OPTOELECTRONICS LETTERS Vol.7 No.5, 1 September 2011

SARG04 decoy-state quantum key distribution based on an unstable source ZHOU Yuan-yuan਼ၯၯ ** and ZHOU Xue-jun਼ᄺ‫ ݯ‬ Electronic Engineering College, Naval University of Engineering, Wuhan 430033, China (Received 6 April 2011) C Tianjin University of Technology and Springer-Verlag Berlin Heidelberg 2011 ƻ A three-state protocol for the SARG04 decoy-state quantum key distribution (QKD) based on an unstable source is presented. The lower bound to the secure key generation rate is derived without using the basic hypothesis of the original decoy-state idea. The three-state SARG04 decoy-state protocol with an unstable parametric down-conversion source is considered in the simulation. The simulation results show that the protocol in this paper with an unstable source gives a key generation rate that is close to that with a stable source, and only slight advantage appears by using a stable source when the transmission distance is long. So the SARG04 decoy-state protocol with an unstable source still can obtain the unconditional security with a slightly shortened final key. Document code: A Article ID: 1673-1905(2011)05-0389-5 DOI 10.1007/s11801-011-1040-9

The unconditional security of quantum key distribution (QKD)[1] has been rigorously proved, even when implemented with imperfect real-life devices[2]. The decoy-state method[3] was proposed and demonstrated as a means to dramatically improve the performance of QKD with unconditional security[4-9]. The essence of the decoy-state idea can be summarized as Eq.(1) given perfect control of photon number from the source Yn (decoy) = Yn(signal) , en (decoy) = en(signal) ,

(1)

where Yn is the yield and en is the quantum bit error rate (QBER) of those n-photon pulses from the decoy source or those from the signal source. However, it is an impossible task to control any real setup perfectly in practice, so Eq.(1) does not always hold for a fluctuating source[10]. Now, a new problem is how to carry out the decoy-state method securely and efficiently with an unstable source. To solve this problem, a lot of theoretical work[11-13] has been done. Especially, without using Eq.(1), a general formula[11] was given for the three-intensity decoystate method with state errors of sources if a few parameters in the states are known with source monitoring. Subsequently, Ref.[13] presented a more tightened formula for a two-intensity decoy-state protocol without monitoring any source * **

fluctuation. The above researches on decoy-state QKD are all about BB84 protocol. In this paper, we consider the SARG04 [14] protocol. The SARG04 protocol uses four exactly identical states of BB84 protocol, but modifies the classical communication between Alice and Bob. It is not difficult to perform the SARG04 scheme once the experiment of the BB84 protocol is available. Thus, it is important to investigate the SARG04 decoy-state protocol with unstable source in practice. In this paper, without using Eq.(1), we develop a three-state protocol for SARG04 scheme with unstable sources and this protocol need not monitor any parameter of the unstable sources. In the three-state protocol, Alice has three sources, vacuum source Sv, signal souce Ss and decoy souce Sd

U is

J

¦a k 0

s ki

k k

, U id

J

¦a k 0

d ki

k k

,

(2)

where |kLjrepresents a k-photon Fork state, and aki is the probability to get a k-photon Fork state and is determined by the intensity of a pulse. J can be either finite or infinite. For simplicity, we assume that every pulse from Sv is exactly in vacuum state, but each single-shot of pulses from Ss and Sd can be in a state slightly different from the expected. Suppose that Alice uses each source of {Sv, Ss, Sd} randomly with probabilities pv, ps, pd, respectively and pv+ps+pd =1.

This work has been supported by the National High Technology Research and Development Program of China (No.2009AAJ128). E-mail: [email protected]

gg Optoelectron. Lett. Vol.7 No.5

We denote Pvi|0, Pi|ks and Pdi|k as the probabilities that the ith pulse containing k photons comes from the vacuum source, signal source and decoy source respectively. s Pvi|0=pvdki , Pdi|k=pdadkidki , Pi|k =psaskidki ,

(3)

With the definitions postulated earlier, we have

so N s

¦n k 0

and N d

s k

¦n k 0

d k

k !0 .

(4)

v

s

¦ Pi|v0

.

(5)

d i|0

ic0

¦p ic0

d

a0di d 0i

¦ ic0

p d a0di v ˜ p d 0i , pv

§p a min ¨¨ v jc © p 0

· v ¸N ¸ ¹

n1s

s

ic1

n1d

§p a max¨¨ v jc © p 0

0

· v ¸N ¸ ¹

n0sL d n0s d n0sU

§ p s a 0s j max¨¨ v jc © p 0

· s ¸ n0 ¸ ¹ .(11)

¦ ic1

1 § pdad 1  min ¨¨ s 1s j jc © p a1 j

,

(12)

· ¸ ¸ ¹

(13)

1

1

consequently § p d a1dj · n0d  min¨¨ s s ¸¸n1sU  Od  G 1 , jc © p a1 j ¹

(14)

n0s  n1s U  Os  G1

(15)

Nd

1

Ns

,

where

Od

J

¦¦ p

d

k 2 ick

J

akid d ki , Os

s

¦¦ p a k 2 ick

s ki

d ki ,

(16)

and n1s  n1sU d 0 .

J

s

¦¦ p a

· v ¸ N .(7) ¸ ¹

J

¦¦ k 2 ick

· v ¸ N .(8) ¸ ¹

s ki

d ki

(17)

We denote G 2

§

J

¦ ¦ ¨¨ p a k 2 ick

1 § pdad 1  min ¨¨ s kjs jc © p akj k

With the same method, we can deduce the bound of ns0 as § p s a 0s j min ¨¨ v jc © p

d 1i d n1sU

§ p d a1dj · sU d d t p a d min ¦ 1i 1i jc ¨¨ p s a s ¸¸n1 , ic 1j ¹ ©

k 2 ick

d 0j

s 1i

¦p a

1

Os d

n0dL d n0d d n0dU

§ pdad · § pdad N d  n0d  max¨¨ s 2s j ¸¸ N s  max¨¨ s 2s j jC jC © p a2 j ¹ © p a2 j d d d d §p a · §p a · max¨¨ s 1s j ¸¸  max¨¨ s 2s j ¸¸  jC j C © p a1 j ¹ © p a2 j ¹

According to Eq.(4), we have

so we can express bound of n by d 0j

(10)

(6)

d 0

d

.

So the bound for the fraction of a single count among all counts caused by the signal source is 's1 ı n1sL/N s. Now we mainly estimate the bound of n2s. Firstly, the upper bound of n1s needs to be known. Recalling the definition of dki in Eq.(4), we have

G1

nd0 can be expressed by

¦P

n1sL

. N , N and N can be directly

ic 0

n0d

¦ Pi|sk

ic k

Now, we derive the formulae for ns1 and ns2 . The lower bound of ns1 is[13]

d

observed in the experiment. Our goal is to find the lower bounds of ns1, ns2 and the upper bounds of es1, es2 without using Eq.(1). We firstly deduce the bounds of nd0 and n0s . With Eq.(3), we have Nv

(9)

k 0

In the protocol, Alice sends Bob M pulses one by one. In response to Alice, Bob observes his detector for M times. We define *={i|i=1, 2,Ă, M}. As Bob’s ith observed result, Bob’s detector can either click or not click. If the detector clicks in Bob’s ith observation, we say that the ith pulse from Alice has caused a count. We disregard how the ith pulse may change after it is sent out. When we say that Alice’s ith pulse has caused a count, we only need that Bob’s detector clicks in Bob’s ith observation. We denote C as the set containing all pulses that have caused a count and ck as the set containing all k-photon pulses that have caused a count. Clearly, C=c0Uc1UĂUckĂUcJ . The number of counts caused by k-photon pulse nk is just the number of elements in set ck. We denote nks and ndk as the numbers of counts caused by a kphoton signal pulse and a k-photon decoy pulse, respectively. These numbers cannot be directly observed in the experiment. We define Nv, Ns and Nd to be the total counts caused by vacuum source, signal source and decoy source respectively, J

¦ n ks , k 0

, n ks

¦ Pi|dk

ic k

1 ­ °° p v  p d a d  p s a s ki ki ® 1 ° °¯ p d a kdi  p s a ks i

J

s , N

d k

asymptotically n kd

J

¦n k 0

where d ki

J

Nd

· ¸ ¸ ¹

©

s

OsU

p s a kis  p d a kid

s ki

.

Os  OsU d 0 and can have

· ¸¸ d ¹

(18)

ZHOU et al.

Optoelectron. Lett. Vol.7 No.5 gg

J

Od

d

¦¦ p a k 2 ick

d ki

§

J

d ki

¦¦ ¨¨1  p a k 2 ick

s

©

§ ¨ J ¨ 1 ¦ ¦ ¨1  d d k 2 ic ¨ 1  min §¨ p a kj s s jc ¨ p a ¨ kj © © k

k

p s akis · ¸  p d akid ¸¹

s ki

n2s t n2sL

· ¸ ¸ ¸ G2 ·¸ ¸ ¸¸ ¹¹

§ p d a3d j · max¨¨ s s ¸¸OsU  G 2  G 3 jc © p a3 j ¹

3

3

, (24)

,

§ pd a3d j · § pd a3dj N d  n0dU  n1dU  max¨¨ s s ¸¸ N s  max¨¨ s s jC jC © p a3 j ¹ © p a3 j d d ª §paj· § p d a dj ·º N s «max¨¨ s 2s ¸¸  max¨¨ s 3s ¸¸» jC jC © p a2 j ¹ © p a3 j ¹¼» ¬«

(19) nsL ǻ2s t 2 s N

§ pdad § p d a 2d j · ¨ s s ¸  min¨ s kjs min ¸ jc ¨ p a jc ¨ p a J 2j ¹ kj © © ¦ ¦ d d k 2 ic § p akj · ¨ ¸ 1  min jc ¨ p s a s ¸ kj ¹ © 2

3

2

2

where G 3

§ p d a3d j · § pd a3d j · N d  n0dU  n1dU  max¨ s s ¸ N s  max¨ s s ¸(n0sU  n1sU ) jc ¨ p a ¸ jc ¨ p a ¸ 3j ¹ 3j ¹ © © d d § p a2 j · § pd a3d j · max¨ s s ¸  max¨ s s ¸ ¸ jc ¨ p a3 j ¸ jc ¨ p a 2j ¹ © © ¹

k

· ¸ ¸ ¹ .

· sU sU ¸(n0  n1 ) ¸ ¹

, (25)

k

k

With Eq.(19), Eqs.(14) and (15) can be expressed as § p d a1dj · § p d a2d j · n0d  min¨¨ s s ¸¸n1sU  min ¨¨ s s ¸¸OsU  jc jc © p a1 j ¹ © p a2 j ¹

Nd

1

2

G1G2G3 ,

(20)

Ns=n0s+n1sU+OsU+G1+G2 ,

(21)

§ p d a 2d j ¨ so with the condition min jc ¨ p s a s 2j ©

· § pdad · ¸  min ¨ s 1s j ¸ , we can ¸ jc ¨ p a ¸ 1j ¹ ¹ ©

2

1

obtain the upper bound of n1s

n1s d n1sU

§ pdad · § pdad · N d  min¨¨ s s2 j ¸¸ N s  min ¨¨ s s2 j ¸¸n0s  n0d jc jc © p a2 j ¹ © p a2 j ¹ § pdad · § pdad · min ¨¨ s 1s j ¸¸  min ¨¨ s s2 j ¸¸ jc jc © p a2 j ¹ © p a1 j ¹ . (22) 2

2

1

§ p d a1dj · max¨¨ s s ¸¸n1sL . jC © p a1 j ¹

where n1dU

In calculating the final key rate, we also need the QBERs ed1 and e2d for single-photon count and 2-photon count from the decoy source and e 1s, e 2s from the signal source. The derivated formulae of ed1 and es1 are the same as those in Ref. [13], so we mainly estimate the bounds of ed2 and es2. We denote Ned and Nes to be the numbers of quantum bit flips among the decoy pulses and among the signal pulses, respectively. The two parameters can be obtained through error test. Since p d a2di e2 i s  p d a2di ic d 2i e , e2s 2 n2d

¦pa s

2

d kj

d

d 3j

§p a §p a · max ¨ s s ¸ d max ¨ s s jc ¨ p a ¸ jc ¨ p a 3j kj ¹ © ©

kı3 ,

(26)

p s a 2s i s t n ¦ 2 s s d d ic p a 2 i  p a 2 i

1 § pdad max¨¨ s s2 j jC © p a2 j

n2d

,

(27)

§ p d a 2d j · § p d a 2d j · ¨ s s ¸ ¨ s s ¸ min max ¸ ¸ jC ¨ p a jC ¨ p a 2j ¹ 2j ¹ d © © d s e2 d d e2 d e2 d § p d a 2d j · § p a2 j · ¨ ¸ max¨¨ s s ¸¸ min jC ¨ p s a s ¸ jC 2j ¹ © © p a2 j ¹

.

(28)

2

· ¸ ¸ ¹

jC

Therefore, we can bound the fraction of 2-photon counts among all counts by the signal source as Eq.(25).

k

,

and

we replace max by max in all the previous formulae.

d

ic2

2

Assuming the condition (23), we can estimate the lower bound of n2s given by Eq.(24) with the same method as above. Because ck is unknown and C is known in the experiment, jck

p s a 2s i e2 i s s  p d a 2di 2i s n2

¦pa

3

d

d 2j

§p a · ¸  max ¨ ¸ jc ¨ p s a 2s j © ¹ 2

d

d 1j

§p a · · ¸, ¸  max ¨ ¸ jc ¨ p s a1s j ¸ © ¹ ¹ 1

(23)

we can get

gg Optoelectron. Lett. Vol.7 No.5

By applying the worst-case estimation method,we have d

d

N n /2  e2d d e d 0d '2 N

,

§ p d a 2d j jC ¨ p s a s 2j ©

where '2d t min ¨

(29)

· s§ Ns · ¸ '2 ¨ . ¸ © N d ¸¹ ¹

The function of final key generation rate is § N es s ©N

 R t  H ¨¨

· ¸  '1s [1  H ( e1s )]  '2s [1  H ( e 2s )] . ¸ ¹

(30)

We will consider the three-state SARG04 protocol with an unstable parametric down-conversion (PDC) source, which is pumped by strong light pulses whose intensity fluctuation can be rather large without special control. In the protocol, mode T is detected by Alice, and mode S is sent out to Bob. Mode S is a signal pulse when Alice’s detector clicks and a decoy pulse when her detector does not click. Suppose u to be the intensity of one mode. At any time i, the intensity of the practical pulse is ui = u(1 + ôi). We denote ô to be the upper bound of intensity fluctuation, namely, |ôi |İô. We use the experimental data mainly from Ref.[15]. The observed experimental data has been listed in Tab.1, where

Qs and E d

Ns psM

, Qd

Nd , pd M

Nv , Es pvM

Q0

Fig.1 Plots of R/R0 (a) in terms of intensity error upper bound W and (b) in terms of transmission distance

Fig.2 shows that our protocol with an unstable source gives a key generation rate that is close to that with a stable source. Slight advantage appears by using a stable source only when the transmission distance is over 140 km.

N es psM

N ed . pd M Tab.1 Experimental parameters

u

Q

s

Qd

Q0

Es

Ed

0.639 3.885 h 10-4 1.386 h 10-4 1.7 h 10-6 7.012% 7.563%

Fig.2 Plots of the key generation rate in terms of transmission distance with stable source and unstable source

In Fig.1, R is the key generation rate with source intensity fluctuation, and R0 is the key generation rate without intensity fluctuation. Fig.1(a) shows that the bad influence of the source fluctuation in normal range on the method in this paper is slight. But the key rate can be lower if the source fluctuation is large, in which case one may observe poor values of Ns and Nd. From Fig.1(b), we notice that the influence on the key generation rate is more and more bad with transmission distance increasing.

In summary, we have presented a three-state protocol for the SARG04 decoy-state QKD with a fluctuating source. The lower bound to the secure key generation rate is derived without using the basic hypothesis of the original decoy-state idea. The protocol with an unstable PDC source is considered in the simulation. The simulation results show that our protocol with an unstable source gives a key generation rate that is close to that with a stable source. Slight advantage appears by using a stable source when the transmission distance is

ZHOU et al.

long. The analytic method in this paper is applicable to SARG04 decoy-state protocols with any unstable source. References

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