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Optics and Spectroscopy, Vol. 94, No. 5, 2003, pp. 684­690. Translated from Optika i Spektroskopiya, Vol. 94, No. 5, 2003, pp. 743­749. Original Russian Text Copyright © 2003 by Burlakov, Krivitskioe, Kulik, Maslennikov, Chekhova.

ENTANGLED STATES IN OPTICS

Measurement of Qutrits
A. V. Burlakov, L. A. Krivitskioe, S. P. Kulik, G. A. Maslennikov, and M. V. Chekhova
Faculty of Physics, Moscow State University, Moscow, 119899 Russia e-mail: postmast@qopt.phys.msu.su
Received November 25, 2002

Abstract--A method of optical realization of a protocol of restoration of the density matrix of an unknown state of a three-level system that represents an arbitrary polarization state of a single-mode biphoton field is proposed. The method is applied to a set of pure states of qutrits, with this set being determined by the properties of SU(2) transformations performed by polarization transformers (retardation plates). © 2003 MAIK "Nauka / Interperiodica".

INTRODUCTION In the physics of quantum information, systems whose complete description is possible in terms of three orthogonal states are called qutrits (q-trits). In the case of a pure state, the wave function of a three-level system can be written in the form = c 1 |1 + c 2 |2 + c 3 |3 , (1) where |1, |2, and |3 are the orthogonal basis states. The complex coefficients ci are called the amplitudes of the basis states |i . These amplitudes are related by the following normalization condition:


i=1

3

c

2 i

= 1.

(2)

Representation (1) is a generalization of the notion of a qubit (q-bit) for a space whose dimensionality d = 3. Several ways of experimentally realizing multilevel states in quantum optical systems are known. In one of them [1], an interferometric method of state preparation is used, in which attenuated laser pulses are sent into a multiple-arm interferometer. The number of arms defines the dimensionality of the system. The basis states are identified either via the delay of the pulses (the temporal basis) or by the presence of constructive interference in the interferometer arm located in the detection system (the energy basis). Another example is an optical field formed by pairs of correlated photons belonging to different polarization modes. The preparation of such fields and their unitary transformations are considered in [2, 3]. Multilevel systems are of great interest in quantum cryptography, where their use is associated with an increase in security with respect to a certain class of eavesdropping attacks [4­6] (the socalled symmetric incoherent individual attacks). Adequate measurement of the parameters of quantum states is a basic problem in quantum information

science. The solution of this problem is associated with the possibility of realizing information output devices, protocols of error correction, quantum repeaters, and other quantum communication devices. Fundamentally, the problem of the minimal set of measurements sufficient for the complete description of the state of a system is very important. It is worth noting that, in some cases, there is no need to perform a complete set of measurements to determine the purity of a system [7]. Of course, different types of measurement procedures are used for different types of quantum states. Thus, homodyne tomography methods are being developed for squeezed states of light [8]. In principle, these techniques allow one to restore the density matrix of nphoton Fock states [9]. For polarization squeezed [10] and scalar [11] light, the fluctuations of Stokes parameters are analyzed and the quasi-probability function is restored [12]. In the case of two-photon fields, the set of fourth-order field moments in different spatial and polarization modes is determined [13]. We note that, in the context of every experimental procedure, a priori information about the properties of the state under consideration plays an important role. BIPHOTONS AS QUTRITS This paper is devoted to the optical realization of a protocol of restoration of the density matrix of an unknown state of a three-level quantum system. The object of study is the polarization state of a two-photon field belonging to a single spatial and frequency mode. In [2], it was shown that the pure state of such a field can be represented in the form = c 1 |2, 0 + c 2 |1, 1 + c 3 |0, 2 . (3) Two-photon Fock states in the two orthogonal H and V polarization modes serve as the basis states. Thus, for example, the second term in (3) corresponds to the existence of a single photon each in the H and in the V

0030-400X/03/9405-0684$24.00 © 2003 MAIK "Nauka / Interperiodica"


MEASUREMENT OF QUTRITS

685

modes with the probability |c2 |2. The vacuum component |0, 0 in (3) is disregarded, since, in measuring the field by the method of coincidence of photon counts, the contribution of this component to the correlation functions is zero. The arguments of the imaginary parts of the complex-valued coefficients ci (i = 1, 2, 3) are the phases of the basis states. Since the total phase is inessential, one of the phases can be excluded from consideration by introducing relative phases, for example, 12 = 1 ­ 2 and 32 = 3 ­ 2. For describing the polarization properties of a biphoton field, the so-called polarization matrix, or coherence matrix, was introduced in [14]: A D E K4 = D* C F . (4) E* F* B The elements of this fourth-order matrix are the normally ordered fourth moments: A a a ,
2 2 2

and condition (7) yields E * = ABC / DF , D
2

F

2

= BC ,

= C(1 ­ B ­ 2C).

(10)

For mixed states, the density matrix involves an additional averaging with the classical distribution function P over possible states of the system, with P satisfying the condition P = 1. In this case, the i=1 i density matrix acquires the form





mn

= cm c* . n

(11)

B b b ,
2 2

C a b ab , F a b a .
2

D a ab ,


E a b ,
2 2

Here, a a H , b a V , a aH, and b aV are the operators of creation and annihilation of photons in the H and V polarization modes, respectively. It can be clearly seen that the diagonal components of the matrix K4 are real-valued quantities. They characterize the intensity fluctuations in parallel (A and B) or orthogonal (C) polarization modes. The off-diagonal elements D, E, and F are generally complex-valued. Since the state of a biphoton field can be fully described by the fourth-order moments with respect to the field, the elements of the matrix K4 can be expressed in terms of the elements of the density matrix of the biphoton field. For example, for the pure state (3), by * definition [15], mn = c m c n and
11

=c

2 1

= A /2 ,
33


2

22

=c

2 2

= C,


12

(5)

=c

3

= B /2 ,
13

* = c 1 c 2 = D */ 2 ,

* = c 1 c 3 = E */2 ,

* 23 = c 2 c 3 = F */ 2 . The purity condition of a state =
2

(6)

(7) (8)

and its normalization condition Sp ( ) = 1 impose certain constraints on the elements of the matrix K4. Thus, it follows from (8) that A + B + 2C = 2,
OPTICS AND SPECTROSCOPY Vol. 94 No. 5

DETERMINATION OF THE STATE OF QUTRITS The question arises as to how many (real-valued) parameters should be measured to completely characterize an unknown state of a biphoton field. From the definition and properties of the density matrix, it follows that, in the case of a pure state, the number of realvalued parameters that determine the state of a system with the dimensionality d is equal to 2d ­ 2 and, in the case of a mixed state, it is equal to d 2 ­ 1. Accordingly, one needs to know four real-valued numbers in the first case and eight in the second. Taking into account that only unnormalized state amplitudes are measured experimentally and that one needs to determine conditions (8) and (9) after the measurement of all three diagonal elements of the matrix K4, we have that five moments should be measured in the case of the pure state and nine moments in the case of the mixed state. Before proceeding to discussion of the protocol proposed, we should note that the measurement procedure is always accompanied by an inevitable destruction of the quantum state due to its interaction with a classical measuring device, in our case, with the photodetector. Therefore, when speaking about an input state, we bear in mind that this state is represented by a sufficiently large set (ensemble) of copies, some part of which is destroyed upon measurement. Accordingly, inferences made from the results of the measurements are referred to the remaining part of the ensemble. Such a measurement procedure lies at the heart of the ensemble approach to quantum measurements. In quantum optics, a Brown­Twiss system is commonly used for measuring the fourth-order moments of a field. This system consists of a beam splitter with photodetectors in its output ports (Fig. 1). Polarization transformations in each of the two spatial modes are performed with retardation plates (/2, /4) and polarization filters (polarizers) É. Consider normally ordered fourth-order moments of the field R12(1, 1, 2, 2) ~ '' b 1 b 2 b '1 b '2 that are measured by the system shown in Fig. 1 at a specified orientation of the plates and at a fixed polarization transmitted by the polarizers. Our aim is to search for the minimal set of such moments,


(9)
2003


686
RP /4 ^^ a, b BS /2 P ^ b1 '

BURLAKOV et al.

is the matrix that describes the action of the nonpolarizing beam splitter;
D1

GV = 0 0 01

(14)

RP /4 /2 P ^ b2 ' D2



is the matrix of the polarizer that transmits the vertical component of the field; and G
R12
/4, /2

= t r ­r * t *

(15)

are the matrices of the retardation plates, with the coefficients t and r being equal to t = cos + i sin cos 2 , r = i sin sin 2 , (16)

Fig. 1. Schematic diagram for measuring the moments forming the polarization matrix K4. (BS) beam splitter; (RP) /4 and /2 retardation plates, characterized by the parameters /2, /4 and /2, /4; (P) polarizers transmitting a vertical polarization; (D1, D2) photodetectors.

i.e., experimentally measured parameters, from which one can compose all the elements of the polarization matrix K4. In this case, the input field will be transformed by the plates and polarizers so that the moments detected can be expressed in terms of the elements of the matrix K4. The polarization transformations performed by the plates are unitary, and each polarizer plays the role of a polarization filter that specifies the polarization state of light measured by the corresponding detector. This idea underlies standard measuring systems for determining the Stokes parameters [16]. It was also used in [13] for measurement of the polarization properties of both spatial modes of biphoton light. It should be noted that the choice of quarter- and halfwave plates as polarization transformers is obviously not unique; this choice is dictated exclusively by convenience (such wave plates are most commonly used in polarization experiments) and by the straightforwardness of the transformations. The successive action on a signal (idler) photon of the beam splitter, the two wave plates, and the polarizer transmitting the vertical polarization is described by the following matrix transformations: a' = G G G G a . V /2 /4 BS b' b (12)

( is the optical thickness (optical path length), and is the angle between the optic axis and the vertical direction (V)). For the quarter- and half-wave plates, /4 = /4, /4 , /2 = /2, and /2 . Correspondingly, these coefficients have the form t
/4

= 1/ 2 ( 1 + i cos 2 ) , r
/4

= i / 2 sin 2 , = i cos 2 , = i sin 2 .

(17)

t

/2 /2

r

(18)

From the definition of the matrix K4, one can clearly see how to measure the diagonal elements of this matrix and the moments B, A, and C. In the first case, the optic axes of all the wave plates are aligned vertically, along the direction of transmission of both polarizers P. In the second case, the direction of polarization in both beams should be rotated by 90°, which can be achieved by setting 1 = 0°, 1 = 45°, 2 = 0°, and 2 = 45°. In the third case, the polarization is rotated in only one beam, and, due to the symmetry of the system, it does not matter in which: 1 = 0°, 1 = 45°, 2 = 0°, and 2 = 0°. More complex transformations are needed in the case of measurement of the off-diagonal elements of the matrix K4. For example, consider the action of the following combination of plates on the input state. Let 1 = 0°, 1 = 45°, 2 = 45°, and 2 = 22.5°. It is easy to verify that, in this case, (19) 2 2 2 2 = 1/8 [ a a ­ a ab ­ a b a + a b ab ] = 1/8 [ A + C ­ 2Re D ] . The moment measured in this experiment contains the contributions from three elements of the coherence matrix. Two of them are the real-valued diagonal elements A and C, and the third one is the real part of the (complex-valued) off-diagonal element D. This examOPTICS AND SPECTROSCOPY Vol. 94 No. 5 2003

Here, a and b are the operators of annihilation of the input state in two orthogonal polarization modes H and V at the input of the system, respectively; a' and b' are the corresponding operators of annihilation at the output of the transformers; the vectors of both states are written in terms of the Jones representation; G = 1/ 2 0 0 1/ 2 (13)

R

12

'' = b 1 b 2 b '1 b '2





BS


MEASUREMENT OF QUTRITS The protocol of measurement of a set of moments forming a polarization matrix No. 1 2 3 4 5 6 7 8 9 Plate /4 (I) 1, deg 0 0 0 45 45 45 45 ­ 45 45 Polarizer (I) 1, deg 90 90 0 0 45 45 0 ­22.5 ­ 45 Plate /4 (II) 2, deg 0 0 0 0 0 0 0 45 45 Polarizer (II) 2, deg 90 0 0 0 0 90 90 22.5 45 Moment A/4 C/4 B/4 1/8(B + C + 2ImF) 1/8(B + C ­ 2ReF) 1/8(A + C ­ 2ReD) 1/8(A + C + 2ImD) 1/16(A + B ­ 2ImE) 1/16(A + C ­ 2ReE)

687

ple shows that, since the cosine and sine of the phases of the 1, 2, or 3 states are measured directly, rather than the phases themselves, the number of necessary measurements increases. In the real experiment, which is described below, instead of a set of two wave plates and a fixed polarizer, we used a simpler configuration in each arm of the Brown­Twiss system. We took into consideration the fact that, in measuring the fourth-order moments, the polarization transformation performed by the halfwave plate and the fixed polarizer is equivalent to the action of a polarizer alone, whose orientation is specified by the angle . In this case, the angles of rotation of the half-wave plate and the polarizer are related by the formula = ­2 . (20)

The angles of orientation of the quarter-wave plates (1, 2) and polarizers (1, 2) in both arms of the system are presented in the table, along with the corresponding values of the moments measured. This table, essentially, serves as a protocol of the restoration of the initial (input) state of the field, represented in the form of a biphoton (a qutrit). It can be seen that, generally, the number of measurements is equal to nine. The first seven measurements realize the protocol for the pure input state. Two additional measurements are necessary to determine the values of the cosines (sines) of the corresponding phases. The eighth and ninth rows of the table show how to find the real and imaginary parts of the complex-valued moment E, which, in the case of the pure state, according to (10), can be expressed in terms of the remaining moments. We note that, for the determination of each of the off-diagonal elements of

351 nm He­Ne -laser 632 nm

1

LiIO3 1 BS

| = |V, V = |0, 2

P1
D
1

702 nm 3

2

P0 I

5 CC 6 5
D
2

4 P2 2 II

ISP-51

III

Fig. 2. Schematic diagram of the experimental setup (explanations in the text): (I) unit of preparation of the input state; (II) measurement unit; (III) spectral unit. OPTICS AND SPECTROSCOPY Vol. 94 No. 5 2003


688 (d) 2 ­ 3, 2 ­ 1, deg 0 2 ­ 1 ­1

BURLAKOV et al.

­2

2 ­ 3

­3 5 c32 0.8

15

25 (c)

35

45

EXPERIMENTAL The experimental setup is shown in Fig. 2. It can be arbitrarily divided into two units, the unit of preparation of the input state and the measurement unit. The first unit contains a CW Ar+ laser 1 operating at 351 nm with a power of 120 mW. The laser serves as a pumping source for a lithium iodate crystal, in which the generation of a biphoton field takes place. This unit also includes a system of adjustable mirrors, a quartz plate 2 whose orientation can be continuously varied, and an interference filter 3 with a central wavelength of 702 ± 5 nm. Due to spontaneous down-conversion, two-photon states of light emerge from the crystal. A collinear regime of down-conversion degenerated in frequency with type I phase matching realized. The wavelength of the biphoton radiation was equal to s = i = 2p = 702 ± 9 nm. Both photons were vertically polarized. Thus, the biphoton field downstream from the crystal was in the state = c 3 |0, 2 + |v ac .
in

0.4

(21)

0 c22 0.8 0 20 (b) 40

The crystalline quartz plate was used to transform this state to that described by formula (3) (Fig. 2). It is known that all the transformations of the polarization of biphotons performed by retardation plates can be described by unitary square (3 â 3) matrices G [2]: c 1 c2 c3 where
out

0.4

c 1 = G c2 , c3

in

(22)

0 c12 0.8 0 20 (a) 40

2 2 tr t 2 G = G ( , ) = ­ 2 tr * t ­ r 2 2 ­ 2t*r* r*

2 t * r , (23) t *2 r
2

0.4

0 0 20 40 , deg

Fig. 3. (d) 2 ­ rotation

Dependences of the parameters (a­c) |ci |2 and 3 and 2 ­ 1 of the input state on the angle of of the setting retardation plate.

the matrix K4, in our protocol, one needs to know only three moments, which, obviously, is the minimal number of sufficient measurements.

the coefficients t and r were defined by formulas (16). Matrices (23) give irreducible representations of the SU(2) group with 3 â 3 dimensionality in a space of state vectors (3). It should be noted that an arbitrary polarization state of a biphoton field cannot be realized by retardation plates alone. In the general case, such transformations, along with the space of state vectors (3), form a three-dimensional unitary representation of the SU(3) group. The thickness of the setting retardation plate was h = 824 ± 1 µm; therefore, the parameter (/ )(no ­ ne)h was fixed. The second parameter, , was varied in the course of the experiments, which allowed us to specify the state of a biphoton field sent to the input of the measurement block. It is clear that all the states prepared in such a way did not remove the initial (input) biphoton field from the class of the pure states.
OPTICS AND SPECTROSCOPY Vol. 94 No. 5 2003


MEASUREMENT OF QUTRITS

689 (a) ImF

The measurement unit contains the Brown­Twiss system shown in Fig. 1. In our experiments, we used Pockels cells 4 instead of the quarter-wave plates. When necessary, appropriate voltage levels were applied to the cells. The Pockels cells were preferable to the quarter-wave plates, because the cells made possible the remote control of the state of polarization. The spectral distribution of the biphoton radiation was determined with an ISP-51 spectrograph. Pulses from detectors 5 were fed to a standard coincidence circuit 6, which counted the number of coincident photons per unit time, with this number being proportional to the correlators (19). The measurement procedure was as follows. For a certain orientation of the setting quartz plate 2, a measurement was made according to the protocol described in the table. Then, the setting plate was rotated by an angle , which corresponded to changing the input state, and the measurement was repeated. Figure 3 shows the squared magnitudes of the amplitudes of the three states and the two phases in relation to the angle of rotation of the plate. Each experimental point on the plots corresponds to a certain input state specified by the orientation of the plate. The solid lines were calculated using formulas (22) and (23). Since all the three moments measured contributed to the calculation of the phases (table), the errors of the three measurements were added together and the accuracy of the corresponding measurements was low. The principal source of the errors was the poor quality of the Pockels cells and, as a consequence, the inadequacy of the polarization transformations performed with these elements. It seems likely that the use of retardation plates operating in the zeroth order of interference is the only way to overcome this problem. Note that the measured and calculated magnitudes of the amplitudes of the states are in good agreement with each other. All the errors that arise in this case are mainly related to the errors in determining the orientation angles of the polarizers. The representation of the measured states of the field in terms of complex coefficients ci , as in Fig. 3, is

ReF, ImF 0 ­ 0.1 ­ 0.2 ­ 0.3 ­ 0.4 0 ReD, ImD 0.4

ReF

20

40 , deg

(b) ReD

0.2

0

ImD

­ 0.2 0

20

40 , deg

Fig. 4. Dependences of the real and imaginary parts of the moments (a) F and (b) D on the angle of rotation of the setting retardation plate.

possible only for pure states. For mixed states, one needs to measure all the six moments that form the matrix K4. The real and imaginary parts of the moments D and F are shown in Fig. 4. Below, were presented the calculated (in parentheses) and measured elements of the density matrix of the state corresponding to an angle of rotation of the setting plate = 25°:

0.25 ( 0.341 ) 0.32 ­ i 0.09 ( 0.405 + i 0.042 ) ( ­ 0.238 ­ i 0.05 ) = 0.32 + i 0.09 ( 0.405 ­ i 0.042 ) 0.55 ( 0.486 ) ­ 0.28 ­ i 0.04 ( ­ 0.288 ­ i 0.03 ) . ( ­ 0.238 + i 0.05 ) ­ 0.28 + i 0.04 ( ­ 0.288 + i 0.03 ) 0.19 ( 0.173 )

Of course, for pure states (and it was this case that was realized in our experiments), these moments can be expressed in terms of the amplitudes ci . The moment E was not determined because, as was noted above, it can be expressed in terms of the remaining moments.
OPTICS AND SPECTROSCOPY Vol. 94 No. 5 2003

CONCLUSIONS In this study, we proposed a procedure for determining an unknown state of a three-level system, a qutrit, which represents an arbitrary polarization state of a single-mode biphoton field. This procedure is realized for


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BURLAKOV et al. 6. D. Bruss and C. Macchiavello, Phys. Rev. Lett. 88, 127 901 (2002). 7. A. K. Ekert, C. M. Alves, D. K. L. Oi, et al., Phys. Rev. Lett. 88, 217 901 (2002). 8. G. M. D'Ariano, in Quantum Optics and Spectroscopy of Solids, Ed. by A. S. Shumowsky and T. Hakiouglu (Kluwer, Amsterdam, 1997), p. 175. 9. S. Schiller, G. Breitenbach, S. F. Pereira, et al., Phys. Rev. Lett. 77, 2933 (1996). 10. A. S. Chirkin, A. A. Orlov, and D. Yu. Parashchuk, Kvantovaya èlektron. (Moscow) 20, 999 (1993). 11. V. P. Karasev, J. Sov. Laser Res. 12 (5), 147 (1991). 12. A. V. Masalov and V. P. Karasev, Opt. Spektrosk. 91, 558 (2001) [Opt. Spectrosc. 91, 526 (2001)]. 13. D. James, P. Kwiat, W. Munro, and A. White, Phys. Rev. A 64, 052 312 (2001). 14. D. N. Klyshko, Zh. èksp. Teor. Fiz. 111, 1955 (1997) [JETP 84, 1065 (1997)]. 15. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory, 4th ed. (Nauka, Moscow, 1989; Pergamon, New York, 1977). 16. W. A. Shurcliff, Polarized Light: Production and Use (Harvard Univ. Press, Cambridge, Mass., 1962; Mir, Moscow, 1965). 17. G. M. D'Ariano, M. G. A. Paris, and M. F. Sachi, Phys. Rev. A 62, 023 815 (2000).

a set of pure states of qutrits, with this set being determined by the properties of the SU(2) transformations performed by polarization transformers (retardation plates). However, it is of interest to realize a complete protocol of restoration of the density matrix both for pure and for mixed states of qutrits. Such experiments are now in progress, and their results will be published in the near future. Separately, the correction of measurement results by using the maximum likelihood method of numerical determination of the most probable value of [13, 17] will also be considered. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research (projects nos. 02-02-16664, 0302-16444) and INTAS (project no. 01-2122). REFERENCES
1. H. Bechmann-Pasquinucci and W. Tittel, Phys. Rev. A 61, 062 308 (2000). 2. A. V. Burlakov and D. N. Klyshko, Pis'ma Zh. èksp. Teor. Fiz. 69, 795 (1999) [JETP Lett. 69, 839 (1999)]. 3. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, et al., Phys. Rev. A 60, R4209 (1999). 4. H. Bechmann-Pasquinucci and A. Peres, Phys. Rev. Lett. 85, 3313 (2000). 5. N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Phys. Rev. Lett. 88, 127 902 (2002).

Translated by V. Rogovooe

OPTICS AND SPECTROSCOPY

Vol. 94

No. 5

2003