- Email: [email protected]
An improved quantum key distribution protocol based on second-order coherence
Optics Communications 260 (2006) 351–354 www.elsevier.com/locate/optcom
An improved quantum key distribution protocol based on second-order coherence Jin Xiong *, Guihua Zeng, Nanrun Zhou The State Key Laboratory on Fiber-Optic Local Area Networks and Advanced Optical Communication Systems, Department of Electronic Engineering, Shanghai Jiaotong University, Room 602, No. 28, Lane 865, Dongchuan Road, Shanghai 200240, China Received 26 June 2005; received in revised form 13 October 2005; accepted 21 October 2005
Abstract We improve the quantum key distribution protocol proposed by Pereira et al. [S.F. Pereira, Z.Y. Ou, H.J. Kimble, Phys. Rev. A 62 (2000) 042311], by employing the second-order coherence of optical fields, which can be easy experimentally measured with a HanburyBrown and Twiss intensity interferometer. It is shown that eavesdropping can be directly detected without sacrificing extra secret bits as test key. The efficiency of the improved system is enhanced greatly, since no secret bit needs to be discarded. Ó 2005 Elsevier B.V. All rights reserved. PACS: 03.67.Dd; 03.67.HK; 42.50.Ar Keywords: Quantum key distribution; Second-order coherence; HBT intensity interferometer; Quantum cryptography
1. Introduction In recent years, it has attracted much attention about the possibility of using quantum mechanics to transmit signals in such a way that any eavesdropping can be detected by the two legitimate communicating parties [1–3]. This idea has been applied to quantum key distribution (QKD), a technique that allows two remote parties to share a secret key. The original QKD protocols have all involved the transmission of single particles, for example, single photons. In this method, a significant current limitation on the practicality is the poor efficiency of photon counting detectors. And losses limit the transmission distance. Fortunately, it has been shown that continuous variable quantum systems [4–11] could offer an alternative to the usual single photon counting QKD. They can be implemented either by non-classical beams, such as squeezed light [4,5,7] and entangled light [6], or by classical beams, i.e., Gaussian-modulated coherent light [8]. This kind of *
Corresponding author. E-mail address: [email protected] (J. Xiong).
0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.050
QKD schemes are potentially advantageous because they may achieve higher transmission and repetition rates in the near future [12]. As is well known, there exists mode–mode correlation [13] between the two modes of a two-mode squeezed light, which can be generated by non-degenerate optical parametric amplification via non-linear interactions in a crystal [14]. The QKD protocol employing such two-mode squeezed states has been carried out by Pereira et al. [7]. They evaluated the security of the system based on the signal to noise ration obtained by a homodyne detector. If an eavesdropper, Eve, attempts to gain useful information, she will necessarily increase the noise level and error rate at the receiving station. In other words, EveÕs intervene will introduce a well-known penalty of a 3 dB reduction into the signal-to-noise ration. However, similarly to the previous protocols, their protocol suffers a low efficiency since a part of newly generated secret bits must be sacrificed as test key after transmission. Motivated by this, in this paper, we employ the second-order coherence of optical fields to improve the efficiency of the original protocol of Pereira et al.
352
J. Xiong et al. / Optics Communications 260 (2006) 351–354
We structure our paper as the following style. The second-order coherence of optical fields is described in Section 2. And an eavesdropping detecting approach by employing second-order coherence is introduced in Section 3. In Section 4, an improved QKD protocol is summarized and a brief conclusion is given in Section 5. 2. Second-order coherence First, let us give a brief review of second-order coherence of light fields. It is known that the famous YoungÕs interference experiment is a classical optical interference experiment, which is a measurement of the first-order correlation function of the field [13]. However, this kind of experiments cannot be employed for judging the quantum nature and classical nature of light. In the 1950s, Hanbury-Brown and Twiss developed a new kind of correlation experiment for applications in intensity-correlation stellar interferometry [15]. They demonstrated the possibility of measuring the magnitude of the coherence function by means of the correlation between intensity fluctuations at two different points, rather than the correlation between the fields. The Hanbury-Brown and Twiss (HBT) experiment, the first observation of higher-order coherence, can reveal quantum properties of light sources, not seen in YoungÕs interference experiment. A recent work has shown that the second-order coherence and HBT experiment can provide an operational framework for the construction of witness operators that can test classical and non-classical properties of a Gaussian squeezed state [16]. In the QKD system implemented by Pereira et al., the two different two-mode squeezed vacuum states Alice prepared can be expressed by ðÞ
y y
jW12 i ¼ erða1 a2 a1 a2 Þ j01 ; 02 i;
ð1Þ
where 1 and 2 indicate two modes of the states, and a1 ðay1 Þ and a2 ðay2 Þ are the photon annihilation (creation) operators for mode 1 and mode 2, respectively. Here, r is the squeeze parameter representing the degree of squeezing. The vacðþÞ ðÞ uum fluctuations of jW12 i and jW12 i are squeezed in the directions of U1 and U2, respectively, with the definitions of U 1 ¼ 12 ða1 þ ay1 þ a2 þ ay2 Þ and U 2 ¼ 2i1 ða1 ay1 þa2 ay2 Þ. We restrict our attention to the normalized second-order ð2Þ cross-correlation function gð12Þ ð0Þ, which describes the intensity correlation between the two modes ð2Þ
g12 ð0Þ ¼
hay1 a1 ay2 a2 i hay1 a1 ihay2 a2 i
¼2þ
1 ; hni
ð2Þ
the second-order cross-correlation function tends to the constant 2. That is, for strong squeezed vacuum states, the squeezing factor r is not an important parameter in the second-order mode–mode correlation. 3. Eavesdropping detecting As is well known, the beam splitter is ubiquitous in many optical systems and plays a fundamental role in many fields such as interferometry, holography, laser systems, etc. Many of its usages in quantum optics and quantum information are derived from the fact that it can act as an entangler [17–20], i.e., an unentangled input beam can emerge from the device in an entangled state. In quantum cryptography, the beam splitter is often used as an eavesdropping tool [4–6], where Eve employs a beam splitter to split off part of the signal and performs measurements on that part to gain useful information. The abstract model of a lossless beam splitter is a linear four-port device, shown in Fig. 1. Two radiation modes enter the instrument, interfere with each other, and then leave it. The beam splitter acts in three steps: (1) shift the phases of the input modes, (2) mix the modes via a rotation, and (3) shift the phase again. The phase shifts could be eliminated by a proper phase redefinition of the incoming/outcoming modes. The rotation would remain. Mixing of modes is in fact the essential operation of a beam splitter. The action of the beam splitter may be corresponding to the unitary transformation connecting the two input fields and the two output fields. Denoting the annihilation operators of the input modes by a1 and a2 and the output modes by b1 and b2, the input/output relation of the beam splitter can be written as [21] b1 a1 ð3Þ ¼B Bþ ; b2 a2 where the unitary evolution beam splitter operator B in the representation of the angular momentum is
b2
a1
b1
ðÞ
where hni = sinh2 r is the mean photon number of jW12 i. Eq. (2) shows that the second-order intensity correlation between the two modes of the two-mode squeezed vacuum ðÞ states jW12 i is a function of the squeezing factor r. With the increasing of the squeezing degree of the squeezed states, the second-order cross-correlation function decreases. For strong squeezed vacuum states, i.e., r P 2.5, the mean photon number hni = sinh2 r is very large and
a2 Fig. 1. The diagrammatic representation of a beam splitter as an optical element with two input ports and two output ports. a1 and a2 denote two input modes and b1 and b2 two output modes.
J. Xiong et al. / Optics Communications 260 (2006) 351–354
ð4Þ 0.6
pffiffiffiffi T pffiffiffi i/ Re
pffiffiffi i/ ! a1 Re pffiffiffiffi . a2 T
12
ð5Þ
In PereiraÕs protocol, Eve may choose two typical possible methods of eavesdropping. If Eve uses the captureresend eavesdropping strategy, she will introduce errors which can be detected by Alice and Bob [7]. If Eve performs a beam splitter to gain useful information, that is, she sends on to Bob the part of the signal that is transmitted through the beam splitter and performs measurements on the part that is reflected, she will bring disturbance to BobÕs station. In order to check out EveÕs presence without lowing efficiency, we employ the second-order coherence of light fields. By a lossless symmetric beam splitter, in the Schro¨dinger picture the fields Bob received are changed because of EveÕs intervention, as g ðÞ ðÞ jW12 i ¼ BjW12 i. ð6Þ Note that we have introduced the symbol ‘‘’’ to tag the variables influenced under EveÕs disturbance. g g ðþÞ ðÞ For each state jW12 i or jW12 i Bob received, the secondg ð2Þ order cross-correlation function g12 ð0Þ can be obtained with Eqs. (4)–(6) 1 1 g ð2Þ 4Rð1 RÞ 1 þ g12 ð0Þ ¼ 2 þ hni hni ð2Þ
0.4 0.3 0.2
¼ g12 ð0Þ Dg ;
ð7Þ
1 where Dg 4Rð1 RÞð1 þ hni Þ is the disturbance item due g ð2Þ ð2Þ to EveÕs eavesdropping. Eq. (7) shows g12 ð0Þ < g12 (for R 5 0). It means that under EveÕs intervention, the degree of the second-order mode–mode correlation decreases. From Eq. (7), one can see that the disturbance item Dg is a function of the reflection coefficient R and the squeezing factor r. The features of the disturbance due to EveÕs presence can be seen more clearly in Figs. 2 and 3, where the ð2Þ ratio of Dg to g12 ð0Þ is plotted as a function of R and r, respectively. ð2Þ Fig. 2 shows that the figure of Dg =g12 ð0Þ versus R is a parabola, which has maximal value at the point R = 0.5. That is, if the beam splitter used by Eve is a 50:50 one, i.e., T = R = 0.5, the second-order cross-correlation funcg ð2Þ tion g12 ð0Þ decreases to the minimum 1, which can be
0.1 0 0
0.1
0.2
0.3
0.4
0.5 R
0.6
0.7
0.8
0.9
1
4.5
5
ð2Þ
Fig. 2. The figure of Dg =g12 ð0Þ versus R, with r = 1.
1 0.9 0.8 (2)
Bþ ¼ B ; a1 þ B B¼ a2
0.5
g
Here, h is the single angular parameter of the beam splitter which satisfies the equations R ¼ sin2 h2 and T ¼ cos2 h2, where R and T are the reflection and transmission coefficients with the normalization T + R = 1. The parameter / is the phase difference between the reflected and transmitted amplitudes given by the beam splitter. The beam splitter operator B has the following properties:
Δ /g(2)(0)
h y i/ ða1 a2 e a1 ay2 ei/ Þ . 2
Δ g/g12 (0)
B ¼ exp
353
0.7
0.6
0.5 0.4 0
0.5
1
1.5
2
2.5 r
3
3.5
4
ð2Þ
Fig. 3. The figure of Dg =g12 ð0Þ versus r, with R = 0.4.
easily found from Eq. (7). It indicates that the secondorder mode–mode correlation disappears [13]. Under this situation, the disturbance brought by Eve reaches the maximum. For a certain beam splitter, i.e., the reflection coefficient R is given, the disturbance due to EveÕs eavesdropping decreases with the increasing of the squeezing factor, shown in Fig. 3. That is, if the squeezing degree of the two-mode squeezed vacuum states is stronger, the disturbance Eve brought appearing in the second-order modemode correlation is smaller. It is reasonable, since when the squeezing degree of the squeezed vacuum states is large, the squeezing factor r is not an important parameter in the second-order mode-mode correlation, which can be seen in Eq. (2). For strong squeezed vacuum states, i.e., r P 2.5, the disturbance tends to a saturated value. Therefore, if Eve uses a beam splitter to divert a part of light to gain useful information, the results of the second-order cross-correlation function at BobÕs receiving
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J. Xiong et al. / Optics Communications 260 (2006) 351–354
station will absolutely decrease, which indicates that EveÕs intervention is detectable. If Eve uses a 50:50 beam splitter, the disturbance she brought reaches maximum, and the second-order mode–mode correlation disappears. Thus we can use the second-order coherence as a new approach to evaluating whether the eavesdropping exists. This new approach has a great merit of that the second-order coherence can be easy experimentally measured with an HBT intensity interferometer in a lab. 4. Improved QKD protocol With the eavesdropping detection approach suggested in Section 3, eavesdropping can be directly checked out without the requirement of random sampling part of the secret bits as test key. Thus we can improve PereiraÕs QKD protocol [7] as follows: ðÞ jW12 i
(1) Alice prepares two different squeezed states and decides at random which to send on to Bob. (2) At the receiving station, Bob randomly decides whether he should obtain secret bits (3a) or detect eavesdropping (3b) for each state Alice sent. (3a) Bob just chooses to measure one of the two quadratures by performing homodyne detection to get secret bits. (3b) Bob uses an HBT intensity interferometer to detect EveÕs presence.If Eve uses a beam splitter to sample part of the signal to gain information, the results of the second-order cross-correlation function at BobÕs receiving station will absolutely decrease. If Eve uses a 50:50 beam splitter, the disturbance she brought reaches maximum, and the second-order mode–mode correlation disappears. (4) If Bob evaluates Eve is on line, then he tells Alice and stops the communication. Otherwise, the communication continues. (5) After all communications over a quantum channel, Alice and Bob communicate with each other via a classical channel to disclose the quadratures they used. Then they discard their data when the quadratures differ, while the rest list of correlation real numbers is used to build a secret key. Then the communication is successfully terminated. The rigorous security proof of the QKD system with two-mode squeezed states can be found in [22], where the security is based on the increase of error probability due to the eavesdropping. It is shown that the QKD system is secure from opaque eavesdropping using the unambiguous or the error optimum detection even in the case of lossy transmission channels. Furthermore, our improved protocol is more secure, since eavesdropping can be directly checked out in step (3b). In addition, no secret bit requires
to be discarded and the efficiency is enhanced greatly, which is of much significance in the practical application. 5. Conclusion In summary, we have presented an improved QKD protocol, originated by Pereira et al. The improved scheme relies on the second-order coherence of light fields, which can be easy experimentally measured with an HBT intensity interferometer. Eavesdropping can be directly detected without the requirement of extra bits sampled randomly from the newly shared secret bits as test key. In this way, no secret bit has to be discarded, and the two communicators can securely transmit messages with higher efficiency. Acknowledgements The authors thank the editor and the anonymous reviewers for their valuable suggestions and comments on an earlier version of this paper. This research is supported by the National Natural Science Foundation of China (Grant No. 60472018). References [1] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [2] N. Zhou, G. Zeng, J. Xiong, Electron. Lett. 40 (2004) 1149. [3] N.Y. Gordeev, K.J. Gordon, G.S. Buller, Opt. Commun. 234 (2004) 203. [4] T.C. Ralph, Phys. Rev. A 61 (1999) 010303 (R); T.C. Ralph, Phys. Rev. A 62 (2000) 062306. [5] M. Hillery, Phys. Rev. A 61 (2000) 022309. [6] M.D. Reid, Phys. Rev. A 62 (2000) 062308. [7] S.F. Pereira, Z.Y. Ou, H.J. Kimble, Phys. Rev. A 62 (2000) 042311. [8] F. Grosshans et al., Nature 421 (2003) 238. [9] N.J. Cerf, A. Ipe, X. Rottenberg, Phys. Rev. Lett. 85 (2000) 1754. [10] D. Gottesman, J. Preskill, Phys. Rev. A 63 (2001) 022309. [11] Ch. Silberhorn et al., Phys. Rev. Lett. 88 (2002) 167902; Ch. Silberhorn et al., Phys. Rev. Lett. 89 (2002) 167901. [12] S.L. Braunstein, P. van Loock, Rev. Mod. Phys. 77 (2005) 513, and references therein. [13] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. [14] Z.Y. OU, S.F. Pereira, J.H. Kimble, K.C. Peng, Phys. Rev. Lett. 68 (1992) 3663. [15] R. Hanbury-Brown, R.Q. Twiss, Nature. 177 (1956) 27. [16] M. Stobin´ska, K. Wo´dkiewicz, Phys. Rev. A 71 (2005) 032304. [17] S.M. Tan, D.F. Walls, M.J. Collett, Phys. Rev. Lett. 77 (1990) 285. [18] B.C. Sander, Phys. Rev. A 45 (1992) 6811; B.C. Sander, Phys. Rev. A 52 (1995) 735. [19] M.S. Kim, W. Son, V. Buzek, P.L. Knight, Phys. Rev. A 65 (2002) 032323. [20] X.L. Feng, R.X. Li, Z.Z. Xu, Phys. Rev. A 71 (2005) 032335. [21] R.A. Campos, B.E.A. Saleh, M.C. Teich, Phys. Rev. A 40 (1989) 1371. [22] M. Osaki, M. Ban, Phys. Rev. A 68 (2003) 022325.
An improved quantum key distribution protocol based on second-order coherence Jin Xiong *, Guihua Zeng, Nanrun Zhou The State Key Laboratory on Fiber-Optic Local Area Networks and Advanced Optical Communication Systems, Department of Electronic Engineering, Shanghai Jiaotong University, Room 602, No. 28, Lane 865, Dongchuan Road, Shanghai 200240, China Received 26 June 2005; received in revised form 13 October 2005; accepted 21 October 2005
Abstract We improve the quantum key distribution protocol proposed by Pereira et al. [S.F. Pereira, Z.Y. Ou, H.J. Kimble, Phys. Rev. A 62 (2000) 042311], by employing the second-order coherence of optical fields, which can be easy experimentally measured with a HanburyBrown and Twiss intensity interferometer. It is shown that eavesdropping can be directly detected without sacrificing extra secret bits as test key. The efficiency of the improved system is enhanced greatly, since no secret bit needs to be discarded. Ó 2005 Elsevier B.V. All rights reserved. PACS: 03.67.Dd; 03.67.HK; 42.50.Ar Keywords: Quantum key distribution; Second-order coherence; HBT intensity interferometer; Quantum cryptography
1. Introduction In recent years, it has attracted much attention about the possibility of using quantum mechanics to transmit signals in such a way that any eavesdropping can be detected by the two legitimate communicating parties [1–3]. This idea has been applied to quantum key distribution (QKD), a technique that allows two remote parties to share a secret key. The original QKD protocols have all involved the transmission of single particles, for example, single photons. In this method, a significant current limitation on the practicality is the poor efficiency of photon counting detectors. And losses limit the transmission distance. Fortunately, it has been shown that continuous variable quantum systems [4–11] could offer an alternative to the usual single photon counting QKD. They can be implemented either by non-classical beams, such as squeezed light [4,5,7] and entangled light [6], or by classical beams, i.e., Gaussian-modulated coherent light [8]. This kind of *
Corresponding author. E-mail address: [email protected] (J. Xiong).
0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.050
QKD schemes are potentially advantageous because they may achieve higher transmission and repetition rates in the near future [12]. As is well known, there exists mode–mode correlation [13] between the two modes of a two-mode squeezed light, which can be generated by non-degenerate optical parametric amplification via non-linear interactions in a crystal [14]. The QKD protocol employing such two-mode squeezed states has been carried out by Pereira et al. [7]. They evaluated the security of the system based on the signal to noise ration obtained by a homodyne detector. If an eavesdropper, Eve, attempts to gain useful information, she will necessarily increase the noise level and error rate at the receiving station. In other words, EveÕs intervene will introduce a well-known penalty of a 3 dB reduction into the signal-to-noise ration. However, similarly to the previous protocols, their protocol suffers a low efficiency since a part of newly generated secret bits must be sacrificed as test key after transmission. Motivated by this, in this paper, we employ the second-order coherence of optical fields to improve the efficiency of the original protocol of Pereira et al.
352
J. Xiong et al. / Optics Communications 260 (2006) 351–354
We structure our paper as the following style. The second-order coherence of optical fields is described in Section 2. And an eavesdropping detecting approach by employing second-order coherence is introduced in Section 3. In Section 4, an improved QKD protocol is summarized and a brief conclusion is given in Section 5. 2. Second-order coherence First, let us give a brief review of second-order coherence of light fields. It is known that the famous YoungÕs interference experiment is a classical optical interference experiment, which is a measurement of the first-order correlation function of the field [13]. However, this kind of experiments cannot be employed for judging the quantum nature and classical nature of light. In the 1950s, Hanbury-Brown and Twiss developed a new kind of correlation experiment for applications in intensity-correlation stellar interferometry [15]. They demonstrated the possibility of measuring the magnitude of the coherence function by means of the correlation between intensity fluctuations at two different points, rather than the correlation between the fields. The Hanbury-Brown and Twiss (HBT) experiment, the first observation of higher-order coherence, can reveal quantum properties of light sources, not seen in YoungÕs interference experiment. A recent work has shown that the second-order coherence and HBT experiment can provide an operational framework for the construction of witness operators that can test classical and non-classical properties of a Gaussian squeezed state [16]. In the QKD system implemented by Pereira et al., the two different two-mode squeezed vacuum states Alice prepared can be expressed by ðÞ
y y
jW12 i ¼ erða1 a2 a1 a2 Þ j01 ; 02 i;
ð1Þ
where 1 and 2 indicate two modes of the states, and a1 ðay1 Þ and a2 ðay2 Þ are the photon annihilation (creation) operators for mode 1 and mode 2, respectively. Here, r is the squeeze parameter representing the degree of squeezing. The vacðþÞ ðÞ uum fluctuations of jW12 i and jW12 i are squeezed in the directions of U1 and U2, respectively, with the definitions of U 1 ¼ 12 ða1 þ ay1 þ a2 þ ay2 Þ and U 2 ¼ 2i1 ða1 ay1 þa2 ay2 Þ. We restrict our attention to the normalized second-order ð2Þ cross-correlation function gð12Þ ð0Þ, which describes the intensity correlation between the two modes ð2Þ
g12 ð0Þ ¼
hay1 a1 ay2 a2 i hay1 a1 ihay2 a2 i
¼2þ
1 ; hni
ð2Þ
the second-order cross-correlation function tends to the constant 2. That is, for strong squeezed vacuum states, the squeezing factor r is not an important parameter in the second-order mode–mode correlation. 3. Eavesdropping detecting As is well known, the beam splitter is ubiquitous in many optical systems and plays a fundamental role in many fields such as interferometry, holography, laser systems, etc. Many of its usages in quantum optics and quantum information are derived from the fact that it can act as an entangler [17–20], i.e., an unentangled input beam can emerge from the device in an entangled state. In quantum cryptography, the beam splitter is often used as an eavesdropping tool [4–6], where Eve employs a beam splitter to split off part of the signal and performs measurements on that part to gain useful information. The abstract model of a lossless beam splitter is a linear four-port device, shown in Fig. 1. Two radiation modes enter the instrument, interfere with each other, and then leave it. The beam splitter acts in three steps: (1) shift the phases of the input modes, (2) mix the modes via a rotation, and (3) shift the phase again. The phase shifts could be eliminated by a proper phase redefinition of the incoming/outcoming modes. The rotation would remain. Mixing of modes is in fact the essential operation of a beam splitter. The action of the beam splitter may be corresponding to the unitary transformation connecting the two input fields and the two output fields. Denoting the annihilation operators of the input modes by a1 and a2 and the output modes by b1 and b2, the input/output relation of the beam splitter can be written as [21] b1 a1 ð3Þ ¼B Bþ ; b2 a2 where the unitary evolution beam splitter operator B in the representation of the angular momentum is
b2
a1
b1
ðÞ
where hni = sinh2 r is the mean photon number of jW12 i. Eq. (2) shows that the second-order intensity correlation between the two modes of the two-mode squeezed vacuum ðÞ states jW12 i is a function of the squeezing factor r. With the increasing of the squeezing degree of the squeezed states, the second-order cross-correlation function decreases. For strong squeezed vacuum states, i.e., r P 2.5, the mean photon number hni = sinh2 r is very large and
a2 Fig. 1. The diagrammatic representation of a beam splitter as an optical element with two input ports and two output ports. a1 and a2 denote two input modes and b1 and b2 two output modes.
J. Xiong et al. / Optics Communications 260 (2006) 351–354
ð4Þ 0.6
pffiffiffiffi T pffiffiffi i/ Re
pffiffiffi i/ ! a1 Re pffiffiffiffi . a2 T
12
ð5Þ
In PereiraÕs protocol, Eve may choose two typical possible methods of eavesdropping. If Eve uses the captureresend eavesdropping strategy, she will introduce errors which can be detected by Alice and Bob [7]. If Eve performs a beam splitter to gain useful information, that is, she sends on to Bob the part of the signal that is transmitted through the beam splitter and performs measurements on the part that is reflected, she will bring disturbance to BobÕs station. In order to check out EveÕs presence without lowing efficiency, we employ the second-order coherence of light fields. By a lossless symmetric beam splitter, in the Schro¨dinger picture the fields Bob received are changed because of EveÕs intervention, as g ðÞ ðÞ jW12 i ¼ BjW12 i. ð6Þ Note that we have introduced the symbol ‘‘’’ to tag the variables influenced under EveÕs disturbance. g g ðþÞ ðÞ For each state jW12 i or jW12 i Bob received, the secondg ð2Þ order cross-correlation function g12 ð0Þ can be obtained with Eqs. (4)–(6) 1 1 g ð2Þ 4Rð1 RÞ 1 þ g12 ð0Þ ¼ 2 þ hni hni ð2Þ
0.4 0.3 0.2
¼ g12 ð0Þ Dg ;
ð7Þ
1 where Dg 4Rð1 RÞð1 þ hni Þ is the disturbance item due g ð2Þ ð2Þ to EveÕs eavesdropping. Eq. (7) shows g12 ð0Þ < g12 (for R 5 0). It means that under EveÕs intervention, the degree of the second-order mode–mode correlation decreases. From Eq. (7), one can see that the disturbance item Dg is a function of the reflection coefficient R and the squeezing factor r. The features of the disturbance due to EveÕs presence can be seen more clearly in Figs. 2 and 3, where the ð2Þ ratio of Dg to g12 ð0Þ is plotted as a function of R and r, respectively. ð2Þ Fig. 2 shows that the figure of Dg =g12 ð0Þ versus R is a parabola, which has maximal value at the point R = 0.5. That is, if the beam splitter used by Eve is a 50:50 one, i.e., T = R = 0.5, the second-order cross-correlation funcg ð2Þ tion g12 ð0Þ decreases to the minimum 1, which can be
0.1 0 0
0.1
0.2
0.3
0.4
0.5 R
0.6
0.7
0.8
0.9
1
4.5
5
ð2Þ
Fig. 2. The figure of Dg =g12 ð0Þ versus R, with r = 1.
1 0.9 0.8 (2)
Bþ ¼ B ; a1 þ B B¼ a2
0.5
g
Here, h is the single angular parameter of the beam splitter which satisfies the equations R ¼ sin2 h2 and T ¼ cos2 h2, where R and T are the reflection and transmission coefficients with the normalization T + R = 1. The parameter / is the phase difference between the reflected and transmitted amplitudes given by the beam splitter. The beam splitter operator B has the following properties:
Δ /g(2)(0)
h y i/ ða1 a2 e a1 ay2 ei/ Þ . 2
Δ g/g12 (0)
B ¼ exp
353
0.7
0.6
0.5 0.4 0
0.5
1
1.5
2
2.5 r
3
3.5
4
ð2Þ
Fig. 3. The figure of Dg =g12 ð0Þ versus r, with R = 0.4.
easily found from Eq. (7). It indicates that the secondorder mode–mode correlation disappears [13]. Under this situation, the disturbance brought by Eve reaches the maximum. For a certain beam splitter, i.e., the reflection coefficient R is given, the disturbance due to EveÕs eavesdropping decreases with the increasing of the squeezing factor, shown in Fig. 3. That is, if the squeezing degree of the two-mode squeezed vacuum states is stronger, the disturbance Eve brought appearing in the second-order modemode correlation is smaller. It is reasonable, since when the squeezing degree of the squeezed vacuum states is large, the squeezing factor r is not an important parameter in the second-order mode-mode correlation, which can be seen in Eq. (2). For strong squeezed vacuum states, i.e., r P 2.5, the disturbance tends to a saturated value. Therefore, if Eve uses a beam splitter to divert a part of light to gain useful information, the results of the second-order cross-correlation function at BobÕs receiving
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station will absolutely decrease, which indicates that EveÕs intervention is detectable. If Eve uses a 50:50 beam splitter, the disturbance she brought reaches maximum, and the second-order mode–mode correlation disappears. Thus we can use the second-order coherence as a new approach to evaluating whether the eavesdropping exists. This new approach has a great merit of that the second-order coherence can be easy experimentally measured with an HBT intensity interferometer in a lab. 4. Improved QKD protocol With the eavesdropping detection approach suggested in Section 3, eavesdropping can be directly checked out without the requirement of random sampling part of the secret bits as test key. Thus we can improve PereiraÕs QKD protocol [7] as follows: ðÞ jW12 i
(1) Alice prepares two different squeezed states and decides at random which to send on to Bob. (2) At the receiving station, Bob randomly decides whether he should obtain secret bits (3a) or detect eavesdropping (3b) for each state Alice sent. (3a) Bob just chooses to measure one of the two quadratures by performing homodyne detection to get secret bits. (3b) Bob uses an HBT intensity interferometer to detect EveÕs presence.If Eve uses a beam splitter to sample part of the signal to gain information, the results of the second-order cross-correlation function at BobÕs receiving station will absolutely decrease. If Eve uses a 50:50 beam splitter, the disturbance she brought reaches maximum, and the second-order mode–mode correlation disappears. (4) If Bob evaluates Eve is on line, then he tells Alice and stops the communication. Otherwise, the communication continues. (5) After all communications over a quantum channel, Alice and Bob communicate with each other via a classical channel to disclose the quadratures they used. Then they discard their data when the quadratures differ, while the rest list of correlation real numbers is used to build a secret key. Then the communication is successfully terminated. The rigorous security proof of the QKD system with two-mode squeezed states can be found in [22], where the security is based on the increase of error probability due to the eavesdropping. It is shown that the QKD system is secure from opaque eavesdropping using the unambiguous or the error optimum detection even in the case of lossy transmission channels. Furthermore, our improved protocol is more secure, since eavesdropping can be directly checked out in step (3b). In addition, no secret bit requires
to be discarded and the efficiency is enhanced greatly, which is of much significance in the practical application. 5. Conclusion In summary, we have presented an improved QKD protocol, originated by Pereira et al. The improved scheme relies on the second-order coherence of light fields, which can be easy experimentally measured with an HBT intensity interferometer. Eavesdropping can be directly detected without the requirement of extra bits sampled randomly from the newly shared secret bits as test key. In this way, no secret bit has to be discarded, and the two communicators can securely transmit messages with higher efficiency. Acknowledgements The authors thank the editor and the anonymous reviewers for their valuable suggestions and comments on an earlier version of this paper. This research is supported by the National Natural Science Foundation of China (Grant No. 60472018). References [1] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [2] N. Zhou, G. Zeng, J. Xiong, Electron. Lett. 40 (2004) 1149. [3] N.Y. Gordeev, K.J. Gordon, G.S. Buller, Opt. Commun. 234 (2004) 203. [4] T.C. Ralph, Phys. Rev. A 61 (1999) 010303 (R); T.C. Ralph, Phys. Rev. A 62 (2000) 062306. [5] M. Hillery, Phys. Rev. A 61 (2000) 022309. [6] M.D. Reid, Phys. Rev. A 62 (2000) 062308. [7] S.F. Pereira, Z.Y. Ou, H.J. Kimble, Phys. Rev. A 62 (2000) 042311. [8] F. Grosshans et al., Nature 421 (2003) 238. [9] N.J. Cerf, A. Ipe, X. Rottenberg, Phys. Rev. Lett. 85 (2000) 1754. [10] D. Gottesman, J. Preskill, Phys. Rev. A 63 (2001) 022309. [11] Ch. Silberhorn et al., Phys. Rev. Lett. 88 (2002) 167902; Ch. Silberhorn et al., Phys. Rev. Lett. 89 (2002) 167901. [12] S.L. Braunstein, P. van Loock, Rev. Mod. Phys. 77 (2005) 513, and references therein. [13] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. [14] Z.Y. OU, S.F. Pereira, J.H. Kimble, K.C. Peng, Phys. Rev. Lett. 68 (1992) 3663. [15] R. Hanbury-Brown, R.Q. Twiss, Nature. 177 (1956) 27. [16] M. Stobin´ska, K. Wo´dkiewicz, Phys. Rev. A 71 (2005) 032304. [17] S.M. Tan, D.F. Walls, M.J. Collett, Phys. Rev. Lett. 77 (1990) 285. [18] B.C. Sander, Phys. Rev. A 45 (1992) 6811; B.C. Sander, Phys. Rev. A 52 (1995) 735. [19] M.S. Kim, W. Son, V. Buzek, P.L. Knight, Phys. Rev. A 65 (2002) 032323. [20] X.L. Feng, R.X. Li, Z.Z. Xu, Phys. Rev. A 71 (2005) 032335. [21] R.A. Campos, B.E.A. Saleh, M.C. Teich, Phys. Rev. A 40 (1989) 1371. [22] M. Osaki, M. Ban, Phys. Rev. A 68 (2003) 022325.