. If they collude, they don’t know any information about Alice’s inputs, outputs and algorithms. Te last one, entanglement concentration can be used to remove the channel noise. Results BQC protocol based on maximally GHZ entangled states. Photons are the best physical systems for the long-distance transmission of entangled states, thus entangled photons states are used as quantum information carriers in BQC. In this BQC protocol, we use |0〉 and |1〉 to express photons. In entanglement concentration, we use |H〉 and |V〉 to express photons, where |H〉 is equal to |0〉 and |V〉 is equal to |1〉. In this section, we propose the BQC protocol based on maximally GHZ photons entangled states GHZ = 1 ( 001 +++010 100 111 ) ABjjCj 2 (j =…1, 2, , n) (Fig. 1). Te cross-Kerr nonlinear can be used to construct a CNOT gate in ref. 19. Tere are also many other methods to realize it19–23. In the BQC protocol, we suppose that these quantum devices are all ideal. Te client owns quantum channels with two servers and quantum disturbing device.

• Bob generates enough maximally GHZ entangled states GHZ , where the subscripts Aj, Bj and Cj repre- ABjjCj sents photons Aj, Bj and Cj. Bob keeps photons sequences SB = [B1, B2, …, Bn] and sends photons sequences SA = [A1, A2, …, An] and SC = [C1, C2, …, Cn] to Alice successively. Afer receiving photons sequences, Alice disturbs the order of photons sequences SA and SC. The reordered photons sequences are rewritten as S A′ = [A ′, A ′, …, A ′] and S C′ = [C ′, C ′, …, C′], meanwhile GHZ is remarked as 1 2 n 1 2 n ABjjCj |GHZt⟩ ′′(,12tn∈ {1,2,, }). Te orders of photons sequences SA′ and SC′ are diferent and only known ABtj21Ct by Alice. Ten Alice sends photons sequences SA′ and SC′ to Charlie. Due to the efect of quantum channel noise, the maximally entangled states GHZ ′′ evolve into less-entangled states ABtj21Ct 2 2 2 2 GHZ′=′′ αβδη001 ++010 100 + 111 , where |α| + |β| + |δ| + |η| = 1. In order to get states ABtj21Ct GHZ ′′, Bob and Charlie frstly perform entanglement concentration. ABtj21Ct • Bob performs one of four operations {I, σx, iσy, σz} randomly chosen by Alice on photons Bj and GHZ ′′ ABtj21Ct states evolve into one of four states |GHZ1 AB′′C , GHZ2 AB′′C , GHZ3 AB′′C , GHZ4 AB′′C . { tj21t tj21t tj21t tj21t }

I 1 →|GHZ1〉=AB′′C (|〉001 +|010〉+|〉100 +|111〉) tj21t 2 1 ++ =|(0ψφ〉|AB′′〉+CA|〉′′BC|〉1), 22 tj21tt21jt σx 1 →|GHZ2〉=AB′′C (|〉011 +|000〉+|〉110 +|101〉) tj21t 2 1 ++ =|(0φψ〉|AB′′〉+CA|〉′′BC|〉1), 22 tj21tt21jt σ i y 1 →|GHZ3〉=AB′′C (−|011〉+|〉000 −|110〉+|〉101 ) tj21t 2 1 −− =|(0φψ〉|AB′′〉−CA|〉′′BC|〉1), 22 tj21tt21jt σz 1 →|GHZ4〉=AB′′C (|〉001 −|010〉+|〉100 −|111〉) tj21t 2 1 −− =−(0|〉ψφAB′′|〉CA+| 〉|′′BC1)〉. 22 tj21tt21jt (1)

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Since the orders of sequences SA′, SB and SC′ are different, both Bob and Charlie cannot know which state GHZ ′′ (u∈{1, 2, 3, 4}) they shared. uAtj21BCt • Charlie performs measurement on photons C ′ using the basis {|0 , |1 } under the guidance of Alice. Alice t1 〉 〉 randomly chooses θπi ∈…{0,/4, 2/ππ4, ,7 /4} and sends to Charlie. Charlie performs measurement on the − θ basis {0|±e i i|1} and Bob obtains photons states | ± , where c (∈{0, 1}) is Charlie’s measurement ⟩⟩ θπii+c ⟩ i outcome. Because the orders of A ′ and B are diferent, Bob can not know anything even if Charlie tells the t2 j value of θi to Bob. n • Alice, Bob and Charlie repeat (1–3) steps such that Bob obtains single-photon states |± successfully. ⨂ θπii+c ⟩ i=1 Te remaining steps are the same as steps (2–3) of the BFK protocol1 or steps (2–5) of blind topological BQC protocol2. Te blindness of graph states and the correctness of quantum computation have already been exhibited in refs 1 and 2 in detail.

In the step 1 of this BQC, entanglement concentration is used to remove the noise. In the following, the process of entanglement concentration is showed with optical system.

Entanglement concentration of pure maximally GHZ entangled state. In a practical transmission, there exist two kinds of quantum channel noises, i.e. pure maximally entangled states evolve into mixed states or less-entangled states. Entanglement purifcation24–28 is applied to extract high-fdelity maximally entangled states from mixed entangled states. Entanglement concentration29–45 is ofen used to distill less-entangled states into pure maximally entangled states by local operations and classical communication (i.e. LOCC). Bennett et al.29 frstly proposed an entanglement concentration protocol by using Schmidt projection. In 2003, Zhao et al.42 not only demonstrated the entanglement concentration scheme in ref. 30 but also verifed a quantum repeater in experiment. Li et al.39 proposed two protocols to concentrate hyper-entangled GHZ states by using a single-photon state of two freedoms and two less-entangled states respectively. Sheng et al.32 proposed to concentrate arbitrary W states by using two steps. Aferwards, a universal concentration scheme of an arbitrary less-entangled N-photon W state is proposed in ref. 43. Here, we consider a special quantum channel noise, i.e. pure maximally entangled states evolve into less-entangled states, which can be distilled by entanglement concentration. In the following, we give the entanglement concentration of GHZ states that were experimentally prepared in refs 46–48.

Te frst round of entanglement concentration. In the BQC, the maximally GHZ states can be rewritten in the form of 1 GHZ abc = ()HHV +++HVHVHH VVV , 111 2 (2) where we defne |H = |0 and |V = |1 . Te subscripts a , b and c represent the spatial-mode of photons A ′ , B 〉 〉 〉 〉 1 1 1 t2 j and C ′. We consider the noisy model that pure maximally entangled states evolve pure less entangled states. t1 Suppose less-entangled pure photons states are GHZ′=αβHHV ++HVHVδηHH + VVV , ab11c1 (3) where four real numbers α, β, δ, η satisfy |α|2 + |β|2 + |δ|2 + |η|2 = 1. Two identical less-entangled states, which the parameters are all unknown, can distill a maximally entangled state in Eq. (2). Te schematic of entanglement concentration is shown in Fig. 2. Here, only Alice knows whether ′ ′ entanglement concentration is successful and the correct orders of A , B and C . t2 j t1 Afer passing HWP , the state GHZ′ evolves to 90° ab11c1 GHZ′=αβVVH ++VHVHδηVV + HHH , ab22c2 (4)

where polarization photons a1, b1 and c1 are fipped and relabeled as a2, b2 and c2. Te entanglement concentration is divided into two steps. In the frst step, the system composed of six photons is

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Figure 2. Te schematic diagram of polarization-entanglement concentration. Te sources is used to produce polarization-entangled states. Photons a1(a2) and c1(c2) belong to Charlie, where Bob retains photons b1(b2). HWP is half-wave plate which HWP90° fips the horizontal and vertical polarization states. HWP45° just likes a Hadamard operation to rotate horizontal and vertical polarization states. Te polarizing beam splitters (PBSs) are used to transmit horizontal polarization |H〉 and refect vertical polarization |V〉. QNDi (with i = 1, 2, 3) represents quantum nondemolition detections. Detectors D and D belong to Bob, D , D , D and D B1 B2 A1 A2 C1 C2 belong to Charlie.

Figure 3. Schematic diagram of QND49. ±θ = χt represents the cross-Kerr nonlinearity media that introduces the phase shif θ when photons pass through the media. |χ〉 〈χ| is homodyne measurement that can distinguish diferent phase shifs. Te signal photons |α1〉, |α2〉 and |α3〉 are related to a1 and a2, b1 and b2, c1 and c2 respectively. Here x1 and x2 can be specifcally expressed as a1 and a2 (b1 and b2, c1 and c2).

Afer both a1 and a2 (b1 and b2, c1 and c2) pass parity check device (Fig. 3), Bob and Charlie can get some specifc quantum state by choosing phase shifs. Here, we suppose that Bob and Charlie are honest to perform the entanglement concentration. Te concrete process of the parity check device is given in Methods. For b1 and b2, a1 and a2, c1 and c2, if Bob and Charlie all choose ±2θ phase shifs of odd-parity check states, the state is

ϕα=+22HHV VVH β HVHVHV 1 ab11ca12bc22 22 ++δηVHHHVV VVV HHH (6) with the probability p14=+αβ444++δη, where pm represents the probability of obtaining ϕ 11 vj 1 ab11ca12bc22 with the number of rounds v(1vk=…,2,3,,), the number of steps j (j = 1, 2) in vth round and the quantum state m (m = 1, 2, 3, 4) in jth step of vth round. If Bob chooses 0 phase shift of even-parity check states for b1 and b2, Charlie chooses 0 phase shift of even-parity check states for c1 and c2, and ±2θ phase shif of odd-parity check states for a1 and a2, the state is |〉ϕα=|β()HHV〉|VHVH〉+|〉VH |〉VVH 2 ab11ca12bc22 +|δη()VHH〉|HHH〉+|〉VVV |〉HVV (7) 22=+αβ22δη2 with the probability p11 2( ). If Bob chooses ±2θ phase shif of odd-parity check states for b1 and b2, Charlie chooses 0 phase shifs of even-parity check states for a1 and a2, c1 and c2, the state is |〉ϕα=|δ()HHV〉|HVVV〉+|〉HH |〉VVH 3 ab11ca12bc22 +|βη()HVH〉|HHH〉+|〉VVV |〉VHV (8) 32=+αδ22βη2 with the probability p11 2( ). If Bob chooses 0 phase shif of even-parity check state for b1 and b2, Charlie chooses ±2θ phase shif of odd-parity check states for c1 and c2, and 0 phase shif of even-parity check states for a1 and a2, the state is

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(1) 1 2222 ϕα11 =+HHV βδHVHV++HH η VVV , ab11c1 (11) m ()γ =… where ϕvj represents the quantum state with the number of rounds v(1vk,2,3,,), the number of ab11c1 steps j (j = 1, 2) in vth round, the quantum state m (m = 1, 2, 3, 4) in jth step of vth round, and the quantum state (γ) (γ = 1, 2, 3, 4) of PBSs measurement for the states ϕ (ε = 1, 2, 3, 4). ε ab11ca12bc22 If the detectors D , D , D (or D , D , D ) are triggered, we get A1 B1 C2 A2 B2 C1

(2) 1 2222 ϕα11 =−HHV βδHVHV−+HH η VVV . ab11c1 (12)

B A (2) 1 Bob and Charlie perform unitary transformation σσz ⊗ z on photons a1 and b1 of state ϕ11 to get ab11c1 (1) 1 ϕ11 . ab11c1 If the detectors D , D , D (or D , D , D ) are triggered, we will get A1 B2 C1 A2 B1 C2

(3) 1 2222 ϕα11 =− HHV +−βδHVHVHH +.η VVV ab11c1 (13)

A C (3) 1 (1) 1 Charlie performs unitary transformation σσz ⊗ z on photons a1 and c1 of state ϕ11 to get ϕ11 . ab11c1 ab11c1 If the detectors D , D , D (or D , D , D ) are triggered, we will get A1 B2 C2 A2 B1 C1

(4) 1 2222 ϕα11 =− HHV −+βδHVHVHH +.η VVV ab11c1 (14)

(1) 1 (1) 2 (1) 3 (1) 4 The four quantum states ϕ11 , ϕ11 , ϕ11 and ϕ11 are not destroyed by quantum ab11c1 ab11c1 ab11c1 ab11c1 non-demolition detections. Tey are used as the initial states in the second step of the frst round and rewritten as ϕ 1 , ϕ 2 , ϕ 3 and ϕ 4 . 12 ab11c1 12 ab11c1 12 ab11c1 12 ab11c1 In the second step, for quantum state

photons are all fipped by HWP90° and relabeled as a2, b2 and c2. We will get

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Afer parity checks and PBSs measurement, we obtain four quantum states

(1) 1 (2) 1 (3) 1 (4) 1 Te probabilities of getting quantum states ϕ12 , ϕ12 , ϕ12 and ϕ12 are ab11c1 ab11c1 ab11c1 ab11c1 αβ8 ++888δη+ p1 = , 12 ()αβ4 ++444δη+ 2 2(αβ44+ δη44) p 2 = , 12 ()αβ4 ++444δη+ 2 2(αδ44+ βη44) p 3 = , 12 ()αβ4 ++444δη+ 2 2(αη44+ βδ44) 4 = . p12 4 4442 ()αβ++δη+ (19) Tese are all failed cases, but they can be used as the initial states in the second round. For quantum state

2 αβ δη |〉ϕ12 abc =|()HHV〉+|〉HVHV+|()HH〉+|〉VVV , 111 2(αβ22++δη22)2()αβ22 δη22 (20) its process of concentration is the same as ϕ 1 and we can get 12 ab11c1 αβδη |〉ϕ (1)2 =|()HHV〉+|〉HVHV+| HH〉+|〉VVV . 12 ab11ca12bc22 2(αβ22+ δη22) (21) Tis is the maximally GHZ entangled state. Te success and failure probabilities of ϕ 2 are 12 ab11c1 2(αβδη)2 αβ44+ δη44 pp2 = , 2 = , 12,sf()αβ22+ δη222 12, ()αβ22+ δη222 (22) where the subscripts s and f represent the success and failure probabilities respectively. For quantum states

=++2 2 3 3 4 4 Pp1 11pp12,sss11pp12, 11p12, 4(αβδη)42 ()αβδη 2 4(αβδη)2 = + + . 22 22 22 22 22 22 αβ + δη αδ + βη αη + βδ (25)

Discussion Blindness and correctness analysis of the proposed BQC protocol. In the following, we will show that the proposed BQC protocol is secure by analyzing the blindness and correctness. First, we show the blindness of the proposed BQC protocol.

(1) Bob performs one of four Pauli operations randomly chosen by Alice on his photons and the initial states 1 |GHZ〉=′′ (|〉001 +|010〉+|〉100 +|111〉) are correspondingly changed into one of ABtj21Ct 2

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{ GHZ ′′, GHZ ′′, GHZ ′′, GHZ ′′}. Whether Bob colludes with Charlie or not, 1 ABtj21Ct 2 ABtj21Ct 3 ABtj21Ct 4 ABtj21Ct they guess the correct Bell state with the probability of 1 . When this BQC protocol is repeated n 4 n times, the probability of obtaining correct quantum states is lim01 = . n→∞()4 ππ2 3ππ7 (2) Alice randomly chooses the phase θi ∈…0, ,,,, and disturbs the order of photons Aj, Bj, Cj. (){}4 4 4 4 Bob and Charlie know nothing about the states | ±〉 because of the no-signaling principle. Afer θi 1 n repeating n times, the probability of guessing correct θi is lim0= . In the process of entanglement n→∞()8 concentration, Bob and Charlie cannot eavesdropping any useful information by exchanging their results because of diference of orders of three photons. (3) Te structures of blind brickwork states and blind topological states are private for servers. Terefore, Bob and Charlie can’t obtain anything about Alice’s private information whether they communicate with each other or not. Te blindness of BFK single-server protocol and blind topological single-server protocol are showed in refs 1 and 2 in detail respectively.

Second, the correctness of quantum computation in BFK single-server protocol and blind topological single-server protocol are presented in refs 1 and 2 in detail. So this BQC protocol is blind and correct.

Analysis of the success probabilities in iteration. In the above discussion, we have already elaborated the frst round of the entanglement concentration with cross-Kerr nonlinearity in detail. QND provides a strong tool for us to perform a quantum nondemolition measurement that does not destroy entanglement of photons, which ensures that each step can be operated independently. Here, we analyse the second round and the k-th round of entanglement concentration. 2 3 4 For the three cases ϕ21 abc , ϕ21 abc and ϕ21 abc , only the frst step is needed to concentrate the ideal max- 111 111 111 1 imally entangled states GHZ ′′. However, we need to implement two steps for the state ϕ21 abc . We con- ABtj21Ct 111 sider the three states ϕ 2 , ϕ 3 and ϕ 4 frst. 21 ab11c1 21 ab11c1 21 ab11c1 In the second round, for the quantum states

2(αβδη)4 αβ88+ δη88 pp2 = , 2 = . 21,sf()αβ44+ δη442 21, ()αβ44+ δη442 (27) In the k-th (k > 1) round, the success and failure probabilities are

k kk++11 kk++11 2(αβδη)2 αβ22+ δη22 2 = 2 = . ppks1, kk kk, kf1, kk kk ()αβ22+ δη222 ()αβ22+ δη222 (28) For the quantum states

the analyses of entanglement concentration are the same as the Eq. (23), the success and failure probabilities in the second round and the k-th round are

k 2(αβδη)4 2(αβδη)2 3 = 3 = pp21,sk44 4421, ,s kk kk, ()αδ + βη ()αδ22+ βη222 kk++11 kk++11 αδ88+ βη88 αδ22+ βη22 3 = 3 = pp21,fk44 4421, ,f kk kk , ()αδ + βη ()αδ22+ βη222 k 2(αβδη)4 2(αβδη)2 4 = 4 = pp21,sk44 4421, ,s kk kk, ()αη + βδ ()αη22+ βδ222 kk++11 kk++11 αη88+ βδ88 αη22+ βδ22 pp4 = , 4 = . 21,fk44 4421,f 22kk 22kk2 ()αη + βδ ()αη + βδ (29) For the quantum states

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|〉ϕα1 =+HHV βδHVHV++HH η VVV , kj ab11ck1 jkjkj kj (30) (where j = 1, 2) we give the relevant normalized coefcients and the probabilities of relevant quantum states. Te iterative process is the same as the Eq. (3). In the frst step of the k-th round, for the quantum states

ϕα1 =+HHV βδHVHV++HH η VVV , ka1 11bc1 kk11k1 k1 (31) where k > 1 and the coefcients are

22k− α2 α = , k1 21kk−−21 21kk−−21 αβ22++δη22+ 22k− β2 β = , k1 21kk−−21 21kk−−21 αβ22++δη22+ 22k− δ2 δ = , k1 21kk−−21 21kk−−21 αβ22++δη22+ 22k− η2 η = . k1 −−−− 2221kk21 2221kk21 αβ++δη+ (32) In the second step of the k-th round, for the quantum states

ϕα1 =+HHV βδHVHV++HH η VVV , ka2 11bc1 kk22k2 k2 (33) where the coefcients are

21k− α2 α = , k2 22kk22kk αβ22++δη22+ 21k− β2 β = , k2 22kk 2k αβ22++δη22k + 21k− δ2 δ = , k2 22kk22kk αβ22++δη22+ 21k− η2 η = . k2 2222kk2222kk αβ++δη+ (34) Te probabilities of obtaining four quantum states in the frst step or the second step of the k-th round are

14=+αβ44++δη4 pkj kj kj kj kj , 22=+αβ22δη2 pkj 2( kj kj kj kj), 32=+αδ22βη2 pkj 2( kj kj kj kj), p 42=+2(αη22βδ2), kj kj kj kj kj (35) where j = 1, 2. Te success probability of the kth round is

= 2 2 2 2 2 2 Ppk 11pp12,fff21, pp31,  (1kf− )1, pks1, + 3 3 3 3 3 pp11 12,ffpp21,  (1kf− )1, pks1, ++4 4 4 4 4 pp11 12,ffpp21,  (1kf− )1, pks1, + 1 2 2 2 2 2 pp11( 12pp21,ff31, pp(1kf− )1, ks1, + 3 3 3 3 3 pp12 21,ffpp31,  (1kf− )1, pkf1, + 4 4 4 4 4 pp12 21,ffpp31,  (1kf− )1, pkf1, ) ++1 1 1 1 1 1 1 1 2 2 ++3 3 4 4 . pp11 12pp21 22pp31 32 pp(1kk−−)1 (1)2()ppkk1 2,skpp1 ks2, ppkk1 2,s (36)

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Figure 4. Te success probability P of getting maximally entangled GHZ state relies on the initial coefcients β and δ. Here, we let α = 1 , β ∈ {0,}3 , δ ∈−{0,}3 β 2 , η =−3 βδ22− . n (n = 1, 2, 3, 4) 2 2 4 4 represents the number of iterations.

n Te total probability is PPtotal = ∑k=1 k, which depends on the number of iterations and parameters of the initial states. Te relationship of the total success probability, parameters and the number of iterations is shown in Fig. 4. It can be seen that the total success probability has kept increasing with the parameters β and δ in the  3  range of 0, . When n = 4, the success probability has already reached 0.9196. When n = 9, the success prob-  2  ability has already reached 0.9975. Terefore, the entanglement concentration is successful in theory. In this paper, we only consider the ideal CNOT gate19–23. In experiment, there exist many nonideal factors such as the double efect of parameter conversion, the imperfect matching of the crystal lattice and phases, and so on. Te probabilities of intrinsic error of experimental methods are unavoidable, such as QND measurements and CNOT operations. Tus optimizing the experimental system is a very meaningful research direction. In the BQC protocol, we only give the concrete quantum channel noise model but not universal. So, we will further study entanglement purifcation of GHZ states. Methods Te optical devices are used to complete the entanglement concentration, where the parity check devices are based on cross-Kerr nonlinearity that can construct QND38, 39, 41 to improve the successful probability. Te cross-Kerr nonlinearity medium is described by the Hamiltonian,

†† Ha= χ ssaappa (37)

† † where as and ap are the creation operators, as and ap are the annihilation operators, a Fock state |n〉 and a coherent state |αc〉 interact. Te whole system evolves into

iθ Ut()ncααcc=+0101ceαc (38)

− θ ++ where U()te= iassaappa , θ = χt is the phase shif and t is the interaction time (c = 1, 2, 3). θ is proportional to the number of photons in the signal state |αc〉. X-quadrature measurement can recognize the phase shif of signal states |αc〉. Te cross-Kerr nonlinearity can measure the number of photons but do not destroy the photons. For the parity check device in Fig. 3, we give an example. Two polarization photons are initially prepared with the τµ=+µ τλ=+λ α = forms of k 01HV and k 01HV that interact with a coherent beam | c〉 (c 1, 2, 3),where 1 2 2 2 2 2 real numbers μ0, μ1, λ0 and λ1 satisfy the normalization condition |μ0| + |μ1| = 1, |λ0| + |λ1| = 1, respectively. Ten the composite quantum system ϒ=ττ⊗⊗α evolves to 1 kk12c

SCientifiC REPOrTs | 7:11104 | DOI:10.1038/s41598-017-06777-w 9 www.nature.com/scientificreports/

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