Robust quantum key distribution with two-photon polarization states

Robust quantum key distribution with two-photon polarization states

Physics Letters A 359 (2006) 126–128 Robust quantum key distribution with two-photon polarization states Ming Gao ∗ , Lin...

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Physics Letters A 359 (2006) 126–128

Robust quantum key distribution with two-photon polarization states Ming Gao ∗ , Lin-Mei Liang, Cheng-Zu Li, Chen-Lin Tian Department of Physics, National University of Defense Technology, Changsha 410073, PR China Received 25 March 2006; received in revised form 27 June 2006; accepted 29 June 2006 Available online 5 July 2006 Communicated by P.R. Holland

Abstract We present a quantum key distribution scheme, based on polarization encoding, that is robust against collective noise by sending orthogonal polarization modes of each photon through different channels and postselection. It has lower bit error rate and only needs local polarization measurement. Compare with similar schemes is discussed. © 2006 Elsevier B.V. All rights reserved. PACS: 03.67.Dd; 03.67.Hk Keywords: Quantum key distribution; Two-photon polarization state; Collective noise; Postselection

Due to its unconditional security and simplicity of technological realization, quantum key distribution (QKD) has become an important branch in the field of quantum information. Although many QKD experiments [1] have been realized through air and optical fiber, decoherence has been a principal impediment. In optical QKD protocols, the major noise is due to birefringence effect of the optical fiber which randomly alters the polarization states of photons. When the coherence time of the photon is large compared to the delay caused by polarization mode dispersion, the birefringence corresponds to a unitary transformation on the polarization space [2]. The alteration of polarization states in optical fiber is hard to control so that the most successful QKD experiments were not based on polarization coding but on phase coding. Even for phase coding, a good control of the polarization modes is necessary to obtain a better visibility since some components like phase modulators are polarization dependent. Typically the variation of the birefringence is slow in time, so that the alteration of the polarization is considered to be same over the sequence of several photons. This type of noise is referred to as collective noise. To overcome such noise, several QKD schemes have been proposed

* Corresponding author.

E-mail address: [email protected] (M. Gao). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.06.080

[2–6]. An important QKD scheme is the plug-and-play system [3] based on phase coding in which all optical and mechanical fluctuations are automatically and passively compensated for via two-way quantum communication. However, the use of two-way quantum communication makes the system more sensitive to a kind of eavesdropping strategy known as the Trojan horse attack. To circumvent the threat of the Trojan horse attack, Wang [6] has recently proposed a fault tolerant protocol based on twophoton polarization states by one-way quantum communication. Another one-way scheme [7] proposed by Yamamoto et al., which sends each polarization state assisted by an additional qubit, is robust to Trojan horse attack. Here we present a novel distribution scheme by one-way communication, which encodes information in two-photon polarization states and sends orthogonal polarization modes of each photon through different channels. Although the states which the sender Alice prepares and transmits have been disturbed, the receiver Bob can correctly distinguish them by means of such postselection which considers both polarization mode and spatial mode of the photons. Let us first introduce our distribution scheme. Alice prepares chosen from |H V , |V H , √ √ two-photon states randomly (1/ 2)(|H V  + |V H ), (1/ 2)(|H V  − |V H ), where |H  and |V  represent horizontal and vertical polarization states.

M. Gao et al. / Physics Letters A 359 (2006) 126–128


Fig. 1. The schematic diagram of the proposed distribution scheme PR: polarization rotator, D: single photon detector.

For example, |H V  represents the first photon in the horizontal polarization state and the second photon in the vertical polarization state. The time delay t between the two photons, which is fixed by Alice and known by Bob, is large enough such that Bob’s detection apparatus can differentiate them. As shown in Fig. 1, these photons which initially are in the same spatial mode are split into two spatial modes by a polarizing beam splitter (PBS), which transmits |H  and reflects |V . Alice then sends the photons in |H  and |V  through channel 1 and 2, respectively. Each channel is composed of optical fiber. Bob receives these photons and mixes them by a PBS. The lengths of the fibers must be adjusted to ensure that the polarization modes of a photon, which have passed through different channels, can arrive at the first PBS on Bob’s side at the same time. To see how we discriminate the states on Bob’s side which have been changed by the collective noise, suppose that there are any collective polarization rotations which occur in two channels independently. Each collective polarization rotation transforms the polarization states as |H  → δ1 |H  + γ1 |V  and |V  → δ2 |H  + γ2 |V  in channels 1 and 2, respectively, where |δ1 |2 + |γ1 |2 = 1 and |δ2 |2 + |γ2 |2 = 1. So the transformations of these rotations on the states are |H V  → δ1 δ2 |H3 H4  + δ1 γ2 |H3 V3  + δ2 γ1 |V4 H4  + γ1 γ2 |V4 V3 ,


|V H  → δ1 δ2 |H4 H3  + δ1 γ2 |V3 H3  + δ2 γ1 |H4 V4  + γ1 γ2 |V3 V4 , (2) √   (1/ 2 ) |H V  + |V H  √    → (1/ 2 ) δ1 δ2 |H3 H4  + |H4 H3      + δ1 γ2 |H3 V3  + |V3 H3  + δ2 γ1 |V4 H4  + |H4 V4    + γ1 γ2 |V4 V3  + |V3 V4  , (3) √   (1/ 2 ) |H V  − |V H  √    → (1/ 2 ) δ1 δ2 |H3 H4  − |H4 H3      + δ1 γ2 |H3 V3  − |V3 H3  + δ2 γ1 |V4 H4  − |H4 V4    + γ1 γ2 |V4 V3  − |V3 V4  , (4)

where the numbers in the subscripts represent the spatial modes in Fig. 1. For example, |H3 H4  denotes that the first photon emerges at port 3 and the second photon at port 4. From (1) and (2) it is easy to see that the state which initially is |H V  or |V H  can be discriminated by considering both polarization modes and spatial modes of the photons. From (3) we could project the state initially in the state √ √ (1/ 2 )(|H V  + |V H ) onto the state (1/ 2 )[δ1 γ2 (|H3 V3  + |V3 H3 ) + δ2 γ1 (|V4 H4  + |H4 V4 )] with a probability of |δ1 γ2 |2 + |δ2 γ1 |2 . From√(4) we could project the state initially √ in the state (1/ 2 )(|H V  − |V H ) onto the state (1/ 2 )[δ1 γ2 (|H3 V3  − |V3 H3 ) + δ2 γ1 (|V4 H4  − |H4 V4 )] 2 2 polarization with a probability of |δ √1 γ2 | + |δ2 γ1 | . After taking √ rotations |H  → (1/ 2 )(|H  + |V ), |V  → (1/ 2 )(|H  − |V ) on polarization modes of each photon, which are performed by the polarization rotators (PR), the states     δ1 γ2 |H3 V3  + |V3 H3  + δ2 γ1 |V4 H4  + |H4 V4      → δ1 γ2 |H3 H3  − |V3 V3  + δ2 γ1 |H4 H4  − |V4 V4  (5) and     δ1 γ2 |H3 V3  − |V3 H3  + δ2 γ1 |V4 H4  − |H4 V4      → δ1 γ2 |H3 V3  − |V3 H3  + δ2 γ1 |V4 H4  − |H4 V4  . (6) Note that the states at the right side of (5) cause single detector in one spatial zone triggered twice, while the states at the right side of (6) cause each detector in one spatial zone triggered once. It is easy to see in this case we can discriminate them through different detection events. However the states at the right side of (3) and (4), which cause one detector in each spatial zone triggered once, are useless √ for the discrimination. So √ we can discriminate between (1/ 2 )(|H V  + |V H ) and (1/ 2 )(|H V  − |V H ) with a success probability of |δ1 γ2 |2 + |δ2 γ1 |2 . In addition there are fluctuations in the optical path length. As long as these fluctuations are slower than the interval t , they only cause an overall phase factor and do not influence the discrimination between the states. We have described how to distinguish the states transmitted over a collective-noise channel and then implement a protocol similar to BB84 [8].


M. Gao et al. / Physics Letters A 359 (2006) 126–128

(1) Alice generates a number of two-photon states, each of √ 2 )(|H V + which is randomly chosen from |H V , |V H , (1/ √ |V H ), √ (1/ 2 )(|H V  − |V H ). She encodes bit 0 with |H V  and (1/ 2 )(|H V  + |V H ), bit 1 with the other two states. She writes down “Z basis” according √ to the state |H V  and 2 )(|H V  + |V H ) and |V H , “X basis” according to (1/ √ (1/ 2 )(|H V  − |V H ). (2) Alice sends these two-photon states to Bob. (3) Bob receives the photons and randomly chooses between Z basis and X basis to measure the states. When choosing the X basis, Bob only keeps the results where there is either single detector triggered twice or each of two detectors triggered once in one spatial zone. (4) Alice and Bob compare measurement basis through public channel. If both of them use Z basis, all results will be kept. If the both use X basis, only the results kept in step 3 will be reserved. (5) From the obtained bits Alice selects a random string of bits half of which are the results of the X basis measures. Then she tells Bob which bits were selected. They announce and compare the value of the selected bits to estimate the error rate. If more than an accepted number of errors are found, they abort the protocol. The procedure of the protocol implies that it is equivalent to the BB84 protocol except for the efficiency, so the protocol has also the unconditional security like that of BB84. If the quantum efficiency of the detectors is η, the efficiency of the original protocol [8] is η/2 while ours efficiency is (1√+ |δ1 γ2 |2 + |δ2 γ1 |2 )η2 /4 since √ the discrimination between (1/ 2 )(|H V  + |V H ) and (1/ 2 )(|H V  − |V H ) can only be achieved with success probability |δ1 γ2 |2 + |δ2 γ1 |2 (< 1). Although the scheme is less efficient than the original one, it has lower bit error rate. It is easy to note that the efficiency is dependent on the noise parameters. In practice, one can use the method mentioned in Ref. [7] to average |δ1 γ2 |2 + |δ2 γ1 |2 to 1/2. A kind of physical realization of the source is given in Ref. [6]. Finally let us compare our scheme with the similar ones. Comparing with the practical protocol (protocol 2 in Ref. [6]), the encoding states are the same, but his decoding method is parity check on the polarization states, in which the distorted states in Z basis will be discarded if the outcomes are out of the subspace {|H V , |V H }. And that the states in X basis only are used to estimate the phase-flip rate and do not generate the key. His scheme even uses the classical CSS code [9] to correct bitflip error in those survived codes originally in |H V  or |V H . Therefore his decoding method is less efficient than ours. Comparing with the QKD protocol in Ref. [7] both schemes employ two-photon states and send orthogonal polarization modes of each photon through different channels, but their states are separable states, i.e., reference photons in the state

√ by signal photons in√one of the (1/ 2 )(|H  + |V ) followed √ four states |H , |V , (1/ 2 )(|H  + |V ) and (1/ 2 )(|H  − |V ), and their decoding method is through the detection events of one of two photons to judge which the state is. Due to their decoding method demanding compensation of temporal difference t between two photons, which results in an additional loss factor up to 1/4, the overall success probability of discrimination between two states in each base is maximally up to η2 /8 even after using a deterministic two-qubit operation. While our decoding method has no requirement of this kind, the overall success probabilities of discrimination in two bases are η2 and η2 /2, respectively. In addition, to avoid the problem of multiphoton emission, their source needs to employ parametric down conversion (PDC), which reduces the simplicity of their scheme. Briefly speaking, though their scheme does not need to prepare entangled states, our scheme has the advantage on the decoding method. In conclusion, we present a one-way quantum key distribution scheme using the two-photon polarization states, which is immune to the collective noise. By two quantum channels and postselection which takes account of both the spatial modes and polarization modes of the photons, we can faithfully distinguish the states which have been disturbed by the collective noise in the optical fiber. Although our scheme is less efficient than the original one, it has lower bit error rate and only needs local polarization measurements in spite of using entangled states. The implementation of our scheme can be achieved by present technology and does not need the stabilization of optical paths with the precision of the order of wavelength. Acknowledgements We thank Wei Wu, Wei-Tao Liu and Shuang Wu for helpful discussions. This work was funded by National Funds of Natural Science (Grant No. 10504042). References [1] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [2] J.C. Boileau, R. Laflamme, M. Laforest, C.R. Myers, Phys. Rev. Lett. 93 (2004) 220501. [3] A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, N. Gisin, Appl. Phys. Lett. 70 (1997) 793. [4] Z.D. Walton, A.F. Abouraddy, et al., Phys. Rev. Lett. 91 (2003) 087901. [5] J.C. Boileau, D. Gottesman, R. Laflamme, et al., Phys. Rev. Lett. 92 (2004) 017901. [6] X.-B. Wang, Phys. Rev. A 72 (2005) 050304. [7] T. Yamamoto, J. Shimamura, S.K. Özdemir, M. Koashi, N. Imoto, Phys. Rev. Lett. 95 (2005) 040503. [8] C.H. Bennett, G. Brassard, Quantum cryptography: Public-key distribution and coin tossing, in: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, IEEE, New York, 1984, pp. 175–179. [9] P.W. Shor, J. Preskill, Phys. Rev. Lett. 85 (2000) 441.