Using a Low-pass Filter to Implement Source Monitoring in Quantum Key Distribution

Yucheng Qiao, Zhengyu Li, Xiang Peng and Hong Guo

State Key Laboratory of Advanced Optical Communication System and Network,

School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, P. R. China. E-mail:{glqyc251, lizhengyu, xiangpeng, hongguo}@pku.edu.cn

Abstract

Quantum key distribution (QKD) has been proven to be unconditionally secure in theory. The light source of QKD is assumed to be trusted in the standard security analysis. However, this assumption may be deviated in a real system, which will cause ‘untrusted source’ problem. To solve the problem, several source monitoring schemes have been proposed, in which passive schemes are more practical than active schemes. In this paper, we first review of these passive schemes, and then propose a new demonstration of the passive scheme with two-threshold detection using a low-pass filter, which can be well implemented in practical.

1. Introduction

Quantum key distribution (QKD) is a method that can allow two communicating parties (Alice and Bob) to share same keys with unconditional security, which is guaranteed by quantum mechanics in theory [1]. However, some assumptions of the theory are not satisfied in practical systems. The GLLP [2] theory is proposed to take some practical factors into account, such as the imperfect single photon source which has the probability of emitting multi photons. However, it still assumes that the source of QKD system is trusted, which is hard to be satisfied in the practical system, especially in the two-way plug&play system. Because in that system, Eve can easily take a Trojan horse attack to fully control the source [3,4].

There are several source monitoring schemes to solve this problem [3-8]. An active scheme had been proposed to show the security of QKD with untrusted source [3]. However, it requires a high-speed random optical switch and an ideal detector (100% efficiency), which are not practical. A passive scheme had been proposed and verified experimentally [5]. In this kind of passive scheme, only a beam splitter and an imperfect detector are needed.

In this paper, we review three specific schemes of the passive schemes [5,6,8], including inverse-Bernoulli transformation scheme, two-threshold detection scheme, and photon-number-resolving (PNR) detector scheme. Among these three schemes, the two-threshold detection method is the most practical one. In this scheme, an integrator is needed after the photodiode to integrate the signal. However, the optical pulse is very narrow, so an ultra-high-speed integrator is necessary, but this is hard to be realized. In this paper, we propose a new demonstration using a low-pass filter instead of the integrator, since low-pass filtering is approximately equal to integrating.

2. Review of the Passive Schemes

In the passive schemes, a beam splitter (BS) is set on Alice’s side to split part of the signal from the source, and a detector is used to detect this part of the signal. The result of the detection is used in security analysis.

2.1 Passive Scheme Using the Inverse-Bernoulli Transformation

The schematic of this scheme is shown in Fig. 1 (a) [5]. Suppose 1( )P n is the photon number distribution (PND) of position 1, and 2 ( )P n is the PND of position 2, then 1( )P n is the inverse-Bernoulli transformation of

2 ( )P n . The photodiode (PD) can detect the energy of optical pulse, and by analyzing the statistics of its detection results, the PND of position 2 can be got. Once 2 ( )P n is known, 1( )P n can be calculated through inverse-Bernoulli transformation. Then the PND of the position 3 3( )P n can be calculated according to 1( )P n .

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(a) Fig.1. (a) The untrusted source pass throf the BS is ,BSη and the efficiency o

a voltage V, and the voltage V1 and V2comparators C1 and C2 output “11”.

2.2 Passive In a practical system, it is hard tinverse-Bernoulli transformation problethat when the probability that position guaranteed [6]. In order to monitor thiFig.1 (b). For simplicity, consider the c

‘11’, it means that the photon number

probability that position 1’s photon num

2.3 Pa

The structure of this scheme is sn from 0,1,2n = and 3.n ≥ Whe2 are known due to the PNR detectionprotocol [7,8].

Fig.3. The untrusted source pass througthe other is sent to the PNR detector. Th This method has a more precipractical experiments, the high-speed PN

3. Two-thresho

The two-threshold detector schecommon. However, there is still a probalways very narrow, as shown in Fig. 4, In this section, we will show thatan integrating, thus it’s reasonable to ch

(b)

rough a BS, and one beam is sent to the photon detectof the detector is .dη (b) When a signal emitted, the

2 corresponds to the photon number minN , maxN . Wh

e Scheme with a Two-threshold Detect

to get 3 ( )P n for every n , because the resolution of them is an ill-posed problem on computer. In another r1’s photon number min max[ , ]n N N∈ is known, th

is probability, the two-threshold detector is used instcase that (1 )BS d BSη η η− = which leads to 1( )P n P=r of position 1 belongs to min max[ , ]N N . By countin

mber min max[ , ]n N N∈ can be obtained directly.

assive Scheme with a PNR Detector

shown in Fig. 3, with a PNR detector which can discrien 1 2( ) ( )P n P n= , the probability of the photon numbn results, so the final secure key rate can be estimate

gh a BS after attenuating and encoding, one beam is she transmission of the BS is ,BSη and the efficiency

ise result because the probability of 0,1,2n = is NR detector is still hard to achieve.

ld Detector Scheme Using a Low-pass

eme is practical because its structure is simple, and thblem in the experiment. In a QKD system, the optica, so it is hard to do the integration in this short time. t for a narrow pulse, the effect of applying a low-pass

hoose a low-pass filter instead of the integrator.

tor (PD). The transmittance integrator transform it into

hen min max[ , ]n N N∈ , the

tor

he detector is finite and the respect, it has been proven he security of QKD is still tead of a PD, as shown in

2( )P n . When the output is

ng the number of ‘11’, the

iminate the photon number ber 0,1,2n = in position d by using the decoy-state

sent out to the channel and of the detector is .dη

directly detected. But in

s Filter

he devices it used are very al pulse from the source is

filter is almost the same as

Fig.4. The waveform of the laser pulse.

For an input signal ( ),f t by

( ) ( ) ( )lF F Hω ω ω= ⋅ , where ( )F ω =

the pulse width is very narrow, the conlow-pass filter and τ is the pulse widt

When using the ideal window fu

( )lf t becomes

1( ) ( )

2i t

l lf t F e dωωπ

+∞

−∞

= ∫

Thus the inequality: 2 2

0 0

0

16

τω τ ωπ

−⎛ ⎞⎜ ⎟⎝ ⎠

∫

( )lf t . Thus when 0 1ω τ⋅ , l mf is

For a general filter, It can 0

00| || ( ) | ( )d dH H

ω

ωω ωω ω

+∞>>∫ ∫

with 0

( )f t dtτ

∫ when 0 1ω τ⋅ .

The key condition 0 1ω τ⋅ cais shown in Fig. 5. Then the maximal w

Fig.5. The intensity modulator (IM) is cgate signal.

The schematic of this experimenpower meter is used to measure the powthe oscilloscope is used to get the averag

The full width half maximum (FWHM) is only about

3.1 Theoretical Analysis

y using the Fourier transform, the signal after fil

1( )

2i tf t e dtω

π−+∞

−∞= ∫ and ( )H ω is the transfer fu

ndition 0 1ω τ⋅ can be satisfied, where 0ω is tth.

unction filter with transfer function((

1, | |( )

0, | |H

ωω

ω

≤=

>

⎧⎨⎩

['

0

1( ) (

sin1(

') '2

i t i td H f t e dt deω ωτ

ωω

πω ω

π

+∞ +∞−

−∞ −∞

= =⋅ ∫∫ ∫

0

0

( ') ' ( ') 'l mf t dt f f t dtτω

π< <∫ ∫ satisfied, where l mf

approximately linear with0

( )f t dtτ

∫ .

also be proved that as long as the filter m

and [ ]arg ( ) |2

| Hπ

ω < for [ ]0 0, ,ω ω ω∈ − l mf is

an be satisfied by applying an intensity modulator (IMwidth of the optical pulse is constrained by the width of

controlled by a gate signal. The width of the pulse is

3.2 Experimental Verification

nt is shown in Fig. 6 (a). The 125MHz PD is used as awer of the signal which is linear with the energy whenge value of l mf .

60 ps.

ltering can be written as

unction of the filter. Since

the cutoff frequency of the

))

0

0

ω

ω

≤

>, the output signal

]0 ( ')( ') '

')t t

f t dttt

ω −−

.

m is the peak value of

meet the conditions that

still approximately linear

M) in front of the BS, which f IM’s control signal.

limited by the width of the

a low-pass filter, the optical n the repetition is fixed, and

When using the linear fitting met

R = 0.99998 and the standard deviation

(a) Fig.6. (a) The diagram of measuring thfitting method.

4 In the review part we can seeexperiment, while it still meets a problelow-pass filter instead of the integrator c In section 3 we discuss about thThe experiments show that the result olow-pass filter can replace the integrator

This work is supported by the N61225003), National Natural Science Foand Development (863) Program.

1. C. H. Bennett and G. Brassard, “QuanIEEE International Conference on Comp 2. D. Gottesman, H. K. Lo, N. Lutkenhadevices,” Quantum Inf. Comput. 4, 325 3. Y. Zhao, B. Qi, and H. K. Lo, “Quant052327 (2008). 4. N. Gisin, S. Fasel, B. Kraus, H. Zbindsystems,” Phys. Rev. A 73, 022320 (200 5. X. Peng, H. Jiang, B. J. Xu, X. Ma, anOpt. Lett. 33, 2077 (2008). 6. X. Peng, B. J. Xu, and H. Guo, “Passdistribution,” Phys. Rev. A 81, 042320 7. X. B. Wang, C. Z. Peng, J. Zhang, L. with source errors,” Phys. Rev. A 77, 04 8. B. J. Xu, X. Peng, and H. Guo, “Passuntrusted source in a plug-and-play qua

thod to deal with the data, the result is shown in Fig. 6

s = 0.00393, which shows that l mf is linear with0

(fτ

∫

(

he peak value l mf and the power of the signal. (b) T

4. Discussion and Conclusion

that the scheme of using a two-threshold detectorem that to integrate a narrow pulse in a very short timcan be a better way to implement the source monitorinhis new demonstration of the passive scheme briefly of filtering is linear enough with the average energy or.

5. Acknowledgments

National Science Fund for Distinguished Young Schoundation of China (Grant No. 61101081), the Nationa

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