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Sci Rep. 2017; 7: 11104.
Published online 2017 September 11. doi:  10.1038/s41598-017-06777-w
PMCID: PMC5594006

Greenberger-Horne-Zeilinger states-based blind quantum computation with entanglement concentration

Abstract

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.

Introduction

Blind quantum computation (i.e. BQC)17 is still a challenging research field, 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 Affleck-Kennedy-LiebTasaki (i.e. AKLT) state3. Some BQC protocols1, 46 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 suffer quantum channel loss due to the influence of noisy channel. To solve this problem, Morimae and Fujii4 proposed a method of entanglement distillation to extract high-fidelity 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 protocols1, 46 are all to obtain the single-qubit states θi with θi{0,π4,2π4,,7π4} to create the blind brickwork 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 efficiently and simply in linear optics with biphotons8. Recently, more and more interesting BQC protocols are proposed918. 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.

The new BQC protocol is based on maximally GHZ entangled states, where there are three participants (a client Alice, two servers Bob and Charlie). The 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 parity check and project measurements. Third, Bob performs Pauli operations on his photons under Alice’s instruction. Then Charlie performs measurement on one photon with the basis {|0right angle bracket, |1right angle bracket}. Alice randomly chooses θ i ({0,π4,,7π4}) and sends to Charlie. Charlie performs measurement on the other photon and Bob gets the single-qubit. Finally, Alice and Bob perform single-server BQC protocol.

This 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. The task of preparing initial entangled states can be assigned to Bob. Third, 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. The 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 |0right angle bracket and |1right angle bracket to express photons. In entanglement concentration, we use |Hright angle bracket and |Vright angle bracket to express photons, where |Hright angle bracket is equal to |0right angle bracket and |Vright angle bracket is equal to |1right angle bracket. In this section, we propose the BQC protocol based on maximally GHZ photons entangled states |GHZAjBjCj = ½(|001〉 + |010〉 + |100〉 + |111〉) (j = 1, 2, n) (Fig. 1). The cross-Kerr nonlinear can be used to construct a CNOT gate in ref. 19. There are also many other methods to realize it1923. In the BQC protocol, we suppose that these quantum devices are all ideal. The client owns quantum channels with two servers and quantum disturbing device.

  • Bob generates enough maximally GHZ entangled states |GHZAjBjCj, where the subscripts A j, B j and C j represents photons A j, B j and C j. Bob keeps photons sequences S B = [B 1, B 2, …, B n] and sends photons sequences S A = [A 1, A 2, …, A n] and S C = [C 1, C 2, …, C n] to Alice successively. After receiving photons sequences, Alice disturbs the order of photons sequences S A and S C. The reordered photons sequences are rewritten as S A = [A1, A2, …, An] and S C = [C1, C2, …, Cn], meanwhile |GHZAjBjCj is remarked as |GHZAt2BjCt1(t1,t2{1,2,,n}). The orders of photons sequences S A and S C are different and only known by Alice. Then Alice sends photons sequences S A and S C to Charlie. Due to the effect of quantum channel noise, the maximally entangled states |GHZAt2BjCt1 evolve into less-entangled states |GHZAt2BjCt1=α|001+β|010+δ|100+η|111, where |α|2 + |β|2 + |δ|2 + |η|2 = 1. In order to get states |GHZAt2BjCt1, Bob and Charlie firstly perform entanglement concentration.
  • Bob performs one of four operations {I, σ x, y, σ z} randomly chosen by Alice on photons B j and |GHZAt2BjCt1 states evolve into one of four states {|GHZ1At2BjCt1, |GHZ2At2BjCt1, |GHZ3At2BjCt1, |GHZ4At2BjCt1}.

I|GHZ1At2BjCt1=12(|001+|010+|100+|111)=122(|ψ+At2Bj|0Ct1+|φ+At2Bj|1Ct1),σx|GHZ2At2BjCt1=12(|011+|000+|110+|101)=122(|φ+At2Bj|0Ct1+|ψ+At2Bj|1Ct1),iσy|GHZ3At2BjCt1=12(|011+|000|110+|101)=122(|φAt2Bj|0Ct1|ψAt2Bj|1Ct1),σz|GHZ4At2BjCt1=12(|001|010+|100|111)=122(|ψAt2Bj|0Ct1+|φAt2Bj|1Ct1).
1

Since the orders of sequences S A, S B and S C are different, both Bob and Charlie cannot know which state |GHZuAt2BjCt1 (u[set membership]{1, 2, 3, 4}) they shared.

  • Charlie performs measurement on photons Ct1 using the basis {|0right angle bracket, |1right angle bracket} under the guidance of Alice. Alice randomly chooses θi{0,π/4,2π/4,,7π/4} and sends to Charlie. Charlie performs measurement on the basis {|0⟩±eiθi|1⟩} and Bob obtains photons states θi+ciπ, where c i ([set membership]{0, 1}) is Charlie’s measurement outcome. Because the orders of At2 and B j are different, Bob can not know anything even if Charlie tells the value of θ i to Bob.
  • Alice, Bob and Charlie repeat (1–3) steps such that Bob obtains single-photon states i=1n|±θi+ciπ successfully. The remaining steps are the same as steps (2–3) of the BFK protocol1 or steps (2–5) of blind topological BQC protocol2. The blindness of graph states and the correctness of quantum computation have already been exhibited in refs 1 and 2 in detail.

Figure 1
Schematic diagram of BQC protocol is based on maximally GHZ entangled states with z-basis ({|0right angle bracket, |1right angle bracket}) measurement M 1, basis |0〉 ± eiθi|1〉 measurement M 2, Pauli operations U, Bell measurement BM, Charlie’s measurement outcome ...

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 purification2428 is applied to extract high-fidelity maximally entangled states from mixed entangled states. Entanglement concentration2945 is often used to distill less-entangled states into pure maximally entangled states by local operations and classical communication (i.e. LOCC). Bennett et al.29 firstly 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 verified 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. Afterwards, 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 4648.

The first round of entanglement concentration

In the BQC, the maximally GHZ states can be rewritten in the form of

|GHZa1b1c1 = ½(|HHV〉 + |HVH〉 + |VHH〉 + |VVV〉), 
2

where we define |Hright angle bracket = |0right angle bracket and |Vright angle bracket = |1right angle bracket. The subscripts a 1, b 1 and c 1 represent the spatial-mode of photons At2, B j and Ct1. We consider the noisy model that pure maximally entangled states evolve pure less entangled states. Suppose less-entangled pure photons states are

|GHZ′〉a1b1c1α|HHV〉 + β|HVH〉 + δ|VHH〉 + η|VVV〉, 
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). The schematic of entanglement concentration is shown in Fig. 2. Here, only Alice knows whether entanglement concentration is successful and the correct orders of At2, B j and Ct1.

Figure 2
The schematic diagram of polarization-entanglement concentration. The sources is used to produce polarization-entangled states. Photons a 1(a 2) and c 1(c 2) belong to Charlie, where Bob retains photons b 1(b 2). HWP is half-wave plate which HWP 90° ...

After passing HWP 90°, the state |GHZ′〉a1b1c1 evolves to

|GHZ′〉a2b2c2α|VVH〉 + β|VHV〉 + δ|HVV〉 + η|HHH〉, 
4

where polarization photons a 1, b 1 and c 1 are flipped and relabeled as a 2, b 2 and c 2.

The entanglement concentration is divided into two steps. In the first step, the system composed of six photons is

|Ψa1b1c1a2b2c2=|GHZa1b1c1|GHZa2b2c2=[α2|HHVVVH+β2|HVHVHV+δ2|VHHHVV+η2|VVVHHH]+[αβ(|HHVVHV+|HVHVVH)+δη(|HHVHVV+|VHHVVH)]+[αδ(|HVHHHH+|VVVVHV)+βη(|VHHHHH+|VVVHVV)]+[αη|HHVHHH+|VVVVVH+βδ(|HVHHVV+|VHHVHV)]
5

After both a 1 and a 2 (b 1 and b 2, c 1 and c 2) pass parity check device (Fig. 3), Bob and Charlie can get some specific quantum state by choosing phase shifts. Here, we suppose that Bob and Charlie are honest to perform the entanglement concentration. The concrete process of the parity check device is given in Methods.

Figure 3
Schematic diagram of QND49. ±θ = χt represents the cross-Kerr nonlinearity media that introduces the phase shift θ when photons pass through the media. |χright angle bracket left angle bracketχ| is ...

For b 1 and b 2, a 1 and a 2, c 1 and c 2, if Bob and Charlie all choose ±2θ phase shifts of odd-parity check states, the state is

|ϕ1a1b1c1a2b2c2=α2|HHV|VVH+β2|HVH|VHV+δ2|VHH|HVV+η2|VVV|HHH
6

with the probability p111=α4+β4+δ4+η4, where pvjm represents the probability of obtaining |ϕ1a1b1c1a2b2c2 with the number of rounds v (v = 1, 2, 3, k), 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 b 1 and b 2, Charlie chooses 0 phase shift of even-parity check states for c 1 and c 2, and ±2θ phase shift of odd-parity check states for a 1 and a 2, the state is

|ϕ2a1b1c1a2b2c2=αβ(|HHV|VHV+|HVH|VVH)+δη(|VHH|HHH+|VVV|HVV)
7

with the probability p112=2(α2β2+δ2η2).

If Bob chooses ±2θ phase shift of odd-parity check states for b 1 and b 2, Charlie chooses 0 phase shifts of even-parity check states for a 1 and a 2, c 1 and c 2, the state is

|ϕ3a1b1c1a2b2c2=αδ(|HHV|HVV+|VHH|VVH)+βη(|HVH|HHH+|VVV|VHV)
8

with the probability p113=2(α2δ2+β2η2).

If Bob chooses 0 phase shift of even-parity check state for b 1 and b 2, Charlie chooses ±2θ phase shift of odd-parity check states for c 1 and c 2, and 0 phase shift of even-parity check states for a 1 and a 2, the state is

|ϕ4a1b1c1a2b2c2=αη(|HHV|HHH+|VVV|VVH)+βδ(|HVH|HVV+|VHH|VHV)
9

with the probability p114=2(α2η2+β2δ2).

We give an example for PBSs measurement. After passing through HWP 45°, |ϕ1a1b1c1a2b2c2 evolves into

(α2|HHV+β2|HVH+δ2|VHH+η2|VVV)a1b1c1×(|HHH+|VVV)a2b2c2+(α2|HHVβ2|HVHδ2|VHH+η2|VVV)a1b1c1×(|HHV+|VVH)a2b2c2+(α2|HHV+β2|HVHδ2|VHH+η2|VVV)a1b1c1×(|HVH+|VHV)a2b2c2+(α2|HHVβ2|HVH+δ2|VHH+η2|VVV)a1b1c1×(|HVV+|VHH)a2b2c2.
10

If the detectors DA1, DB1, DC1 (or DA2, DB2, DC2) are triggered, we will get

|ϕ11(1)a1b1c11=α2|HHV+β2|HVH+δ2|VHH+η2|VVV,
11

where |ϕvj(γ)a1b1c1m represents the quantum state with the number of rounds v (v = 1, 2, 3, k), the number of 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 |ϕεa1b1c1a2b2c2 (ε = 1, 2, 3, 4).

If the detectors DA1, DB1, DC2 (or DA2, DB2, DC1) are triggered, we get

|ϕ11(2)a1b1c11=α2|HHVβ2|HVHδ2|VHH+η2|VVV.
12

Bob and Charlie perform unitary transformation σzBσzA on photons a 1 and b 1 of state |ϕ11(2)a1b1c11 to get |ϕ11(1)a1b1c11.

If the detectors DA1, DB2, DC1 (or DA2, DB1, DC2) are triggered, we will get

|ϕ11(3)a1b1c11=α2|HHV+β2|HVHδ2|VHH+η2|VVV.
13

Charlie performs unitary transformation σzAσzC on photons a 1 and c 1 of state |ϕ11(3)a1b1c11 to get |ϕ11(1)a1b1c11.

If the detectors DA1, DB2, DC2 (or DA2, DB1, DC1) are triggered, we will get

|ϕ11(4)a1b1c11=α2|HHVβ2|HVH+δ2|VHH+η2|VVV.
14

Bob and Charlie perform unitary transformation σzBσzC on photons b 1 and c 1 of state |ϕ11(4)a1b1c11 to get |ϕ11(1)a1b1c11.

For the three states |ϕ2a1b1c1a2b2c2, |ϕ3a1b1c1a2b2c2 and |ϕ4a1b1c1a2b2c2, we have the similar results

|ϕ11(1)a1b1c12=αβ(|HHV+|HVH)+δη(|VHH+|VVV),|ϕ11(1)a1b1c13=αδ(|HHV+|VHH)+βη(|HVH+|VVV),|ϕ11(1)a1b1c14=αη(|HHV+|VVV)+βδ(|HVH+|VHH).
15

The four quantum states |ϕ11(1)a1b1c11, |ϕ11(1)a1b1c12, |ϕ11(1)a1b1c13 and |ϕ11(1)a1b1c14 are not destroyed by quantum non-demolition detections. They are used as the initial states in the second step of the first round and rewritten as |ϕ12a1b1c11, |ϕ12a1b1c12, |ϕ12a1b1c13 and |ϕ12a1b1c14.

In the second step, for quantum state

|ϕ12a1b1c11=1α4+β4+δ4+η4(α2|HHV+β2|HVH+δ2|VHH+η2|VVV),
16

photons are all flipped by HWP 90° and relabeled as a 2, b 2 and c 2. We will get

|ϕ12a2b2c21=1α4+β4+δ4+η4(α2|VVH+β2|VHV+δ2|HVV+η2|HHH).
17

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

|ϕ12(1)a1b1c11=1α4+β4+δ4+η4[α4|HHV+β4|HVH+δ4|VHH+η4|VVV],|ϕ12(2)a1b1c11=1α4+β4+δ4+η4[α2β2(|HHV+|HVH)+δ2η2(|VHH+|VVV)],|ϕ12(3)a1b1c11=1α4+β4+δ4+η4[α2δ2(|HHV+|VHH)+β2η2(|HVH+|VVV)],|ϕ12(4)a1b1c11=1α4+β4+δ4+η4[α2η2(|HHV+|VVV)+β2δ2(|HVH+|VHH)].
18

The probabilities of getting quantum states |ϕ12(1)a1b1c11, |ϕ12(2)a1b1c11, |ϕ12(3)a1b1c11 and |ϕ12(4)a1b1c11 are

p121=α8+β8+δ8+η8(α4+β4+δ4+η4)2,p122=2(α4β4+δ4η4)(α4+β4+δ4+η4)2,p123=2(α4δ4+β4η4)(α4+β4+δ4+η4)2,p124=2(α4η4+β4δ4)(α4+β4+δ4+η4)2.
19

These are all failed cases, but they can be used as the initial states in the second round.

For quantum state

|ϕ12a1b1c12=αβ2(α2β2+δ2η2)(|HHV+|HVH)+δη2(α2β2+δ2η2)(|VHH+|VVV),
20

its process of concentration is the same as |ϕ12a1b1c11 and we can get

|ϕ12(1)a1b1c1a2b2c22=αβδη2(α2β2+δ2η2)(|HHV+|HVH+|VHH+|VVV).
21

This is the maximally GHZ entangled state. The success and failure probabilities of |ϕ12a1b1c12 are

p12,s2=2(αβδη)2(α2β2+δ2η2)2,p12,f2=α4β4+δ4η4(α2β2+δ2η2)2,
22

where the subscripts s and f represent the success and failure probabilities respectively.

For quantum states

|ϕ12a1b1c13=αδ2(α2δ2+β2η2)(|HHV+|VHH)+βη2(α2δ2+β2η2)(|HVH+|VVV),|ϕ12a1b1c14=αη2(α2η2+β2δ2)(|HHV+|VVV)+βδ2(α2η2+β2δ2)(|HVH+|VHH),
23

the success and failure probabilities of |ϕ12a1b1c13 and |ϕ12a1b1c14 are respectively

p12,s3=2(αβδη)2(α2δ2+β2η2)2,p12,f3=α4δ4+β4η4(α2δ2+β2η2)2,p12,s4=2(αβδη)2(α2η2+β2δ2)2,p12,f4=α4η4+β4δ4(α2η2+β2δ2)2.
24

The total success probability of the first round is

P1=p112p12,s2+p113p12,s3+p114p12,s4=4(αβδη)2α2β2+δ2η2+4(αβδη)2α2δ2+β2η2+4(αβδη)2α2η2+β2δ2.
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 |GHZAt2BjCt1=12(|001+|010+|100+|111) are correspondingly changed into one of {|GHZ1At2BjCt1, |GHZ2At2BjCt1, |GHZ3At2BjCt1, |GHZ4At2BjCt1}. Whether Bob colludes with Charlie or not, they guess the correct Bell state with the probability of ¼. When this BQC protocol is repeated n times, the probability of obtaining correct quantum states is limn(14)n=0.
  2. Alice randomly chooses the phase θ i ({0,π4,2π4,3π4,,7π4}) and disturbs the order of photons A j, B j, C j. Bob and Charlie know nothing about the states θi because of the no-signaling principle. After repeating n times, the probability of guessing correct θ i is limn(18)n=0. In the process of entanglement concentration, Bob and Charlie cannot eavesdropping any useful information by exchanging their results because of difference of orders of three photons.
  3. The structures of blind brickwork states and blind topological states are private for servers. Therefore, Bob and Charlie can’t obtain anything about Alice’s private information whether they communicate with each other or not. The 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 first 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.

For the three cases |ϕ21a1b1c12, |ϕ21a1b1c13 and |ϕ21a1b1c14, only the first step is needed to concentrate the ideal maximally entangled states |GHZAt2BjCt1. However, we need to implement two steps for the state |ϕ21a1b1c11. We consider the three states |ϕ21a1b1c12, |ϕ21a1b1c13 and |ϕ21a1b1c14 first.

In the second round, for the quantum states

|ϕ21a1b1c12=α2β22(α4β4+δ4η4)(|HHV+|HVH)+δ2η22(α2β4+δ4η4)(|VHH+|VVV),
26

its analysis is the same as the Eq. (20). The success and failure probabilities are

p21,s2=2(αβδη)4(α4β4+δ4η4)2,p21,f2=α8β8+δ8η8(α4β4+δ4η4)2.
27

In the k-th (k > 1) round, the success and failure probabilities are

pk1,s2=2(αβδη)2k(α2kβ2k+δ2kη2k)2,pk1,f2=α2k+1β2k+1+δ2k+1η2k+1(α2kβ2k+δ2kη2k)2.
28

For the quantum states

|ϕ21a1b1c13=α2δ22(α4δ4+β4η4)(|HHV+|VHH)+β2η22(α4δ4+β4η4)(|HVH+|VVV),|ϕ21a1b1c14=α2η22(α4η4+β4δ4)(|HHV+|VVV)+β2δ22(α4η4+β4δ4)(|HVH+|VHH),

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

p21,s3=2(αβδη)4(α4δ4+β4η4)2,pk1,s3=2(αβδη)2k(α2kδ2k+β2kη2k)2,p21,f3=α8δ8+β8η8(α4δ4+β4η4)2,pk1,f3=α2k+1δ2k+1+β2k+1η2k+1(α2kδ2k+β2kη2k)2,p21,s4=2(αβδη)4(α4η4+β4δ4)2,pk1,s4=2(αβδη)2k(α2kη2k+β2kδ2k)2,p21,f4=α8η8+β8δ8(α4η4+β4δ4)2,pk1,f4=α2k+1η2k+1+β2k+1δ2k+1(α2kη2k+β2kδ2k)2.
29

For the quantum states

|ϕkja1b1c11=αkj|HHV+βkj|HVH+δkj|VHH+ηkj|VVV,
30

(where j = 1, 2) we give the relevant normalized coefficients and the probabilities of relevant quantum states. The iterative process is the same as the Eq. (3).

In the first step of the k-th round, for the quantum states

|ϕk1a1b1c11=αk1|HHV+βk1|HVH+δk1|VHH+ηk1|VVV,
31

where k > 1 and the coefficients are

αk1=α22k2α22k1+β22k1+δ22k1+η22k1,βk1=β22k2α22k1+β22k1+δ22k1+η22k1,δk1=δ22k2α22k1+β22k1+δ22k1+η22k1,ηk1=η22k2α22k1+β22k1+δ22k1+η22k1.
32

In the second step of the k-th round, for the quantum states

|ϕk2a1b1c11=αk2|HHV+βk2|HVH+δk2|VHH+ηk2|VVV,
33

where the coefficients are

αk2=α22k1α22k+β22k+δ22k+η22k,βk2=β22k1α22k+β22k+δ2k+η22k,δk2=δ22k1α22k+β22k+δ22k+η22k,ηk2=η22k1α22k+β22k+δ22k+η22k.
34

The probabilities of obtaining four quantum states in the first step or the second step of the k-th round are

pkj1=αkj4+βkj4+δkj4+ηkj4,pkj2=2(αkj2βkj2+δkj2ηkj2),pkj3=2(αkj2δkj2+βkj2ηkj2),pkj4=2(αkj2ηkj2+βkj2δkj2),
35

where j = 1, 2. The success probability of the kth round is

Pk=p112p12,f2p21,f2p31,f2p(k1)1,f2pk1,s2+p113p12,f3p21,f3p(k1)1,f3pk1,s3+p114p12,f4p21,f4+p(k1)1,f4pk1,s4+p111(p122p21,f2p31,f2p(k1)1,f2pk1,s2+p123p21,f3p31,f3p(k1)1,f3pk1,f3+p124p21,f4p31,f4p(k1)1,f4pk1,f4)++p111p121p211p221p311p321p(k1)11p(k1)21(pk12pk2,s2+pk13pk2,s3+pk14pk2,s4).
36

The total probability is Ptotal=k=1nPk, which depends on the number of iterations and parameters of the initial states. The 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 range of [0,32]. When n = 4, the success probability has already reached 0.9196. When n = 9, the success probability has already reached 0.9975. Therefore, the entanglement concentration is successful in theory.

Figure 4
The success probability P of getting maximally entangled GHZ state relies on the initial coefficients β and δ. Here, we let α = ½, β{0,32}, δ{0,34β2}, η=34β2 ...

In this paper, we only consider the ideal CNOT gate1923. In experiment, there exist many nonideal factors such as the double effect of parameter conversion, the imperfect matching of the crystal lattice and phases, and so on. The probabilities of intrinsic error of experimental methods are unavoidable, such as QND measurements and CNOT operations. Thus 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 purification of GHZ states.

Methods

The 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. The cross-Kerr nonlinearity medium is described by the Hamiltonian,

H=χasasapap
37

where as and ap are the creation operators, a s and a p are the annihilation operators, a Fock state |nright angle bracket and a coherent state |α cright angle bracket interact. The whole system evolves into

U(t)|n〉|αc〉 = c0|0〉|αc〉 + c1|1〉|αceiθ
38

where U(t)=eiθas+asap+ap, θ = χt is the phase shift and t is the interaction time (c = 1, 2, 3). θ is proportional to the number of photons in the signal state |α cright angle bracket. X-quadrature measurement can recognize the phase shift of signal states |α cright angle bracket. The 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 forms of |τk1μ0|H〉 + μ1|V and |τk2λ0|H〉 + λ1|V that interact with a coherent beam |α cright angle bracket (c = 1, 2, 3),where real numbers μ 0, μ 1, λ 0 and λ 1 satisfy the normalization condition |μ 0|2 + |μ 1|2 = 1, |λ 0|2 + |λ 1|2 = 1, respectively. Then the composite quantum system |ϒ1〉 = |τk1 ⊗ |τk2 ⊗ |αc evolves to

|ϒ2〉 = μ0λ1|HV〉|αce−2iθ〉 + μ1λ0|VH〉|αce2iθ〉 + (μ0λ0|HH〉 + μ1λ1|VV〉)|αc
39

From the Eq. (39), we can pick up a phase shift 0 related with |HHright angle bracket and |VVright angle bracket, and phase shift 2θ related with |HVright angle bracket and |VHright angle bracket. One can distinguish |HHright angle bracket and |VVright angle bracket from |HVright angle bracket and |VHright angle bracket by different phase shifts, however, the states |α1e±2iθ can not be distinguished by the setup.

Acknowledgements

This work was supported by National Science Foundation of China (Grant Nos 61472165, 61373158, 61672014 and 61502200), Guangdong Provincial Engineering Technology Research Center on Network Security Detection and Defence (Grant No. 2014B090904067), Guangdong Provincial Special Funds for Applied Technology Research and development and Transformation of Important Scientific and Technological Achieve (Grant No. 2016B010124009), the Zhuhai Top Discipline–Information SecurityGuangzhou Key Laboratory of Data Security and Privacy Preserving, Guangdong Key Laboratory of Data Security and Privacy Preserving, the Transformation Project of Sci-tech Achievements of SYSU, Natural Science Foundation of Guangdong Province, China, under Grant Nos 2016A030313090 and 2014A030310245, and Science and Technology Planning Project of Guangdong Province, China, under Grant No. 2013B010401018, Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) (No. nsfc2015_180), the Natural Science Foundation of Guangdong (No. 2016A030313350), the Special Funds for Science and Technology Development of Guangdong (No. 2016KZ010103), the Fundamental Research Funds for the Central Universities (No. 16lgjc83), and Scientific and Technological Achievements Transformation Plan of Sun Yat-sen University.

Author Contributions

Author Contributions

X.Q. Zhang and J. Weng proposed and wrote the main manuscript text. J. Weng, W. Lu, X.C. Li, W.Q. Luo and X.Q. Tan reviewed the manuscript. J. Weng, W. Lu and X.Q. Tan provided funding support.

Notes

Competing Interests

The authors declare that they have no competing interests.

Footnotes

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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