Self-error-rejecting multipartite entanglement purification for electron systems assisted by quantum-dot spins in optical microcavities

Yong-Ting Liu(刘永婷), Yi-Ming Wu(吴一鸣), and Fang-Fang Du(杜芳芳)

Science and Technology on Electronic Test and Measurement Laboratory,North University of China,Taiyuan 030051,China

Keywords: quantum communication,entanglement purification,electron-spin system

1. Introduction

Entanglement lies at the heart of quantum mechanics and is a fundamental resource in quantum information processing(QIP), especially for accelerating quantum computation[1–3]and creating secure quantum communication, such as quantum teleportation,[4]quantum key distribution,[5–10]quantum secure direct communication,[11–19]quantum secret sharing,[4,20,21]and so on. Multipartite entangled systems[23–26]shared by the detached parties in remote locations are in a maximally entangled state for the security and the efficiency of quantum communication. However,in a practical transmission,the multipartite propagated away from each other are bound to sustain channel noises, which will inevitably degrade the entanglement or even make the maximally entangled state change into a mixed one,making quantum communication insecure. In order to establish an extensive quantum network,the quantum repeater is used to restrain the decoherence caused from the environmental noise.[27]Entanglement purification, which is one of the key constituents for quantum repeaters in the quantum communication, can distil some high-fidelity entangled quantum systems from the mixed entangled ones.[28–32]

The initial entanglement purification protocol (EPP) by Bennettet al.[33]and that by Deutchet al.[34]were presented,resorting to perfect controlled-not(CNOT)gates. Subsequently, an EPP[35]based on linear optical elements and single-photon detectors and an efficient EPP[36]with a currently obtainable parametric down-conversion (PDC) source assisted by cross-Kerr nonlinearity were introduced. EPPs have been presented in various ways,mainly consisting of conventional entanglement purification protocols (CEPPs)[33–40]and the deterministic entanglement purification protocols(DEPPs).[41,42]The latter referred to two-step DEPP based on hyerentanglement[41]and one-step DEPP based on the spatial entanglement of a pragmatic PDC source and linear optical elements.[42]Recently, much attention has been drawn to hyerentanglement purification protocols(hyper-EPPs).[43–47]For instance, Renet al.[43]proposed the first two-photon hyper-EPP in the mixed polarization-spatial hyperentanglement Bell states for bit-flip errors,resorting to the nonlinear optical property of a nitrogen-vacancy center embedded in a photonic crystal cavity.

However, most of the EPPs[33–42]and hyper-EPPs[43–47]have been aimed at bipartite entangled and hyperentangled systems, respectively, and there are only few multipartite entanglement purification protocols (MEPP), including highdimension HEPPs.[48–50]The first MEPP with CNOT gates to purify multipartite entangled systems in a Werner-type state by Muraoet al.,[48]and then the other one to purify high-dimensional multipartite quantum systems by Cheonget al.[49]were proposed. Sequentially, a feasible MEPP[50]and an efficient MEPP[51]were proposed in a Greenberger–Horn–Zeilinger(GHZ)state with nondestructive quantum nondemolition detectors(QND),which were available to perform iteratively the MEPPs.

The EPP for electron-spin systems also plays a significant role in the quantum communication and quantum computation. For example, Shenget al.[52]presented an MEPP for electron-spin states. The original fidelity of the MEPP was required to be lager than 0.5, meanwhile, much entangled quantum resource was discarded, leading to the relatively low efficiency. Recently, semiconducting quantum dot(QD) embedded in a optical microcavities is the best service for solid-state qubit especially in QD-spin QIP,[53]owing to the electronic spin confined in a charged QD possessing μs coherence time[54]and ps time-scale single-qubit manipulation[55,56]for controlling and measuring the spin state. Many researchers have devoted much effort to improving photon-QD-spin interactions by the integration of charged QDs amalgamated with nanophotonic micropillar cavities in experiment recently.[56–58]Besides, an entangled beam splitter,[59]a flexible two-electron-spin EPP,[60]and an optical Faraday rotation[61]can be generated with an electronspin QD coupled to a microcavity.

In this article, we present an efficient MEPP forNelectron-spin systems in a GHZ state by exploiting the singleside cavity-spin-coupling system. First,we can obtain a highfidelityN-electron-spin ensemble directly with fidelity-nearunity parity-check devices(PCDs),similar to the conventional MEPPs with perfect controlled-not gates.[48]Subsequently,the recycling MEPP with the entanglement link is used to reproduce someN-electron-spin entangled systems from subspaces. In detail, the parties in quantum communication first distil some entangledM-electron-spin subsystems (2≤M ＜N), which are discarded in the previous MEPPs,[48–50]and then they reproduce someN-electron-spin entangled systems with entanglement link.

2. Establishing an error-heralded parity-check device by QD-cavity system

can be obtained by solving the Heisenberg–Langevin equations of the motion in Eq. (2) for the cavity field operator ˆaand negative excitonX-operator ˆσ-driven by the input field operators ˆain,and combing the relation between the input field operators ˆainand the output field operators ˆaoutin the weak excitation approximation〈ˆσz〉⋍-1,[61–67]where the QD spin dominantly occupies the ground state

Here,ωc,ωX-,andωrepresent the microcavity frequency,the transition frequency of negative excitonX-and the input photon frequency,respectively.gis the coupling strength between the QD and the single-sided microcavity,κ,κs, andγare the decay rate of single-sided microcavity,leakage rate of singlesided microcavity, and decay rate of negative excitonX-, respectively. In the case of coupling strengthg=0(uncoupled cavity),the reflection coefficient becomes

Fig.1. Schematic diagram of a single-sided QD-cavity system,and the optical transitions of a QD.

We can construct a error-heralded QD block by combing the above QD-cavity system and linear optical elements,as shown in Fig. 2(a). Suppose that the single photon in the

We input a single photon in left-polarized state|L〉into a quantum circuit of a error-heralded PCD. After the photon passes through VBS,the left-polarized photon|L〉in the lower mode 1 subsequently passing through the error-heralded QD1block,X1, QD2block,X2, and BS, meanwhile in the upper mode 2 combining again at the BS, is detected by the detector D1′or D2′.

Fig. 2. (a) Schematic diagram of the error-heralded QD block. (b) Schematic diagram of error-heralded parity-check device (PCD) on electron-spin system.HPi(i=1,2)is a half-wave plate that performs Hadamard operation on the photon,i.e.,|R〉→(|R〉+|L〉)/2.CPBS represents a circular polarization beam splitter,which transmits a right-polarized photon|R〉and reflects a left-polarized photon|L〉. Dj (j=1,2,1′,2′)is a single-photon detector. VBS represents a non-equilibrium beam splitter with a transmission coefficient of(1+P4)-1/2 and a reflection coefficient of P2/[(1+P4)]1/2. BS is a 50:50 beam splitter. Xk (k=1,2)is a half-wave plate,which performs bit-flip operation on the photon σpx =|R〉〈L|+|L〉〈R|.

During the single photon scattering processes,if the photon is reflected by the error-heralded QD-cavity block, it will trigger the detector either D1or D2, which means the failure of the PCD. In detail, when the single photon detector D1triggers, the error occurs in the parity measurement task of this round,another single photon can be input to complete the parity outcome of the two-electron spins. On the premise of ignoring the photon scattering with inherent losses channeled into its environment in first QD block,the photon passes through the half-wave plateX1when the D1does not respond.When the single photon detector D2responds,after the phaseflip operationσex=|↑〉〈↑|-|↓〉〈↓| on the second electron spin, another single photon is input. If the input photon is transmitted through the two error-heralded QD blocks, there is no click of the single-photon detector D1or D2. At last,the two modes of the photon,passing though different optical paths, will come together again through the BS and be detected,completing the PCD of the two-electron spins. The BS performs operations

3. High-efficiency three-electron-spin EPP for bit-flip errors

For three-electron-spin entangled states, there are eight GHZ states written as follows:

Here,the subscriptsA,B,andCdenote the three-electron spins sent to Alice, Bob, and Charlie, respectively. Suppose that the original three-electron-spin GHZ state transmitted among the three parties is|φ+0〉ABC. As we know, the noisy channel will inevitably change a pure entangled state ensemble into a mixed one. In other words,when the initial|φ+0〉ABCturns into|φ+i 〉ABCby taking place a bit-flip error on thei-th(i=1,2,3)qubit, meanwhile|φ+0〉ABCevolves to|φ-0〉ABCdue to appearing a phase-flip error. Sometimes, both a bit-flip error and a phase-flip error will take place on the three-electron-spin system with the state|φ-i 〉ABC. Generally, a phase-flip error can be transformed into a bit-flip error assisted by a bilateral local operation. Therefore, we only discuss the MEPP for bit-flip errors of three-electron-spin mixed states in detail.

Suppose that Alice, Bob, and Charlie share a threeelectron-spin ensembleρafter the transmission of qubits over the noisy channels,

3.1. The first step of three-electron-spin EPP for bit-flip errors with PCDs

Table 1. The states of the three-electron-spin systems obtained from identity-combination items and the corresponding probabilities(suppose x=A1B1C1 and y=A2B2C2 for simplification).

Fig. 3. (a) The principle of the first step of our three-electron-spin EPP for bit-flip errors with parity-check devices (PCDs). (b) The principle of the entanglement link for reproducing a three-electron-spin entangled system from two two-electron-spin entangled systems. Dj (j =A1′,A2′,B1′,B2′,C1′,C2′) represent a single-photon detector. The black symbol‘·’represents a QD spin.

Table 2. The states of the two electron-spin systems obtained from the cross-combination terms and the corresponding probabilities(suppose that x=A1B1C1 and y=A2B2C2 for simplifciation).

3.2. The second step of three-electron-spin EPP for bit-flip errors with HL

Table 3. The three-electron-spin systems reobtained from two-electron-spin systems and the corresponding probabilities(suppose p=A1B1 and q=A2C1 for simplification).

3.3. The efficiency of three-electron-spin EPP for bit-flip errors

After the above two purified steps, the first round of the MEPP process is accomplished. The purified fidelity of the three-electron-spin systemA1B1C1can be improved further by repeating multiple rounds of the MEPP. Furthermore, the efficiency of obtaining the efficiency of three-electron-spin systemA1B1C1after the first purification step isη1, while the efficiency of obtaining the three-electron-spin systemA1B1C1after introducing the second purification step with HL isη2,

wheref1=f2=f3=(1-f0)/3.η1andη2are shown in Fig. 4, which easily find that the efficiencyη2of our MEPP is greatly increased. For the initial fidelityf0＜0.5,η2is far larger than 2η1.Obviously,the initial fidelityf0is smaller 0.5,the second purified step with quantum HL plays an important role in the first round of the MEPP process.

Fig.4. The efficiencies η1 and η2 versus the initial fidelity f0.

In addition, the efficiencyη1of our MEPP is double as those in Refs. [48–50] due to taking all the cases in which all the parties obtain either even or odd parity into account for obtaining high-fidelity three-electron-spin entangled systems. Two purified steps are independently in the next round.That is, they can first purify two-electron-spin systems with the fidelityf0and then produce high-fidelity three-electronspin systems with entanglement link. Besides,as all quantum operations, the PCDs and HL,will work with a near-unity fidelity, the MEPP here will be performed faithfully and work without the influence from every purified operation.

4. Discussion and summary

Furthermore, our three partite EPP can be directly extended to purifyN-electron-spin entangled systems,resorting to the self-error-rejecting parity-check devices(PCDs)and entanglement link. There are 2NGHZ states for anN-electronspin systems and can be written as

Fig.5. The principle of the entanglement link for producing a four-electron spin entangled system from(a)a three-electron spin entangled subsystem and a two-electron spin entangled subsystem with a PCD,(b)two three-electron-spin entangled subsystems with two PCDs,(c)three two-electron-spin entangled subsystems with two PCDs.

Fig. 6. The efficiency η of error-heralded PCD in the case κs/κ = 0,κs/κ =0.1, and κs/κ =0.2 represented by the green solid line, the red dot–dash line, and the blue dot line, respectively, with ω =ωc =ωX- and γ/κ =0.1.

We construct the self-error-rejecting non-destructive PCD by using the error-heralded QD blocks. During the measurement process, if there is an error, the detector either D1or D2responds, and the parity outcome of electron-spin quantum state can be repeated until success. If the detector D′1or D′2responds, the parity outcome of PCD is successful. The fidelity of the PCD is robust, and immunes to the coupling strengthg,the microcavity decay rateκ,the microcavity leakage rateκsand the exciton decay rates ofγ,which reduces the requirements for the experimental conditions of the scheme.However, the efficiencyη=β2=2P4/(1+P4) of PCD is largely affected by the all this mentioned above factors, as shown in Fig. 6. In the case ofω=ωc=ωX-,γ/κ=0.1,andg/(κ+κs)＞2, the efficiency becomesη ＞97.52%,η ＞78.55%, andη ＞62.48%, respectively, in the conditionκs/κ=0,κs/κ=0.1,andκs/κ=0.2,respectively. Furthermore,the efficiencyηcould be further improved via increasing the effective QD-cavity couplingg/(κ+κs)and decreasing the side leakageκs/κof the cavity.

In summary, we have proposed a high-efficiency MEPP forN-electron-spin systems in the GHZ state, resorting to fidelity-robust PCDs and entanglement link, which contains two steps. One is the MEPP with fidelity-robust PCDs to obtain not only high-fidelityN-electron-spin entangled systems

Acknowledgements

Project supported in part by the National Natural Science Foundation of China (Grant No. 61901420), the Shanxi Provincial Science Foundation for Youths (Grant No. 201901D211235), and the Scientific and Technological Innovation Program of Higher Education Institutions of Shanxi Province,China(Grant No.2019L0507).