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J. Opt. B: Quantum Semiclass. Opt. 5 (2003) S530­S534

PII: S1464-4266(03)60206-5

Practical realization of a quantum cryptography protocol exploiting polarization encoding in qutrits
G A Maslennikov , A A Zhukov , M V Chekhova and S P Kulik
Chair of Quantum Electronics, Department of Physics, Moscow M V Lomonosov State University, Moscow 119992, Russia E-mail: postmast@qopt.phys.msu.su

Received 26 February 2003, accepted for publication 13 June 2003 Published 6 August 2003 Online at stacks.iop.org/JOptB/5/S530 Abstract We propose and discuss a specific scheme allowing realization of a quantum cryptography qutrit protocol. This protocol exploits the polarization properties of single-frequency and single-spatial-mode biphotons.
Keywords: Qutrits, biphotons, quantum key distribution

(Some figures in this article are in colour only in the electronic version)

1. Introduction
The art of quantum cryptography (QC) could well be the first practical application of quantum information at the singlequanta level. It provides two distant users, Alice and Bob, with random secret keys, which later can be used for encrypting messages in a `one-time pad' scheme--the only cryptographic scheme that is mathematically proven to be secure. The simplest method is the use of polarized single photons as the quantum carriers between Alice and Bob. The orthogonal polarization states represent bit values 0 and 1. The first QC protocol based on such systems was described by Bennett and Brassard [1]. It requires the use of four qubits prepared in four different states that belong to two mutually unbiased bases. This means that any state vectors |ei , |e j that belong to different bases must satisfy the following condition: | ei |e j |2 = 1/2. One may also extend this protocol, which is commonly called BB84, to the case where three mutually unbiased bases are used. This extension leads to the improvement of the security against eavesdropping in the caseofso-called symmetric attacks [2, 3]. Various methods are used to implement these protocols in experiment (for details, see the review [4]) and a lot of experiments have been carried out in recent years. It is important to mention that some of the experiments also show excellent results outside the laboratory in real surroundings [5]. Possible kinds of eavesdropper's attacks on these protocols are reviewed in [2­4, 6]. But even with present technologies, the most promising QC systems suffer from a low bit rate of some hundred hertz, compared to
1464-4266/03/040530+05$30.00 © 2003 IOP Publishing Ltd

thousands of megahertz achieved in classical systems. There are some ways to enhance the bit rate, such as by the use of single-photon sources (`photon guns'), where one can be sure that there is only one photon emitted at a time, by improvement of the detector's efficiency, and by using systems of higher dimensions than qubits. In the last case, the amount of information that is carried will be proportional to d n , where d is the dimension of the system and n the total number of systems. The possibility of increasing the system's dimension wasproposed in [7]. The authors used four-level systems (ququarts) as an example and it was shown that the key generation rate is higher than in the case of qubits, and their protocol is also more sensitive to the intercept-resend eavesdropping strategy. The other example of extending the system's space is by the use of three-level systems (qutrits); a QC protocol that exploits such systems was introduced in [8]. This protocol uses twelve states and four mutually unbiased bases to encode the information. The optimal eavesdropping on this protocol was studied in [9]. An entanglement-based protocol for qutrits was also suggested recently and found to be more robust against quantum cloning machine attacks [10, 11]. In this paper we will discuss a proposal for possible optical realization of three-level systems, based on the polarization state of the biphotons--superpositions of two-photon Fock states and the vacuum--and propose an experimental set-up for the protocol suggested in [8]. S530

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Practical realization of a quantum cryptography protocol exploiting polarization encoding in qutrits

2. Biphotons as qutrits
Several ways for experimentally realizing multilevel quantum optical systems are known. In one of them [7], the interferometric procedure of state preparation is used, where attenuated laser pulses are sent into a multi-arm interferometer. The number of arms is equal to the system's dimensionality. Identification of the basis states is done either through the pulse delay (temporal basis) or from the presence of constructive interference in a certain arm of the interferometer that is put into the registration system (energy basis). Another example is the optical field that consists of pairs of correlated photons (biphotons) that belong to different polarization modes. The preparation of such fields and their unitary transformations are described in [12, 13], and the measurement procedure (tomography) for such quantum objects can be found in [14]. The pure polarization state of a single-mode biphoton field (by saying `single-mode' we mean that the photons that form a biphoton have equal frequencies and propagate along the same direction) can be written as | = c1 |2, 0 + c2 |1, 1 + c3 |0, 2 = c1 | + c2 | + c3 | , (1) where the first and the second position in the brackets indicate the polarization modes (say, horizontal H and vertical V), the total number of photons in this mode is N = 2, and ci = |ci | exp(ii ) are the complex probability amplitudes for finding the biphoton in the corresponding state. The states |2, 0 and |0, 2 correspond to type-I phase matching where both photons in a biphoton have the same polarization, and the state |1, 1 is obtained via type-II phase matching where these photons are polarized orthogonally. One can see that the state | describes a three-level system. Equation (1) presents three possible states of a biphoton written in the so-called `HV basis'. For the realization of the protocol that was proposed in [8], one needs three more bases that can be prepared from the natural HV bases according to the rule 1 | = (| + | + | ), 3 2 i 2 i 1 | +exp - | | = | +exp 3 3 3 | 2 i 2 i 1 | +exp | = | +exp - 3 3 3 , (2) .

and | | | 2 i 1 | + | + | = exp - 3 3 2 i 1 | + | = | +exp - 3 3 , , (4)

1 2 i = | + | +exp - | . 3 3 All these bases are mutually unbiased. In the case of threedimensional systems this means that | ei |e j |2 = 1/3. As was shown in [15], the polarization state of the biphoton can be ´ imaged on the surface of the Poincare sphere. The main idea is to image the two-photon polarization state as two points on the surface of the sphere, where each point corresponds to the polarization state of a single photon from the pair and its position is described using a well-known technique. Sometimes such a mapping helps one to understand better the polarization properties of single-mode biphotons. It follows from the representation of the state vector in the form | = a (, )a ( , )|vac , a (, )a ( , )|vac (5)

where a (, ) and a ( , ) are the creation and annihilation operators for the photon in the corresponding polarization mode, a (, ) = cos( /2)a + ei sin( /2)aV , a ,V H H are photon creation operators in the horizontal (H) and vertical (V) polarization modes, , [0, 2 ] and , [0, ] are the azimuthal and polar angles that define the position of ´ a photon on the Poincare sphere. The primed and unprimed indices correspond to different photons from the pair. In this case, , = - 3 1 |c2 |2 ± arccos 2 2 2|c1 ||c3 | 1+ |c2 |4 |c2 |2 - cos(22 - 3 ) , 4|c1 |2 |c3 |2 |c1 ||c3 |
2 2

(6)

, = arccos |c1 |2 -|c3 |

± 2 [|c2 |2 -|c1 ||c3 | cos(2 - 3 )]2 -|c1 |2 |c3 |
-1

â 1+ |c2 |2 - 2|c1 ||c3 | cos(2 - 3 ) . (7) The degree of polarization of a biphoton can be shown to be P= 2cos(/2) , 1+cos2 (/2) (8)

These states form the second basis and the other two bases can be taken as | | | 2 i 1 | + | + | = exp 3 3 2 i 1 | + | = | +exp 3 3 1 2 i | = | + | +exp 3 3 , , , (3)

where is the angle between the lines that connect the position of each photon on the surface with the centre (see figure 1(a)). It is important to note that using linear transformations one cannot change the polarization state of the biphoton to a state with different a degree of polarization. As will be shown later, this greatly reduces the possibility of practical implementation of the QC protocol discussed. In figure 1(b) all 12 states that ´ are used in the protocol are shown on the Poincare sphere, according to (6), (7). According to the protocol, Alice randomly chooses one of these four bases in which she will prepare her qutrit, then sends the qutrit to Bob, who measures it in a randomly chosen basis. After the base reconciliation, Alice and Bob will have the sifted key, from which they can distil the secret key by means of error correction and privacy amplification techniques. S531


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(a) (b)

´ Figure 1. (a) Graphical representation of an arbitrary polarization state of the biphoton field on the Poincaresphere ( , are the azimuthal ´ and polar angles that are used in the standard Poincaresphere technique; is the angle that determines the degree of polarization of a ´ biphoton). (b) Mapping of the 12 states which are used in the protocol on the Poincaresphere.

combining biphotons using beam splitters, we propose to use the two-arm version where the type-II non-linear crystal is not located directly in the interferometer. The scheme works as follows: the laser beam is split by a non-symmetric beam splitter. The transmitted part (66%) goes through the long arm of the interferometer and is used for pumping two type-I non-linear crystals, whose axes are oriented orthogonally to each other. All crystals are oriented to produce the biphotons in the frequency-degenerate, collinear regime. In this arm we generate a superposition of the form |c1 ||2, 0 +e
Figure 2. Preparation of an arbitrary polarization state of a single-mode biphoton (qutrit) with given parameters. (/2)1,2 are the half-wave plates used to change the amplitudes of the basis states; P.S.1 is the phase shifter for shifting between the states |2, 0 and |0, 2 ; P.S.2 is the phase shifter for the |1, 1 state with respect to the superposition obtained using two type-I crystals; DM is a dichroic mirror that transmits the biphoton field coming from the long arm and reflects the pump coming through the short arm; F is the pump cut-off filter.
i
31

|c3 ||0, 2 ,

(9)

3. Preparation and measurement of qutrits in four given bases
3.1. Preparation and transformation of qutrits (Alice's station) For the quantum key distribution it is necessary that Alice can send Bob one randomly prepared qutrit at a time, to establish good time synchronization to ensure that Bob measures the correct qutrit. We propose an interferometric set-up, similar to one that was used in [13]. It is a Mach­Zehnder interferometer, with three arms and with non-linear crystals of the corresponding type in each arm. In this set-up one can generate a truly arbitrary polarization state of the biphoton field, including those that are described by formulae (1)­(4). The selection between the twelve states is done by the phase shifters (PS), which imply a certain phase shift of the states, according to table 1. The possible practical realization of the interferometer is shown in figure 2. To improve the stability of the interferometer and prevent the losses introduced by S532

where 31 is the relative phase of the states |2, 0 and |0, 2 . The amplitudes of these states can be changed by rotating a half-wave plate (/2)1 , located before the crystals, and the relative phase can be set by using the PS. A cut-off filter eliminates the pump. A moving mirror that is driven by a piezoelectric driver is used to apply a certain phase shift 2 = 31 - 2 between the superposition (9) and the state |1, 1 , which is obtained by pumping the type-II crystal with the pump from the short arm of the interferometer through the dichroic mirror (DM) that reflects the pump from the short arm and transmits the biphotons fromthe long arm. The amplitude |c2 | is varied by means of a (/2)2 plate. Thus, at the output of the interferometer we have the coherent superposition that is given by (1), and by varying four parameters ((/2)1 , (/2)2 , 31 , and 2 ) we can generate all twelve states that are needed for the QC protocol. We should mention here that the proposed interferometric scheme allows one to introduce timing using a short-pulsed pump. In this regime, approximately one pair of photons, i.e. a single qutrit, is emitted at given time, determined by the repetition rate and pulse duration of the pump laser. Possible implementation of the cascade-crystal configuration is discussedin[16]. 3.2. Measurement of qutrits (Bob's station) The measurement of the arriving qutrits is performed at Bob's station where he has to measure them in a randomly chosen


Practical realization of a quantum cryptography protocol exploiting polarization encoding in qutrits

Figure 3. Aschematic drawing of the Bob's station. RNG is the random number generator that defines the basis in which Bob measures the incoming qutrits; the choice of the state may be made using the symmetric three-output beam splitter; WP are the wave plates forming the polarization filters; A are the analysers; D are the detectors; C.C. are the double-coincidence schemes. Table 1. Amplitudes and phases of the states that are used in the QC protocol. |c1 | | | | | | | | | | | | | 1 0 0
1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3

|c2 | 0 1 0
1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3

|c3 | 0 0 1
1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3



1

0 0 0 0 0 0

2

0 0 0 0

3

P 0 0 0 0 1 0 1

120 -120 0 120 0 0 -120 0

-120 120 0 0 120 0 0 -120

120 0 0 -120 0 0

22 3 22 3 22 3 22 6 22 6 22 6 22 6 22 6 22 6

basis. The problem arising here is that one cannot switch between the proposed states, as they have different degrees of polarization. This is so because usage of only linear polarization transformations, like retardation plates, does not change the degree of polarization. Therefore, one has to use non-linear interactions to switch from one basis to another and this does not seem practical at the moment. The solution is to use the random number generator to decide in which basis we are going to measure our qutrits. Certainly, this is a weak point in the experimental realization of the protocol, because the randomness of the choice is given by a computational and not a physical process. But considering the recent developments in technology, one can build a random number generator where randomness is guaranteed by the physical process and the first such devices have already been introduced [17]. The second

problem that arises here is that even if we have decided about the basis, we still have to distinguish the states within these bases from one another with unity probability. This leads to the problem of filtering the orthogonal polarization states of the biphoton from one another. As was suggested in [18], this can be done by observing the anticorrelation effect of general type, i.e., for biphotons with an arbitrary degree of polarization in the Brown­Twiss scheme with polarization filters in its arms (figure 3). This filter consists of a pair of phase plates (/2 and /4) and a polarization analyser. Changing the orientation of the plates relatively to the vertical axis (parameters /2 ,/4 ) one can achieve any polarization state of single photons forming the biphoton. Assume that one sets the filters in such a way that in each arm, complete transmittance of thecorresponding photon from a pair is observed. In this case, we will be saying that the Brown­Twiss scheme is `tuned' to a certain polarization state of the biphoton. The criterion of the orthogonality of two biphotons is identical to the observation of zero coincidence rate in the case where the input of the Brown­Twiss scheme obtains a biphoton that is orthogonal to the one to which the scheme is `tuned'. This allows us to distinguish between the orthogonally polarized biphotons that constitute the basis in the protocol from one another, making it possible to distil the ternary raw key. Bob's station can be built as in figure 3. It consists of three Brown­Twiss schemes with 50% non-polarizing beam splitters and polarization filters. In the first stage, Bob has to randomly choose the basis in which he will measure the incoming qutrit. In practice, this can be achieved by randomly setting out the certain parameters of the polarization filters in the Brown­Twiss schemes. For example, if we choose to measure in the basis that is given by the states | , | ,and | , then the first scheme is `tuned' to the state | . This means that filters in each arm transmit horizontal polarization. The second scheme must be set to | ; i.e. the first arm transmits horizontal polarization and the second transmits the vertical S533


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one and so on. The incoming qutrit from Alice can appear at the input of each Brown­Twiss scheme with equal probability with the help of the symmetric three-output splitter. By the procedure of basis reconciliation, unfortunate results, where Bob did not guess the basis correctly, can be eliminated. The positive side of this set-up, that makes it preferable to the other proposed interferometric set-ups (see [7] for example), is that we are working with only a two-armed interferometer that does not require high stability over time and temperature. Moreover, this interferometer is located only in the preparation side of the set-up, i.e. at Alice's station. We also want to point out that the achievement of higher dimensionality is based on using the specific states of quantum light fields (biphotons as qutrits), rather than increasing the Hilbert space dimension of the single photons. The combination of two methods indicated above may also produce a reasonable and practical resource for the further increase of the system's dimension. Finally, we would like to discuss the possible losses which are present in the scheme under consideration. Definitely this scheme suffers from losses that cannot in principle be eliminated. First of all, Bob may not guess the basis correctly, which leaves us1/4of the number of qutrits. This kind of loss can be considered as inevitable, because it is caused by the nature of the protocol. Then, in the stage of the state separation at the three-output beam splitter, one may receive a biphoton at the input of the Brown­Twiss scheme with a probability of 1/3. Considering the action of the non-polarizing beam splitter on the biphoton, we are left with 1/2of the cases where it may result in a coincidence. And finally, the action of the polarizing filters on the halves of the biphoton may also filter out 1/2 of the fortunate cases. It is worth mentioning that such action of the filters gives us the upper bound of the losses. The lower bound is given when the filters act in the same way on the halves of the biphoton. Then the coincidence rate does not depend on how the biphoton was split at the beam splitter. Therefore the total losses in a proposed scheme vary from 96 to 98%. Subtracting inescapable losses arising from the random basis choice (75%), one gets about 8% of events with successful registration by Bob of qutrits sent in a given basis by Alice. All of these are stipulated by accidental splitting of light by beam splitters. The presented numbers show that the practical implementation of this protocol will hardly enhance the bit rate of the key transition, but the higher dimensionality of the carriers gives us higher security than the traditional qubit protocols [8, 9].

properties of single-mode biphotons. The examination of the set-up showed that the losses that will be induced by various elements of the set-up are quite high for the implementation of such polarization-based systems in practice. The positive side is that it is in principle possible to prepare a polarizationbased optical qutrit without any interferometric scheme that requires stability over time. This can be done in the scheme with three non-linear crystals located in the common pump one after another. This scheme is now being considered. Still, the proposed possibility of preparing an optical three-level system using three non-linear crystals with quite high efficiency and no principal losses at Alice's station seems very attractive and merits further investigation.

Acknowledgments
This work was supported in part by the Russian Foundation for Basic Research (02-02-16843, 03-02-16444) and INTAS (grant 2122-01).

References
[1] Bennett C H and Brassard G 1984 Proc. IEEE Int. Conf. on Computers, Systems and Signal Processing (Bangalore, India, 1984) (New York: IEEE) p 175 [2] Bruss D 1998 Phys. Rev.Lett. 81 3018 [3] Bechmann-Pasquinucci H and Gisin N 1999 Phys. Rev. A 59 4238 [4] Gisin N, Ribordy G, Tittel W and Zbinden H 2002 Rev. Mod. Phys. 74 145 [5] Ribordy G, Gautier J-D, Gisin N, Guinnard O and Zbinden H 2000 J. Mod. Opt. 47 517 [6] Fuchs C A, Gisin N, Griffiths R B, Niu C S and Peres A 1997 Phys. Rev. A 56 1163 [7] Bechmann-Pasquinucci H and Tittel W 2000 Phys. Rev. A 61 062308 [8] Bechmann-Pasquinucci H and Peres A 2000 Phys. Rev.Lett. 85 3313 [9] Bruss D and Machiavello C 2002 Phys. Rev.Lett. 88 127901 [10] Kaszlikowski D, Oi D K L, Christandl M, Chang K, Ekert A, KwekL Cand Ou C H 2003 Phys. Rev. A 67 012310 [11] Durt T, Cerf N J, Gisin N and Zukowski M 2003 Phys. Rev. A 67 012311 [12] Burlakov A V and Klyshko D N 1999 JETP Lett. 69 795 [13] Burlakov A V, Chekhova M V, Karabutova O A, Klyshko D N and Kulik S P 1999 Phys. Rev. A 60 R4209 [14] Burlakov A V, Krivitskiy L A, Kulik S P, Maslennikov G A and Chekhova M V 2002 Preprint quant-ph/0207096 [15] Burlakov A V and Chekhova M V 2002 JETP Lett. 75 432 [16] Kim Y H, Chekhova M V, Kulik S P, Rubin M and Shih Y H 2001 Phys. Rev. A 63 062301 [17] Stefanov A, Gisin N, Guinnard O, Guinnard L and Zbinden H 1999 Preprint quant-ph/9907006 [18] Zhukov A A, Maslennikov G A and Chekhova M V 2002 JETP Lett. 76 696

4. Conclusions
We propose the first concrete experimental set-up for the realization of a QC protocol that exploits the polarization

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