Kai Yu (喻凱), Chun-Hui Zhang (張春輝)Xing-Yu Zhou (周星宇)

1 Institute of Quantum Information and Technology,Nanjing University of Posts and Telecommunications,Nanjing 210003, China

2 Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, China

3 National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

4 The two authors contributed equally to this work.

Abstract In quantum key distribution (QKD), the passive decoy-state method can simplify the intensity modulation and reduce some of side-channel information leakage and modulation errors.It is usually implemented with a heralded single-photon source.In Wang et al 2016(Phys.Rev.A 96 032312), a novel passive decoy-state method is proposed by Wang et al, which uses two local detectors to generate more detection events for tightly estimating channel parameters.However,in the original scheme, the two local detectors are assumed to be identical, including the same detection efficiency and dark count rate,which is often not satisfied in the realistic experiment.In this paper,we construct a model for this passive decoy-state QKD scheme with two mismatched detectors and explore the effect on QKD performance with certain parameters.We also take the finite-size effect into consideration, showing the performance with statistical fluctuations.The results show that the efficiencies of local detectors affect the key rate more obviously than dark count rates.

Keywords: quantum key distribution, passive decoy state, heralded single-photon source,detector mismatch

1.Introduction

Quantum key distribution (QKD) [1] enables two terminals Alice and Bob to establish a secure communication system in the presence of the eavesdropper Eve.Its security is based on the principles of quantum physics instead of the complexity of mathematical problems.In the past decades,many experimental QKD protocols have been proposed and implemented[2–21],which makes QKD become the most prospective implementation of quantum communication.Unfortunately,real-life singlephoton sources will produce multiphoton pulses in a certain probability and suffer photon number splitting (PNS) [22]attacks inevitably.To counter PNS attacks,different protocols and methods have been proposed,and one of the most widely used methods is the decoy-state method[23–25].

In the decoy-state method, Alice will randomly prepare pulses into different intensities.Then it will select one of the produced pulses as the signal state and the others as decoy states.The multi-intensity decoy-state methods commonly make one intensity correspond to one signal state or decoy state,which inevitably introduces side-channel information leakage and modulation errors due to intensity modulation.As in[26, 27], some methods are able to change or distinguish the characters of different light intensities, such as time and waveform,which is conductive to eavesdroppers to identify the decoy states and signal states, and therefore undermine the security of a QKD system.The passive decoy state method[28,29]can solve this problem skillfully,which usually uses a heralded single-photon source (HSPS) to generate different local detection events as decoy and signal states.Besides,HSPS is also helpful to decrease the effect of the dark count and raise the transmission distance of QKD [30, 31].Meanwhile, in existing schemes, a novel passive decoy-state method is proposed to tightly estimate channel parameters and enhance the system performance [31].It designs a specific local detection structure consisting of a beam splitter and two identical detectors.But in real-life implementation, it is difficult to unify the parameters of two local detectors,including detection efficiency and dark count rate.Therefore, it is necessary to estimate the performance of this passive decoy-state QKD scheme with mismatched local detectors.

In this paper, we will introduce the theoretical model of the passive decoy-state QKD scheme with mismatched local detectors.Besides, we consider the system performance in finite-size regime, calculating the key rate with statistical fluctuations.In numerical simulation, we show the effect of different transmission efficiencies and dark count rates for infinite case and finite case.Finally, we will summarize our simulation results and give conclusions.

2.Theoretical model

The passive decoy-state QKD scheme is presented as in [32],and the schematic diagram is shown in figure 1.In this system,a pump light focuses on a nonlinear crystal (NC) and generates the signal light(S)and the idler light(I)through a spontaneous parametric down-conversion (SPDC) process [33].Then the idler light will be split into two parts by a beam splitter(BS)and sent to two different local detectors(D1,D2),respectively[34].

Figure 1.A schematic diagram of the passive decoy-state QKD[32]:NL, nonlinear crystal; BS, beam splitter; PR, polarization rotator;PBS,polarization beam splitter;D1 and D2,single-photon detectors.

The clicking events of the two detectors can be divided into four events Xi(i=1,2,3,4).When an event Xioccurs,the signal state will be projected to the state,y,z,w), where l is the state corresponding to the event Xi.The definitions of each event and corresponding probability are shown in table 1.As presented in [32], we can deduce the probability of each event:

When it is projected to the vacuum state, the local detector Diwill click with a probability of di(the probability of the dark counts)or not click with a probability of(1 ?di).The probabilities of the clicking event Xiin local detectors are shown in table 2.

Table 1.The definitions of each event and corresponding probability.

Considering the detection efficiency of Di(ηi) and the transmittance of local beam splitter (tv), the conditional probabilityPs1s2∣mare given in table 3.Finally we can get the probability of x, y, z event (the probability of w state is negligible) in equation (1) with the Poisson distribution

Similarly, we can get

Table 2.Conditional probability PX i ∣s 1 s 2that the event Xi when the idler light is projected to state |s1s2〉.

Table 3.Conditional probability Ps1 s 2∣ m that an m-photon state is projected into state |s1s2〉.

where l, r ?{x, y, z} and l ≠r.According to [35], given that these inequalities being respected by our passive source, the single-photon yield of our source can now be lower bounded.

Denote Yias the yield of the i-photon state,and Y0as the background count rate including the dark count rate of Bob’s detectors and channel background contributions.Using the linear channel model,Yiand error rate of i-photon state eican be given by

where edrepresents the probability that a signal photon hit the erroneous detector, hereηi= 1 ? (1 ?η)iis the transmittance of the i-photon state, where η=tABηBobdenotes the total transmission and detection efficiency.It consists of the channel transmittance tAB=10?αs/10and the transmittance in Bobs side ηBob=tBobηD, where α and s represent the loss coefficient (dB km?1) and the length of the fiber (km), ηDis the detector efficiency at Bob’s side.

Based on the passive HSPS and channel model, the overall gain Qland quantum bit error rate (QBER) Elcan be written as

With the above formula and inequality (6), we can estimate the yield and error rate of single-photon pulses [35]:

In the previous analysis,we just discussed the derivation ofandwhen the emitted pulses is infinite.But in real life, the number of pulses used for QKD systems is always finite.The finite pulses will inevitably lead to finite-size effect on the results when we analyze a QKD system quantitatively.Next, we will analyze the above variables with statistical fluctuations [32].

In BB84 protocol,we need to select two bases to code the signal light,namely X basis and Z basis.Denoteandas the probability of choosing X basis in Alice’s side and Bob’s side, respectively.We can optimize two basis selection probability when we calculate secure key generation rate.If the totalnumber of pulses we transmit is N0, thenandwill be the number of pulses which can be successfully aligned to X and Z basis.Combined with analysis in[32],the counting rate Qland overall error rate should satisfy

where

Otherwise,we can consider statistical fluctuations using other methods in[37,38].Consequently we can get the lower bound ofand the upper bound ofin finite cases:

With the above deduction,we can calculate the secure key generation rate Rlas

where f is the error correction efficiency,H(x) = ?xlog2(x)?(1 ?x) log2(1 ?x)is the binary Shannon function.Finally we can calculate the secure key generation rate:

3.Numerical simulations

In this section, we will display the numerical simulations to discuss the effect of detection efficiency ηiand dark count rate diof local detector on system performance.Meanwhile, the key rate under infinite case and finite case will be discussed and compared.All simulations use the device parameters:tv=0.25, ηD=0.5, d0=6.02×10?6, e0=0.5, ed=0.03,f=1.16, N0=109and γ=5.3 corresponding to a failure probability 10?7[32].

First, we fix the dark counting rate as d1=d2=10?6to discuss the effect of local detection efficiency.Figure 2 shows the trend of the key generation rate R with the increase of transmission distance under different detection efficiencies η2.Here η1is set as 0.5.Obviously better detection efficiency will lead to higher key generation rate R at the same distance.This result is quite consistent with our objective cognition.And when considering the finite-size effect, key rate decays faster compared with the infinite case.

Figure 2.Relationship between security key generation rate R and transmission distance s for the infinite (IF) and finite case (F)(N0=109),where the detection efficiency of two local detectors ηi is divided into three groups: (1) η1=0.5, η2=0.3;(2) η1=0.5,η2=0.5;(3) η1=0.5, η2=0.7.

Meanwhile,η1and η2can be jointly optimized to explore the effects on key rate in figure 3.As presented in the figure,increasing the value of η1or η2will improve R significantly.To be noted, if we fix the sum of η1and η2, key rate will perform better when η2is greater than η1because the transmittance of local beam splitter tvis 0.25.As a result, the number of photons entering D1will be more than entering D2.Consequently,the influence of η2on the key generation rate R will be greater than η1.

Figure 3.Joint optimization graph of η1 and η2 at a transmission distance of 50 km when the number of total pulses is finite.

Accordingly, we can simulate the relationship between dark count rate d2and the key generation rate R in figure 4.The curves corresponding to different cases in the figure are almost overlapped.Obviously, the dark count rate diof two detectors at Alice’s side has little effect on the key generation rate R.Meanwhile,we display the joint effect of diin figure 5 as well.Increasing the values of d1and d2will only decrease the value of R on a small scale.

Figure 4.Relationship between security key generation rate R and transmission distance s for the infinite (IF) and finite case (F)(N0=109), where the dark count rate of two local detectors di is divided into three groups: (1) d1=106, d2=10?5;(2) d1=10?6,d2=10?6;(3) d1=10?6, d2=10?7.

Figure 5.Joint effect on key rate of d1 and d2 at a transmission distance of 100 km when the number of total pulses is N0=109.

4.Conclusions

In summary, we have studied the effect of local detection efficiency and dark count rate on the performance of passive decoy-state QKD when local detectors’ parameters are asymmetric.Through mathematical analysis, we present the secure key generation rate R for both infinite-size and finitesize cases with the mismatched detector parameters.Based on the theoretical model,we display the relationship between key rate and two detectors’ parameters.As a result, the detection efficiencies of local detectors affect the key generation rate more obviously than dark count rate.Therefore,our work will provide a reference value for a further research of passive decoy-state QKD systems with mismatched local detectors.

Otherwise,in this paper,we only discuss the case of two local mismatched detectors.But in realistic experiment, the parameters of detectors at Bob’s side are often asymmetric as well, which will affect the solution of Yiand ei.Reference[39] has provided a linear channel loss model to explore the influence of detectors at Bob’s side for the case of equal (or unequal) prepared basis and measurement basis.Combined with this model, we can extend our method to be more applicable to the real-life implementation.

Acknowledgments

We gratefully acknowledge the financial support from the National Key R&D Program of China (2018YFA0306400,2017YFA0304100), the National Natural Science Foundation of China (12074194, 11774180, U19A2075, 12104240,62101285), the Leading-edge technology Program of Jiangsu Natural Science Foundation (BK20192001), Natural Science Foundation of Jiangsu Province(BK20210582),and NUPTSF(NY220122, NY220123).

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