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Analysis of polarization state in quantum key distribution via single-photon two-qubit states
Optik 125 (2014) 1522–1525
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of polarization state in quantum key distribution via single-photon two-qubit states Gu-hao Zhao a,∗ , Shang-hong Zhao a , Zhou-shi Yao b , Chen-Lu Hao c , Wen Meng a , Xiang Wang a , Zhi-hang Zhu a , Feng Liu b a
School of Information and Navigation, Air Force Engineering University, Xian 710077, Shaanxi, China Xi’an Branch of China Academy of Space Technology, Xian 710000, Shaanxi, China c School of Air and Missile Defense, Air Force Engineering University, Xian 710077, Shaanxi, China b
a r t i c l e
i n f o
Article history: Received 2 March 2013 Accepted 3 July 2013
PACS: 03.67.Dd 84.40.Ua 77.22.Ej
a b s t r a c t An improved quantum key distribution scheme via single-photon two-qubit states is proposed. The input–output model of the polarization state is established. And the influence of the interferometers to the polarization state is analyzed. Quantum bit error rate of polarization coding caused by birefringent and coordinate system difference between incident light and the fast and slow axes in fiber interferometer is simulated. Furthermore, maintaining conditions of polarization state are given on this basis. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Quantum key distribution Mach–Zehnder interferometers Phase coding Polarization coding Stability conditions of polarization
1. Introduction Since Bennett and Brassard presented the first quantum key distribution (QKD) scheme in 1984 [1], quantum cryptography is attracting researchers’ attention [2–4]. A lot of schemes and theories have been proposed and improved [5–8]. These QKD schemes and theories have been implemented in a number of groups both in fiber-based systems [9–11] and in free space arrangements [12–16]. Most existing QKD systems rely on either polarization or phase of faint laser pulses as an information carrier. For long-distance free space distribution, the main difficulty of practical systems is the key generation rate. Because of the link loss, “Bob” can just capture a few photons which “Alice” sends to. It is not a significant issue for experimental demonstrations but a vital disadvantage for practical applications. Several schemes have been proposed to improve the key generation rate [17,18]. Single-photon two-qubit states possess a deterministic nature that can be exploited for direct secure communication [19]. Paper [20] proposed a scheme for quantum key
∗ Corresponding author. E-mail address: [email protected] (G.-h. Zhao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.07.072
distribution, in which a single photon is encoded with polarization state and phase state. But the implementation system of this scheme is a little bit complicated, and it has just analyzed the security and implementation roughly. In this article, an improvement scheme of free-space QKD via single-photon two-qubit states is proposed. The implementation system of this scheme is concise. And the quantum bit error rate (QBER) of polarization coding caused by fiber M–Z interferometer is discussed. 2. Improvement scheme of two-qubit QKD The implementation system in paper [21] needs four interferometers and also needs as many photon detectors and beam splitters as two polarization QKD systems. So we improve the scheme and implementation system as shown in Fig. 1. The two-qubit QKD scheme setups are shown in Fig. 1. Only four semiconductor lasers (LD1–LD4) send out one pulse carried polarization message at a time. The laser pulse passed through Alice’s interferometer and was modulated phase information and attenuated by the attenuator (A). Then, the pulse was launched into free space. The pulse was caught by Bob’s antenna and passed through an interferometer. Then enters the detection unit that has eight
G.-h. Zhao et al. / Optik 125 (2014) 1522–1525
detectors (D) on the output of Bob’s interferometer to distinguish what phase and polarization message the photon takes. L, S represents, respectively, the long and short arms of Alice and Bob’s interferometers. Like the original phase QKD system, here are four possible paths to reach the detectors in the twoqubit QKD scheme; they are L + S, S + S, S + L, L + L. Fig. 2 shows the output pulses of Bob’s interferometer. Because the lengths of L + S and S + L are equal, so these two pulses overlap in time domain (pulse B), and the interference occurs. For the pulses of path S + S and path L + L, here is no interference. So they are kept out of the door in original phase QKD scheme. The two-qubit QKD scheme is proposed to codes the phase and polarization information in one photon. Specifically, the interference pulse (pulse B) carries both phase and polarization information, and the pulses (A and C) which are discarded in the original phase QKD scheme can be coded in polarization information. The principle of key distribution in the improvement scheme is as follows. (1) The polarization coding unit of Alice randomly activates for each photon pulse to rotate the state √of polarization √ to one of the√ four states,√which are −45◦ ((1/ 2)|H − (1/ 2)|V ), +45◦ (1/ 2|H + 1/ 2|V ), 0◦ (|H ), 90◦ (|V ), then sends them into the M–Z interferometer. (2) The phase modulator randomly encodes 0, , /2 or 3/2 states to pulse in the long arm. Then the photon carries two different quantum key information which are sent to Bob. (3) In much the same way that phase QKD scheme decode does, the Bob’s phase modulator selects 0 or /2 phase to measuring what phase the pulse B takes. (4) All the pulses of A, B and C carry the polarization information. The polarization decoding units decoding the polarization information in the same way as polarization QKD scheme, both in out1 and out2. Bob told Alice what measurement matrix he used in the phase and polarization detection. But Bob did not publish the result he had obtained. Then Alice tells Bob which results are correct. 3. Polarization QBER caused by M–Z interferometers From the standpoint of interference, the difference of transmission matrix between long arm and short arm must be very small. And in some mobile platforms, the polarization state will change with the change of mirror’s position drastically. Typically, the encapsulated fiber interferometer is used in the phase QKD scheme quantum distribution experiment. But the polarization state of photon also changes slowly in the fiber system, largely because of the birefringence. The bit error rate of polarization coding increases. We will discuss the variation regularity of polarization state and QBER under the influence of fiber interferometer in detail below. Take the case ofS + S path, suppose Jones vector of incident Ex1 Ex2 light is EAin = , and the output light is Eout1 = . The Ey1 Ey2 coordinate system of fast and slow axes in the fiber interferometer is (u, v). The coordinate system of incident light is (x, y), and the angle between these two coordinate systems is . Matrix cos sin for coordinate transform is R() = . Jones matrix − sin cos
of fiber is J =
eiSX 0
0 eiSX
= ei(SX +SX )/2
ei
SX /2
0
0 ei
SX /2
1523
fiber-based interferometers, JC is the rotation matrix.
Eout1 =
Ex2
i
= R(−)JSB R(ϕ)Ce JSA R()
Ey2
0 −i e
⎡ i SA i(SA + SA ) ⎢ e 2 ⎢ R(ϕ)Cei e 2 ⎣
Ex1
R()
e
Ey1
0 −i e
0 −i e
SB
+ 2
SA
−e
Ex1
R()
Ey1
e
SA
E ⎥ ⎥ R() x1 ⎦ E y1
2
SB
⎡ i SA ⎥ ⎢e 2 ⎥ R(ϕ) ⎢ ⎦ ⎣
2
⎤ 0 −i e
0
SA
⎥ ⎥ ⎦
2
i(SB + SB ) i(SA + SA ) ei e 2 2
⎡ i( ⎢ e = R(−) ⎢ ⎣ i(
2
⎤
0
SB
⎥ ⎥ ⎦ ⎤
0
i(
SA )
cos ϕ
− 2
SB
e −i(
SB )
sin ϕ
− 2
sin ϕ
+ 2
SA )
SB
e
⎤
SA )
⎥ ⎥ ⎦
cos ϕ
i(SB + SB ) i(SA + SA ) ei e 2 2
where the ˚ represents common phase of the atmospheric chanei 0 nel, C is considered as unitary. Let M = R(−) R(), 0 e−i = ( SB + SA )/2 represent the birefeingent phase difference between fast and slow axes in the path.
M =
cos
− sin
sin
cos
=
=
ei
0
0
e−i
cos
sin
− sin
cos
cos2 ei + sin2 e−i
sin cos (ei − e−i )
sin cos (ei − e−i )
sin2 ei + cos2 e−i
cos + i sin cos 2
i sin sin 2
i sin sin 2
co s − i sin cos 2
The polarization measurement is little affected by the public phase. And the output of these four polarizations in polarization coding could be:
Eout0 = ME0 = =
cos + i sin cos 2
cos2 + sin2 cos2 2|0 +
Eout90 = ME90 = =
i sin2 sin 2
,
SX = SX − SX is the phase difference between v-phase component and u-phase component, and SX = {SA, SB} represent the short arm fibers of interferometer both on Alice’s side and Bob’s side. Considering the difference between coordinate systems of two
⎤
0
Ey1
⎡ i SB i(SB + SB ) ⎢ e 2 ⎢ = R(−)e 2 ⎣
⎡ i SB 2 e ⎢ = R(−) ⎢ ⎣
Ex1
sin2 + sin2 2|1
i sin sin 2 cos − i sin cos 2
sin2 sin2 2|0 +
cos2 + sin2 cos2 2|1
1524
G.-h. Zhao et al. / Optik 125 (2014) 1522–1525
Eout45 = ME45 =
=
+
cos + i sin cos 2 + i sin sin 2 i sin sin 2 + cos − i sin cos 2 2
cos2 + sin2 (cos 2 + sin 2) |0 2
cos2 + sin2 (cos 2 − sin 2) |1 2
Eout−45 = ME−45 =
=
+
cos + i sin cos 2 − i sin sin 2
i sin sin 2 − cos + i sin cos 2 2
cos2 + sin2 (cos 2 − sin 2) |0 2
cos2 + sin2 (cos 2 + sin 2) |1 2
Fig. 4. QBER of polarization coding caused by M–Z interferometer.
The QBER of these polarizations are:
the safety upper limit of QBER (11%) and the QBER (2–5%) caused by the other factors are concerned.
Q0 = Q90 = sin2 sin2 2 1 =1− 2
Q45 = Q−45
cos2 + sin2 (cos 2 + sin 2)
+
2
cos2
2
+ sin (cos 2 − sin 2)
2
4. The polarization maintaining conditions By using the method of matrix optical, the relationship between input and output light can be written as EBout1 = (JSB eiˇB + JLB ei(˛B +ϕB ) )Cei (JSA eiˇA + JLA ei(˛A +ϕA ) )EAin
Excluding the other factors, the polarization coding QBER can be totally represented as: QBERP = 0.25 × (Q0 + Q90 + Q45 + Q−45 ) Here Figs. 3 and 4 show the connections among the polarization QBER, coordinate system difference and the phase difference caused by birefringent. In Fig. 3, the first derivative of the -QBER curve decreased with the increasing. And the first derivative of the -QBER curve increased with the increasing. That means when the fiber is short (like the fiber interferometers), the polarization QBER is greatly affected by the difference between the coordinate systems of the fast and slow axes in fiber and incident light. When the fiber is long (like long distance fiber channel), the phase difference caused by birefringent is the principal factor of the rising of polarization QBER. In Fig. 4, when and are both below 0.2 rad, the polarization QBER is less than 0.015. When and are both below 0.4 rad, the polarization QBER is less than 0.06. The polarization QBER in two-qubit QKD scheme could be accepted as far as
= JSB CJSA EAin ei(ˇB ++ˇA ) + JSB CJLA EAin ei(ˇB ++˛A +ϕA ) + JLB CJSA EAin ei(˛B +ϕB ++ˇA ) + JLB CJLA EAin ei(˛B +ϕB ++˛A +ϕA ) where ˛A and ˛B are the equivalent phase of long arm, and ˇA and ˇB are the equivalent phase of short arm. ϕA and ϕB are the equivalent phase of phase modulator. C is the Jones matrix of atmospheric channel. ˚ is the equivalent phase of atmospheric channel. The first term on right-hand side of the equation represent S + S pulse, and the fourth term represent L + L pulse. These two pulses are abandoned in the original phase QKD scheme. They are modulated polarization information in the two-qubit QKD scheme. So the anti-disturbance conditions of these two pulses are JSB CJSA = JLB CJLA = I The rest two terms are the interference pulses and the antidisturbance conditions of these two pulses are JSB CJLA = JLB CJSA = I There is little birefringence in the atmosphere. Where, C is considered as unitary. Aforementioned anti-disturbance conditions can also be equivalently written as: JLA = JSA ,
Fig. 3. QBER of polarization coding caused by and .
JLB = JSB ,
−1 JLA = JLB ,
−1 JSA = JSB
More specifically, the Jones matrix of each two long arms and the two short arms should be the same. And the Jones matrixes of long arm should be the reciprocal matrix of short arm’s which belongs to both Alice side and Bob side. Contrasting with paper [21], LA = SA and LB = SB also are the anti-disturbance conditions to keep the interference visibilities in the original phase QKD scheme stable. LA = LB −1 and SA = SB −1 are the additional conditions for the polarization coding.
G.-h. Zhao et al. / Optik 125 (2014) 1522–1525
5. Conclusion In summary, the QBER of polarization coding can be influenced deeply by the fiber M–Z interferometers. The theoretic analysis reveals that the jitter of polarization state is not only derived from the birefringent, but also derived from coordinate system difference between incident light and fast and slow axes in fiber interferometer. Some necessary conditions for the QKD via singlephoton two-qubit states are given. However, all the simulation results are obtained in perfect atmosphere channel. The influence of atmosphere turbulence and complicated weather should be considered in the sequent works. And the polarization maintaining conditions in the fiber interferometer maybe is neither stable enough nor easy to realize perfectly. So some active compensation method for polarization should be analyzed. Acknowledgments The authors would like to thank Doctor Tang Feng and Doctor Tian Xiaofei for the helpful advice. References [1] C.H. Bennett, G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, 1984, pp. 175–179.
1525
[2] N. Zhao, C.X. Pei, D. Liu, et al., Acta Phys. Sin. 60 (2011) 1–6, 090307. [3] K. Mattle, H. Weinfurter, P.G. Kwiat, et al., Phys. Rev. Lett. 76 (1996) 4656–4659. [4] B.C. Jacobs, J.D. Franson, Opt. Lett. 21 (1996) 1854–1856. [5] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881–2884. [6] C.H. Bennett, Phys. Rev. Lett. 69 (1992) 3121–3124. [7] F.A.A. El-Orany, M.R.B. Wahiddin, M-A. Mat-Nor, N. Jamil, I. Bahari, Laser Phys. 20 (2010) 1210–1214. [8] D.A. Kronberg, S.N. Molotkov, Laser Phys. 19 (2009) 884–893. [9] Z.F. Han, X.F. Mo, Y.Z. Gui, et al., Appl. Phys. Lett. 86 (2005) 221103. [10] W. Wan, Z.J. Wei, J.D. Wang, et al., Acta Phys. Sin. 60 (2011) 1–6, 094216 (in Chinese). [11] D. Stucki, et al., N. J. Phys. 13 (2011) 1–5, 123001. [12] Y.C. Jeong, Y.S. Kim, Y.H. Kim, Laser Phys. 21 (2011) 1438–1442. [13] T.S. Manderbachet, et al., Phys. Rev. Lett. 98 (2007) 1–5, 010504. [14] M.P. Peloso, et al., N. J. Phys. 11 (2009) 045007. [15] W.T. Buttler, et al., Phys. Rev. Lett. 84 (2000) 5652–5655. [16] D.M. Benton, P.M. Gorman, P.R. Tapster, et al., Opt. Commun. 283 (2010) 2465–2471. [17] K. Inoue, E. Waks, Y. Yamamoto, Rev. Lett. 89 (2002) 037902. [18] B.K. Zhao, F.G. Deng, H.Y. Zhou, Opt. Commun. 282 (2009) 3379–3380. [19] A. Beige, B.G. Englert, C. Kurtsiefer, et al., J. Phys. A: Math. Gen. 35 (2002) L407–L413. [20] JinDong Wang, ZhengJun Wei, et al., J. Phys. B: Atom. Mol. Opt. Phys. 43 (2010) 095504. [21] Z.F. Han, et al., Appl. Phys. Lett. 86 (2005) 221103.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of polarization state in quantum key distribution via single-photon two-qubit states Gu-hao Zhao a,∗ , Shang-hong Zhao a , Zhou-shi Yao b , Chen-Lu Hao c , Wen Meng a , Xiang Wang a , Zhi-hang Zhu a , Feng Liu b a
School of Information and Navigation, Air Force Engineering University, Xian 710077, Shaanxi, China Xi’an Branch of China Academy of Space Technology, Xian 710000, Shaanxi, China c School of Air and Missile Defense, Air Force Engineering University, Xian 710077, Shaanxi, China b
a r t i c l e
i n f o
Article history: Received 2 March 2013 Accepted 3 July 2013
PACS: 03.67.Dd 84.40.Ua 77.22.Ej
a b s t r a c t An improved quantum key distribution scheme via single-photon two-qubit states is proposed. The input–output model of the polarization state is established. And the influence of the interferometers to the polarization state is analyzed. Quantum bit error rate of polarization coding caused by birefringent and coordinate system difference between incident light and the fast and slow axes in fiber interferometer is simulated. Furthermore, maintaining conditions of polarization state are given on this basis. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Quantum key distribution Mach–Zehnder interferometers Phase coding Polarization coding Stability conditions of polarization
1. Introduction Since Bennett and Brassard presented the first quantum key distribution (QKD) scheme in 1984 [1], quantum cryptography is attracting researchers’ attention [2–4]. A lot of schemes and theories have been proposed and improved [5–8]. These QKD schemes and theories have been implemented in a number of groups both in fiber-based systems [9–11] and in free space arrangements [12–16]. Most existing QKD systems rely on either polarization or phase of faint laser pulses as an information carrier. For long-distance free space distribution, the main difficulty of practical systems is the key generation rate. Because of the link loss, “Bob” can just capture a few photons which “Alice” sends to. It is not a significant issue for experimental demonstrations but a vital disadvantage for practical applications. Several schemes have been proposed to improve the key generation rate [17,18]. Single-photon two-qubit states possess a deterministic nature that can be exploited for direct secure communication [19]. Paper [20] proposed a scheme for quantum key
∗ Corresponding author. E-mail address: [email protected] (G.-h. Zhao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.07.072
distribution, in which a single photon is encoded with polarization state and phase state. But the implementation system of this scheme is a little bit complicated, and it has just analyzed the security and implementation roughly. In this article, an improvement scheme of free-space QKD via single-photon two-qubit states is proposed. The implementation system of this scheme is concise. And the quantum bit error rate (QBER) of polarization coding caused by fiber M–Z interferometer is discussed. 2. Improvement scheme of two-qubit QKD The implementation system in paper [21] needs four interferometers and also needs as many photon detectors and beam splitters as two polarization QKD systems. So we improve the scheme and implementation system as shown in Fig. 1. The two-qubit QKD scheme setups are shown in Fig. 1. Only four semiconductor lasers (LD1–LD4) send out one pulse carried polarization message at a time. The laser pulse passed through Alice’s interferometer and was modulated phase information and attenuated by the attenuator (A). Then, the pulse was launched into free space. The pulse was caught by Bob’s antenna and passed through an interferometer. Then enters the detection unit that has eight
G.-h. Zhao et al. / Optik 125 (2014) 1522–1525
detectors (D) on the output of Bob’s interferometer to distinguish what phase and polarization message the photon takes. L, S represents, respectively, the long and short arms of Alice and Bob’s interferometers. Like the original phase QKD system, here are four possible paths to reach the detectors in the twoqubit QKD scheme; they are L + S, S + S, S + L, L + L. Fig. 2 shows the output pulses of Bob’s interferometer. Because the lengths of L + S and S + L are equal, so these two pulses overlap in time domain (pulse B), and the interference occurs. For the pulses of path S + S and path L + L, here is no interference. So they are kept out of the door in original phase QKD scheme. The two-qubit QKD scheme is proposed to codes the phase and polarization information in one photon. Specifically, the interference pulse (pulse B) carries both phase and polarization information, and the pulses (A and C) which are discarded in the original phase QKD scheme can be coded in polarization information. The principle of key distribution in the improvement scheme is as follows. (1) The polarization coding unit of Alice randomly activates for each photon pulse to rotate the state √of polarization √ to one of the√ four states,√which are −45◦ ((1/ 2)|H − (1/ 2)|V ), +45◦ (1/ 2|H + 1/ 2|V ), 0◦ (|H ), 90◦ (|V ), then sends them into the M–Z interferometer. (2) The phase modulator randomly encodes 0, , /2 or 3/2 states to pulse in the long arm. Then the photon carries two different quantum key information which are sent to Bob. (3) In much the same way that phase QKD scheme decode does, the Bob’s phase modulator selects 0 or /2 phase to measuring what phase the pulse B takes. (4) All the pulses of A, B and C carry the polarization information. The polarization decoding units decoding the polarization information in the same way as polarization QKD scheme, both in out1 and out2. Bob told Alice what measurement matrix he used in the phase and polarization detection. But Bob did not publish the result he had obtained. Then Alice tells Bob which results are correct. 3. Polarization QBER caused by M–Z interferometers From the standpoint of interference, the difference of transmission matrix between long arm and short arm must be very small. And in some mobile platforms, the polarization state will change with the change of mirror’s position drastically. Typically, the encapsulated fiber interferometer is used in the phase QKD scheme quantum distribution experiment. But the polarization state of photon also changes slowly in the fiber system, largely because of the birefringence. The bit error rate of polarization coding increases. We will discuss the variation regularity of polarization state and QBER under the influence of fiber interferometer in detail below. Take the case ofS + S path, suppose Jones vector of incident Ex1 Ex2 light is EAin = , and the output light is Eout1 = . The Ey1 Ey2 coordinate system of fast and slow axes in the fiber interferometer is (u, v). The coordinate system of incident light is (x, y), and the angle between these two coordinate systems is . Matrix cos sin for coordinate transform is R() = . Jones matrix − sin cos
of fiber is J =
eiSX 0
0 eiSX
= ei(SX +SX )/2
ei
SX /2
0
0 ei
SX /2
1523
fiber-based interferometers, JC is the rotation matrix.
Eout1 =
Ex2
i
= R(−)JSB R(ϕ)Ce JSA R()
Ey2
0 −i e
⎡ i SA i(SA + SA ) ⎢ e 2 ⎢ R(ϕ)Cei e 2 ⎣
Ex1
R()
e
Ey1
0 −i e
0 −i e
SB
+ 2
SA
−e
Ex1
R()
Ey1
e
SA
E ⎥ ⎥ R() x1 ⎦ E y1
2
SB
⎡ i SA ⎥ ⎢e 2 ⎥ R(ϕ) ⎢ ⎦ ⎣
2
⎤ 0 −i e
0
SA
⎥ ⎥ ⎦
2
i(SB + SB ) i(SA + SA ) ei e 2 2
⎡ i( ⎢ e = R(−) ⎢ ⎣ i(
2
⎤
0
SB
⎥ ⎥ ⎦ ⎤
0
i(
SA )
cos ϕ
− 2
SB
e −i(
SB )
sin ϕ
− 2
sin ϕ
+ 2
SA )
SB
e
⎤
SA )
⎥ ⎥ ⎦
cos ϕ
i(SB + SB ) i(SA + SA ) ei e 2 2
where the ˚ represents common phase of the atmospheric chanei 0 nel, C is considered as unitary. Let M = R(−) R(), 0 e−i = ( SB + SA )/2 represent the birefeingent phase difference between fast and slow axes in the path.
M =
cos
− sin
sin
cos
=
=
ei
0
0
e−i
cos
sin
− sin
cos
cos2 ei + sin2 e−i
sin cos (ei − e−i )
sin cos (ei − e−i )
sin2 ei + cos2 e−i
cos + i sin cos 2
i sin sin 2
i sin sin 2
co s − i sin cos 2
The polarization measurement is little affected by the public phase. And the output of these four polarizations in polarization coding could be:
Eout0 = ME0 = =
cos + i sin cos 2
cos2 + sin2 cos2 2|0 +
Eout90 = ME90 = =
i sin2 sin 2
,
SX = SX − SX is the phase difference between v-phase component and u-phase component, and SX = {SA, SB} represent the short arm fibers of interferometer both on Alice’s side and Bob’s side. Considering the difference between coordinate systems of two
⎤
0
Ey1
⎡ i SB i(SB + SB ) ⎢ e 2 ⎢ = R(−)e 2 ⎣
⎡ i SB 2 e ⎢ = R(−) ⎢ ⎣
Ex1
sin2 + sin2 2|1
i sin sin 2 cos − i sin cos 2
sin2 sin2 2|0 +
cos2 + sin2 cos2 2|1
1524
G.-h. Zhao et al. / Optik 125 (2014) 1522–1525
Eout45 = ME45 =
=
+
cos + i sin cos 2 + i sin sin 2 i sin sin 2 + cos − i sin cos 2 2
cos2 + sin2 (cos 2 + sin 2) |0 2
cos2 + sin2 (cos 2 − sin 2) |1 2
Eout−45 = ME−45 =
=
+
cos + i sin cos 2 − i sin sin 2
i sin sin 2 − cos + i sin cos 2 2
cos2 + sin2 (cos 2 − sin 2) |0 2
cos2 + sin2 (cos 2 + sin 2) |1 2
Fig. 4. QBER of polarization coding caused by M–Z interferometer.
The QBER of these polarizations are:
the safety upper limit of QBER (11%) and the QBER (2–5%) caused by the other factors are concerned.
Q0 = Q90 = sin2 sin2 2 1 =1− 2
Q45 = Q−45
cos2 + sin2 (cos 2 + sin 2)
+
2
cos2
2
+ sin (cos 2 − sin 2)
2
4. The polarization maintaining conditions By using the method of matrix optical, the relationship between input and output light can be written as EBout1 = (JSB eiˇB + JLB ei(˛B +ϕB ) )Cei (JSA eiˇA + JLA ei(˛A +ϕA ) )EAin
Excluding the other factors, the polarization coding QBER can be totally represented as: QBERP = 0.25 × (Q0 + Q90 + Q45 + Q−45 ) Here Figs. 3 and 4 show the connections among the polarization QBER, coordinate system difference and the phase difference caused by birefringent. In Fig. 3, the first derivative of the -QBER curve decreased with the increasing. And the first derivative of the -QBER curve increased with the increasing. That means when the fiber is short (like the fiber interferometers), the polarization QBER is greatly affected by the difference between the coordinate systems of the fast and slow axes in fiber and incident light. When the fiber is long (like long distance fiber channel), the phase difference caused by birefringent is the principal factor of the rising of polarization QBER. In Fig. 4, when and are both below 0.2 rad, the polarization QBER is less than 0.015. When and are both below 0.4 rad, the polarization QBER is less than 0.06. The polarization QBER in two-qubit QKD scheme could be accepted as far as
= JSB CJSA EAin ei(ˇB ++ˇA ) + JSB CJLA EAin ei(ˇB ++˛A +ϕA ) + JLB CJSA EAin ei(˛B +ϕB ++ˇA ) + JLB CJLA EAin ei(˛B +ϕB ++˛A +ϕA ) where ˛A and ˛B are the equivalent phase of long arm, and ˇA and ˇB are the equivalent phase of short arm. ϕA and ϕB are the equivalent phase of phase modulator. C is the Jones matrix of atmospheric channel. ˚ is the equivalent phase of atmospheric channel. The first term on right-hand side of the equation represent S + S pulse, and the fourth term represent L + L pulse. These two pulses are abandoned in the original phase QKD scheme. They are modulated polarization information in the two-qubit QKD scheme. So the anti-disturbance conditions of these two pulses are JSB CJSA = JLB CJLA = I The rest two terms are the interference pulses and the antidisturbance conditions of these two pulses are JSB CJLA = JLB CJSA = I There is little birefringence in the atmosphere. Where, C is considered as unitary. Aforementioned anti-disturbance conditions can also be equivalently written as: JLA = JSA ,
Fig. 3. QBER of polarization coding caused by and .
JLB = JSB ,
−1 JLA = JLB ,
−1 JSA = JSB
More specifically, the Jones matrix of each two long arms and the two short arms should be the same. And the Jones matrixes of long arm should be the reciprocal matrix of short arm’s which belongs to both Alice side and Bob side. Contrasting with paper [21], LA = SA and LB = SB also are the anti-disturbance conditions to keep the interference visibilities in the original phase QKD scheme stable. LA = LB −1 and SA = SB −1 are the additional conditions for the polarization coding.
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5. Conclusion In summary, the QBER of polarization coding can be influenced deeply by the fiber M–Z interferometers. The theoretic analysis reveals that the jitter of polarization state is not only derived from the birefringent, but also derived from coordinate system difference between incident light and fast and slow axes in fiber interferometer. Some necessary conditions for the QKD via singlephoton two-qubit states are given. However, all the simulation results are obtained in perfect atmosphere channel. The influence of atmosphere turbulence and complicated weather should be considered in the sequent works. And the polarization maintaining conditions in the fiber interferometer maybe is neither stable enough nor easy to realize perfectly. So some active compensation method for polarization should be analyzed. Acknowledgments The authors would like to thank Doctor Tang Feng and Doctor Tian Xiaofei for the helpful advice. References [1] C.H. Bennett, G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, 1984, pp. 175–179.
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