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July 2009
EPL, 87 (2009) 10008 doi: 10.1209/0295-5075/87/10008 www.epljournal.org

Quantum state engineering with ququarts: Application for deterministic QKD proto col
A. P. Shurupov1 , S. S. Straupe
1 2 3 1 (a)

, S. P. Kulik1 , M. Gharib2 and M. R. B. Wahiddin

2, 3

Faculty of Physics, Moscow State University - Russia International Islamic University Malaysia - Kuala Lumpur, Malaysia MIMOS Berhad - Kuala Lumpusr, Malaysia received 3 April 2009; accepted in final form 25 June 2009 published online 24 July 2009
PACS PACS PACS

03.67.Dd ­ Quantum cryptography and communication security 42.50.Dv ­ Quantum state engineering and measurements 03.67.Hk ­ Quantum communication

Abstract ­ We discuss the proof-of-principle demonstration of the extended deterministic Quantum Key Distribution (QKD) protocol based on ququarts. The experimental realization is based on the polarization degrees of freedom of two-mode biphotons, making the process of state preparation, transformation and measurement rather simple. The scheme uses only single nonlinear crystal for biphoton generation and linear optical elements for their following transformation and can be used as a base for further practical applications.
Copyright c EPLA, 2009

Intro duction. ­ Unconditional security of QKD is limited in practical implementations by technical imperfections, such as non-perfect detectors efficiency and losses in the communication channel [1]. For example, in the standard BB84 protocol the key distribution is proven to be secure if the total quantum bit error rate (QBER) is less than approximately 11% [2]. In practice that results in limitations on the distance over which the secure communication can be established. One can look for a solution of the problem in different ways. The first one is straightforward and "technical" --it is to improve technical characteristics of the equipment used: increase detectors quantum efficiency, reduce losses in the channel and dark noise of the detectors, etc. The other way is a "physical" one and relies on properties of the physical system used to encode the information. One of the possible solutions following this way is to increase dimensionality of the system where information is encoded or/and to increase the number of bases used in the protocol [3]. Following this line of research, one faces the necessity to experimentally prepare, transform and measure the states of quantum systems with Hilbert space of more than two dimensions (qudits). In other words, one comes to, the so-called, "quantum state engineering" with qudits, an important branch for the whole quantum information science. Optical realizations of qudits are
( a)

E-mail: straups@yandex.ru

obviously preferable for the purposes of QKD, and much effort was made recently to study the properties of such systems [4­6]. Here we would like to stress on one of possible realizations of optical qudits, which turned out to be useful for quantum communication. It is the, so-called polarization ququarts, i.e. the four-dimensional optical qudits implemented using the polarization degrees of freedom of single-beam biphotons. This realization of ququarts seems to be promising, because the usage of polarization degrees of freedom allows one to perform unitary state transformations with linear optical elements. The tools for arbitrary polarization ququart state generation are developed in [6], giving the experimentalist a wide range of opportunities for state engineering. The way to realize a direct extension of BB84 protocol with polarization ququarts was discussed in [5]. It was shown that the polarization ququarts are easily controllable and the experimental scheme to distinguish the orthogonal states for different bases deterministically was proposed. In this letter we discuss the implementation of polarization ququarts in the other group of QKD protocols, the deterministic ones. The first protocol of this kind, called "Ping-Pong" was proposed in [7] and, although proven to be insecure [8,9], gave rise to an elegant modification discussed in [10] and referred to as LM05. Except for the improved key transfer rate due to its deterministic nature, LM05 turned out to be more resilient to a wide class of individual attacks than standard BB84 [11].

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Table 1: 6DP protocol: the possible combinations of qubits sent by Bob, the operations on them with the resulting number of bits flipped and the encoded pair of booleans.

Fig. 1: (Color online) Deterministic QKD protocol.

|y± , z± , |x± , z± , |x± , y± , Encoded

|z± , y± |z± , x± |y± , x± value

I 0 0 0 00

X 2 1 1 01

iY 1 2 1 10

Z 1 1 2 11

The proof-of-principle experimental tests of LM05 were reported in [12]. The extension of LM05 involving a larger number of bases was recently proposed in [13]. This protocol, using 6 states out of three bases, which we will refer to as 6DP, seems to be more secure with respect to LM05 in the same way as BB84 extensions are more secure than the original one [3]. It turns out, that such extension requires two-qubit encoding, which makes polarization ququarts a natural ob ject for the implementation of the protocol. 6DP proto col. ­ Let us first briefly describe the ideas underlying the LM05 protocol to make the notion of a deterministic QKD clearer. Let X, Y , Z be the Pauli matrices: X= 01 , 10 Y= 0 -i , i0 Z= 10 , 0 -1

making deterministic measurement possible again. Identifying the applied transformation with a set of two Boolean values, one gets the principle of encoding in informational ququarts, as shown by table 1. Exp erimental realization. ­ In the proposed implementation 6 states from table 1 are realized as polarization states of photon pairs generated in SPDC process. SPDC may be phenomenologically described as a process of spontaneous decay of a pump photon into a pair of correlated photons with lower frequency due to nonzero quadratic susceptibility [15]. We will consider the, so-called, type-I SPDC, when pump is extraordinary and downconverted photons are ordinary waves in the birefringent nonlinear crystal. Let us choose the orientation of the crystal axis and the polarization of the pump to be horizontal, so both photons would have vertical polarization. In this case the state of the downconverted field, calculated in the first order of perturbation theory has the following form: | = |v ac + dk1 dk2 (k1 , k2 )|V
1

and |x± , |y± , |z± their respective eigenvectors. We will identify these states with the polarization states of photons (circularly, diagonally and horizontaly/verticaly polarized, respectively). The protocol works as follows (fig. 1): the photon in one of the X or Z eigenstates is prepared at Bob's side and sent to Alice. Alice performs a unitary polarization transformation described by one of the operators, either I (identity) or iY (bit-flip), and sends the photon back to Bob. In the first case the state of the qubit remains unchanged while in the second case it is changed to an orthogonal one. Identifying the identity operator with logical 0 and the bit-flip with logical 1, Alice encodes information, which can be transferred to Bob deterministically, since measuring the qubit in the known basis (the one in which it was initially prepared) yields to a deterministic result. To ensure security of the protocol, Alice with some probability performs a pro jective measurement instead of encoding, i.e. the protocol is switched to BB84-like Control Mode. The extension of LM05, considered in this letter, adds one more possible basis: the circular one. The straightforward generalization faces the problem stated by the wellknown no-go theorem: the universal bit-flip operator does not exist [14]. An elegant solution with two-qubit encoding was found in [13]. Now Bob prepares pairs of photons in the states, belonging to different bases. Alice performs one of four transformations: I , X, iY , Z , which now results in either no flips, or a flip of one or two photons in a pair transforming the initial state to an orthogonal one,

|V

2

.

(1)

Here k1 , k2 are wave vectors of the downconverted photons and (k1 , k2 ) is the biphoton amplitude characterizing the frequency-angular spectrum of SPDC radiation. In the stationary case the photon's frequencies satisfy the condition 1 + 2 = p due to the energy conservation. After fixing the frequency-angular degrees of freedom (by using narrow-band filters and pinholes) and postselection induced by coincidence measurement eliminating the vacuum component, the polarization state of the biphoton can be described by a state vector | = |V1 V2 , where indexes 1,2 correspond to different wavelengths and/or different propagation directions. An arbitrary biphoton polarization state can be expressed as a quauart: | = c1 |1 + c2 |2 + c3 |3 + c4 |4 = c1 |H1 H
2

+ c2 |H1 V

2

+ c3 |V1 H

2

+ c4 |V1 V2 .

(2)

Here cj = |cj |eij (j = 1, . . . , 4) are complex probability amplitudes, whereas polarization states are produced by corresponding creation operators acting upon the vacuum state: |Hi = a |0 , |Vi = b |0 , i = 1, 2. The choice of i i above representation in terms of product polarization states (2) is rather natural. Indeed each basic state in (2)

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Quantum state engineering with ququarts: Application for deterministic QKD protocol
Table 2: 6-states preparation algorithm.
M

BOB

H

Diode laser

LiIO3
DBS

State |D1 R2 |R1 D2 |D1 V2 |V1 D2 |R1 V2 |V1 R2

1 +22.5 0 +22.5 0 0 0

1 +45 +45 +45 0 +45 0

2 0 +22.5 0 +22.5 0 0

2 +45 +45 0 +45 0 +45

The action of a wave plate on each of the photons is described by a unitary S U (2) transformation:
Fig. 2: (Color online) Bob's station.

G1 t

,2

=

t1,2 r1,2 , -r1,2 t1,2
1, 2

(5)
1, 2

can be directly achieved in SPDC process from a single crystal. We will identify the states of table 1 with polarization states of photons as follows: |z = |H , |z = |V ,

1, 2 ,2

= cos

+ i sin cos 2
))h

cos 2

1, 2

,

r1

= i sin

1, 2

1, 2

,

e 1, 2 o 1, 2 where = is the plate's optical thick1,2 + - ness, and 1,2 is the angle of plate's optical axis rotation |x+ = |D , |x- = |A , (3) with respect to vertical direction. One can easily show that an arbitrary polarization transformation of a single |y+ = |R , |y- = |L . photon can be implemented using a sequence of quarterAccording to the protocol, we should be able to prepare and half-wave plates rotated at appropriate angles: the 6 states of the form (/4) (/2) G1,2 = G1,2 (1,2 )G1,2 (1,2 ). (4) |D1 R2 , |R1 D2 , |D1 V2 , |V1 D2 , |R1 V2 , |V1 R2 The total transformation of the biphoton field is an outer (on Bob's side), and to apply the transformations, product of transformations of each photon in pair: G = G1 (1 , 1 ) G2 (2 , 2 ) [16]. As well as the required six described by Pauli operators X, Y , Z (on Alice's side). These 6 states can be easily generated out of initial states are product states, exhibiting no entanglement, this |V1 V2 state by means of local transformations. In partic- type of transformations is enough to prepare them all. Let ular we used the quarter- and half-wave plates applied to us describe the transformations explicitly. The states and each of the photons in a pair independently. The setup is corresponding orientation of quarter- and half-wave plates in both arms of Bob's setup are listed in table 2. depicted in fig. 2. Basically to make polarization transformations on The 10 mm LiIO3 crystal, cut for non-collinear, nondegenerate type-I phase-matching, is pumped by a 50 mW Alice's side one can implement the same procedure as diode laser operating at 405 nm wavelength. The crystal at Bob's station using a Mach-Zander interferometer axis is oriented horizontally, the pump is linearly polar- with dichroic beam-splitters. This scheme allows one to ized in the same plane which is ensured by the Glan- split the photons with different wavelengths and perform Thompson prism (H). The wavelengths of the down- local transformations separately with subsequent photon converted photons are selected to be 1 = 750 nm and merging (fig. 3a). However, we exploit here the remarkable 2 = 880.4 nm. The two beams are combined on the property of a polarization ququart: violation of polarizadichroic beamsplitter (DBS) transmitting 880 nm and tion degree invariance under local transformations [5]. reflecting 750 nm. The initially non-collinear regime is This property serves as an effective tool for ququart chosen only for the reasons of experimental simplicity, state engineering with only linear local transformations. allowing to transform each photon independently when In particular the transformations on Alice's side can be they are spatially separated. At the exit of Bob's setup realized in a single-beam configuration, using a set of two both photons propagate collinearly to achieve the state specifically cut quartz plates, acting as half-wave plates of ququart |meas --one of the states in (4). It worths on both wavelengths (but of different order, of course). mentioning, that the setup was designed in a specific In our setup we used two identical quartz plates of "triangular" shape (fig. 2) in order to minimize the inci- 545 mkm length (fig. 3b). The phase shift introduced by dent angles for all mirrors, reducing the depolarization these plates was 1 = 6 + /2 and 2 = 5 + /2 for the effects. A set of quarter- and half-wave plates for appro- chosen wavelengths 1,2 , respectively. To perform identity priate wavelengths is inserted in each arm of the setup to transformation I nothing is to be inserted into the beam. For Z and X transformations only one plate, rotated at realize the polarization transformations.

(n (

) -n (

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Table 3: Experimentally measured fidelity for the states prepared at Bob's station (I column) and the states after Alice's transformations (X , iY and Z columns, respectively).

Fig. 3: (Color online) Alice's station.
DBS PBS

State |D1 L2 |L1 D2 |D1 V2 |V1 D2 |L1 V2 |V1 L2

I 97.93 98.45 98.60 98.40 99.53 99.13

X 98.25 98.86 94.90 96.99 98.25 98.15

iY 98.08 98.78 97.76 96.93 97.21 98.06

Z 97.20 98.99 96.94 97.89 99.18 96.51

D4

D3 PBS D1 D2

coincidence circuit

Fig. 4: (Color online) Deterministic measurement scheme.

0 or 45 respectively, is used. The iY transformation is a superposition of Z and X and is performed when both plates are inserted. The transformed ququart state is measured deterministically by Bob by means of the setup shown in fig. 4. Dichroic beam-splitter separates the photons of different wavelengths, and two sets of quarter- and half-wave plates followed by polarizing beam-splitters allow one to chose the measurement basis. Measurement results in clicks of a pair of detectors, which are registered with a 4-input double-coincidence scheme. The scheme produces only pairwise clicks of the detectors, and no coincident clicks of D1 and D2 (as well as of D3 and D4) are possible, since there is only one photon in each arm after the DBS.

set-up, to be sure that the whole state evolution runs in a desirable way. To estimate the quality of reconstructed state, we calculated the fidelity: F = | meas |theor |2 , where |theor is the expected (theoretical) state vector. Table 3 represents the fidelity for the reconstruction of six states used in the protocol, prepared at Bob's station (i.e. in the case, when Alice performs identity transformation) and after Alice's transformations (X , iY and Z ). The fidelity of reconstructed states is rather high ranging from 0.949 through 0.995. At the same time it is not perfect due to the instrumental errors, caused by the imperfections of optical elements such as variances in quartz plates thickness and small depolarizing effects of dichroic mirrors. Basically the error rate accumulates errors arising both from Bob's preparation scheme and Alice's transformation one. In any case, the achieved fidelity is well above the error limit which is expected to be critical for protocol's security. Conclusion. ­ As a conclusion, we have demonstrated the ability to implement the six-state deterministic QKD protocol using polarization ququarts as information carriers. The experimental results clearly demonstrate that the proposed method allows one to prepare the states used in the protocol and to perform all the necessary unitary transformations with high quality. The simplicity of the protocol implementation is based mainly on the fact that only product states of two qubits are involved in ququart engineering. The reported experiment is a proofof-principle, and some parts of the 6DP protocol stayed beyond the scope of this work, particularly the Control Mode was not actually realized. Nevertheless, there are no principal problems with implementing it using the deterministic measurement scheme presented in the work. It is worthy to mention, that all manipulations with ququarts described above can be implemented with fast polarization transformers, whereas no principal problems are to be expected when using this scheme for free-space key distribution. In that sense the biphoton implementation of qudits seems to be an attractive tool for further research.

Results and discussion. ­ To analyze quality of prepared and transformed states the tools of quantum state tomography (QST) were used [16]. The ququart state |in is sub jected to the linear transformations that are done by a set of the retardation plates. Then pro jective measurements of the transformed state are performed onto the states with vertical polarizations. The state can be statistically reconstructed using the data obtained in the series of measurements with varying plates orientation. The parameters of the plates, i.e. (1,2) and optical thicknesses for different wavelengths j their orientations, are supposed to be known. Statistical analysis of experimental data allows us to reconstruct initial state |meas , i.e. to obtain complex coefficients of (2). Such a procedure is rather standard, fully automatized and can be applied to an arbitrary ququart state. We made This work is supported in part by MIMOS Berhad the state reconstruction in several control points of the (Malaysia), Russian Foundation for Basic Research 10008-p4


Quantum state engineering with ququarts: Application for deterministic QKD protocol (07-02-91581-ASP), and the Leading Russian Scientific Schools (796.2008.2).
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