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JETP Letters, Vol. 78, No. 6, 2003, pp. 352­357. Translated from Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 78, No. 6, 2003, pp. 804­809. Original Russian Text Copyright © 2003 by Bogdanov, Krivitsky, Kulik.

Statistical Reconstruction of the Quantum States of Three-Level Optical Systems
Yu. I. Bogdanov1, *, L. A. Krivitsky2, **, and S. P. Kulik2, **
1

Russian Control System Agency, "Angstrem," Moscow, 124460 Russia *e-mail: bogdan@angstrem.ru 2 Moscow State University, Vorob'evy gory, Moscow, 119992 Russia **e-mail: postmast@qopt.phys.msu.su
Received July 24, 2003

The procedure of measurement followed by the reconstruction tem is implemented for a frequency- and spatially degenerate mation of the quantum state from a solution to the likelihood erties of the obtained estimators is developed. Using the root two-photon (qutrit) wave function is reconstructed from the MAIK "Nauka / Interperiodica". PACS numbers: 03.67.Hk; 42.50.Dv; 42.25.Ja

of the quantum state of a three-level optical systwo-photon field. The method of statistical estiequation and the analysis of the statistical propmethod of estimating quantum states, the initial measured fourth-order field moments. © 2003

Introduction. The ability of measuring quantum states is no doubt of fundamental interest because it provides a tool for the analysis of basic concepts of quantum theory, such as the fundamentally statistical character of its predictions, the superposition principle, the Bohr's complementarity principle, etc. By the measurement of quantum state we will imply a two-step measurement and computation procedure. The first step is a genuine measurement consisting of a set of operations under the representatives of a quantum statistical (pure or mixed) ensemble, as a result of which the experimenter acquires a set of frequencies with which particular events occur. The second step consists of the mathematical procedure of reconstructing the quantum state of an object using the combination of the obtained statistical data. This work is devoted to the state reconstruction for optical three-level systems. Such states are obtained, e.g., in the polarization representation of a frequencyand spatially degenerate biphoton field [1]. The necessity of properly measuring the states of such systems is dictated by the applied problems. For example, increase in the key distribution security in quantum cryptography is associated with the increase in the dimensionality of Hilbert space for the states in use [2]; in this respect, certain hopes are pinned on the three-level systems (qutrits) [3, 4]. 1. Biphotons as three-level systems. A biphoton field is a coherent mixture of the biphoton Fock's states |1k, 1k' and the vacuum state |vac [5]: 1 | = |vac + -2

where the coefficients Fk, k' are called biphoton amplitudes [6], and |1k, 1k' denotes the state with one (signal) photon in the k ks mode and one (idler) photon in the k' ki mode. We will consider the collinear and degenerate regimes, for which ks ki, s i p/2, and s + i = p, where p is the laser pump frequency. The method of producing, transforming, and measuring these states is described in detail in [7­10]. The initial state to be measured and reconstructed has the following form: | = c 1 |2, 0 + c 2 |1, 1 + c 3 |0, 2 . (2)


k, k'

F

k, k'

|1 k, 1 k' ,

(1)

Here, we use the representation of a biphoton-light polarized state (1) in the Fock's basis. For example, the notation |2, 0 indicates that both photons are in the horizontal (H) polarization mode, while no photons are present in the vertical (V) mode. The state of a three-level system in quantum information theory has come to be known as qutrit. The properties of biphoton qutrits and their mapping on the PoincarÈ sphere were described in [8]. The idea of producing and measuring state (2) was put forward in [9, 10]. The system for detecting biphoton qutrits includes a beam splitter and a pair of detectors whose outputs are connected to the photocount coincidence circuit (figure). An event is considered detected if a pulse appears at the output of the coincidence circuit. In approximately one half of all cases, one of the photons (signal, by convention) of a biphoton is led to one of the detectors, while another (idler photon) is led to the second detector. In the remaining cases, both photons occur in the same beam-splitter arm, and these events are not detected because they do

0021-3640/03/7806-0352$24.00 © 2003 MAIK "Nauka / Interperiodica"


STATISTICAL RECONSTRUCTION OF THE QUANTUM STATES

353

not coincide. The polarization transformations are accomplished using a quarter-wave plate and a polarizing prism placed ahead of each detector. It was shown in [9, 10] that, if the qutrit state is unknown, one is forced to make nine moment-projection measurements to reconstruct the initial mixed (in the general case) state. These moments have the form [11]: R
s, i

( b 's ) ( b 'i ) b 's b 'i = R ( s, s, i, i ) ,

(3)

where ( b 'j ) and b 'j are, respectively, the photon creation and annihilation operators for the signal and idler modes j = s, i after the transformation and j and j are the setting parameters (plate and polarizing-prism orientation angles, respectively). The time (as a rule, 100 ms) it took for measuring each of the nine moments was one of the experimental parameters. Each measurement was made in triplicate and consisted in the taking of 700­800 averages, whereupon the scheme was reset; namely, the j and j angles were set according to the tomographic protocol (Table 1), after which the cycle was repeated. Thus, the mean photocount coincidence rates R1, 2, ..., 9 were the output data of the measuring setup. To compare the results of reconstruction with the parameters of the input states, which should be known with a high accuracy, we used the following method of state preparation. The biphotons were produced in the process of collinear frequency-degenerate spontaneous parametric down conversion in lithium iodate crystal. The polarization of both created photons was vertical; i.e., the state |c' = |0, 2 (4)

Schematic of experimental setup. (1) Argon laser (p = 351 nm); (2) lithium iodate crystal in which the biphotons with a central wavelength of 702 nm are generated; (3) quartz phase plate with parameters (, = 32.82); (4) beam splitter directing (conventionally) the signal photons to the right and the idler photons downward; (5) quarter-wave plates (s, i, = /4); (6) polarizing prisms (s, i); (7) detectors operating in the photon counting mode; (8) double coincidence circuit; (9) deflecting mirror; (10) mirror transmitting the radiation at a wavelength of 351 nm and reflecting at a wavelength of 702 nm; and (11) mode selection iris.

was generated. Next, this state was transformed using a quartz plate with a given thickness h = 824 ± 1 µm. Upon turning this plate in the plane perpendicular to the incident biphoton beam (the plate optical axis lied in this plane), the state (2) transformed according to the rule |c where the matrix
2 2 tr t G = ­ 2 tr * t 2 ­ r 2 r *2 ­ 2 t * r * in

= G |c' ,

(5)

2t*r 2 t* r
2

(6)

describes the action of a plate with effective transmittance t and reflectance r, t = cos + i sin cos 2 and r = i sin sin 2. In this expression, the optical thickness = |no ­ ne|h/ and is the angle between the plate optical axis and the vertical. For crystalline quartz, |no - ne| = 0.0089 at the wavelength = 702 nm, whence it follows that = 32.82 ± 0.04.
JETP LETTERS Vol. 78 No. 6 2003

So, unitary transformation (5) gave a set of states cin() that was fed into the measuring unit of the setup. The purpose of this work was to reproduce these states. 2. Statistical reconstruction of biphoton states from the results of mutually complementary measurements. When analyzing the experimental data, we will use the so-called root method of estimating quantum states [12­14]. This method is designed specially for the analysis of mutually complementary measurements.1 The advantages of this method consist in the possibility of reconstructing states in a high-dimensional Hilbert space and posing fundamental limits on the accuracy of reconstruction of an unknown quantum state. The use of asymptotically efficient algorithms allow one to achieve a reconstruction accuracy close to its fundamental limit. The set of mutually complementary measurements of a biphoton-field state was implemented in accordance with the tomographic protocol presented in Table 1. The event-generation intensity R, = 1, 2, ..., 9, is the main quantity accessible for the measurement. The moments R are the coincidence frequencies measured in frequency units (Hz). The number of events occurring in any given time interval obeys the Poisson distribution. Therefore, the quantities R specify the intensities of the corresponding mutually complementary Poisson processes and serve as estimators of the Poisson parameters (see below).
1

In the sense of the Bohr's complementarity principle.


354 Table 1 s 1 2 3 4 5 6 7 8 9 0° 0° 0° 45° 45° 45° 45° ­ 45° ­ 45°

BOGDANOV et al.

Parameters of experimental setup s ­90° ­90° 0° 0° ­45° ­45° 0° ­22.5° ­45° i 0° 0° 0° 0° 0° 0° 0° 45° 45° i ­90° 0° 0° 0° 0° ­90° ­90° ­22.5° 45°

Amplitude of the process M c1 -----2 c2 ---2 c3 -----2 1 i --------- c 2 ­ -- c 3 2 22 1 1 --------- c 2 ­ -- c 3 2 22 1 1 -- c 1 ­ --------- c 2 2 22 1 i -- c 1 ­ --------- c 2 2 22 1 i --------- c 1 + --------- c 22 22 1 1 --------- c 1 ­ --------- c 22 22

3

3

For each process, the event-generation intensity can be represented as a squared absolute value of the corresponding amplitude: R = M* M , = 1, 2, ..., 9. (7) Although the amplitudes of the processes cannot be measured directly, they are of the greatest interest as quantities describing the fundamental relationships of quantum physics. From the superposition principle, it follows that the amplitudes are linearly related to the state-vector components. It is the purpose of quantum tomography to reconstruct the amplitudes and state vectors that are hidden from direct observation. The linear transformation of the state vector c into the amplitude of the process M is described by a certain matrix X, which can easily be obtained from Table 1: 1/ 2 0 0 0 X= 0 1/2 1/2 1/ ( 2 2 1/ ( 2 2 0 1/2 0 1/(2 2) 1/(2 2) ­1 / ( 2 2 ) ­i / ( 2 2 ) ) ) 0 0 1/ 2 ­ i /2 ­ 1/2 . 0 0 i/(2 2) ­1 / ( 2 2 ) 0 0

Then the set of all nine amplitudes of the processes can be expressed by a single matrix equation Xc = M . (9) We call matrix X the instrumental matrix of a set of mutually complementary measurements, by analogy with the conventional instrumental function. In statistical terms, Eq. (9) is the linear regression equation. A distinctive feature of the problem is that only the absolute value of the right-hand side of the equation is measured in the experiment. The estimate of the absolute value of the amplitude is given by the square root of the corresponding experimentally measured coincidence frequency: M
exp

=

k/t,

(10)

(8)

where k is the number of events detected in the th process during exposure time t. It is worth noting that, by the action of root-square procedure on a Poisson random value, one obtains the random variable with a uniform variance, i.e., at the variance stabilization [13, 15]. Note also that, since we deal not with event probabilities but with their frequencies or intensities, it is convenient to use nonnormalized state vectors. These vectors allow the coincidence counting rate (event-generation intensities) to be derived directly from the formulas given in Table 1, without introducing coefficients related to the biphoton generation rate, detector efficiencies, etc. The dimensionality of the vector state thus obtained is 1 / time .
JETP LETTERS Vol. 78 No. 6 2003


STATISTICAL RECONSTRUCTION OF THE QUANTUM STATES

355

The final state vector obtained by the reconstruction procedure, nevertheless, will be normalized to unity. Considering that the variances of different |M|exp are independent and identical, one can apply the standard least-squares estimator to Eq. (11) [16]:
­1 + + ^ ^ c = ( X X) X M.

whence it follows that I Jc = c .
­1

(17)

(11)

Contrary to the traditional least-squares method, the relation obtained cannot be used for explicit estimation of the state vector c, because it is to be solved by the ^ iteration method. The absolute value of M is known exp), and its phase is ^ from the experiment ( M = |M| determined by the iteration procedure (it is assumed that the phase of vector Xc at the ith iteration step deter^ mines the phase of the vector M at the (i + 1)th step). It turns out that, in the Gaussian approximation for the Poisson's quantities, this least-squares estimator coincides with a more exact and rigorous maximum likelihood estimator considered below. 3. Maximum likelihood method. The likelihood function is defined by the product of Poisson probabilities: L=

We will call the latter relationship the likelihood equation. This is a nonlinear equation, because i depends on the unknown state vector c. Because of the simple quasi-linear structure, this equation can easily be solved by the iteration method [12­14]. The operator I-1J can be called a quasi-identity operator. Note that it acts as the identical operator on only one vector in the Hilbert space, namely, on the vector corresponding to solution (17) and representing the maximum possible likelihood estimator for the state vector. The condition for existence of the matrix I­1 is a condition imposed on the initial experimental protocol.2 The resulting set of equations automatically includes the normalization condition, which is written as


i

ki =


i

( i ti ) .

(18)


i

( i ti ) i -------------- e ki!

k

­i t

This condition implies that, for all processes, the total number of detected events is equal to the sum of the products of event detection rates into the exposure time. The elements of Fisher information matrix are defined as the average products of the derivatives of the logarithmic likelihood function [12­14] I
sj

i

,

(12)

where ki is the number of coincidences observed in the ith process during exposure time ti, and i (i = 1, 2, ..., 9) are the unknown theoretical event-generation intensities (photocurrent coincidences), whose estimation is the subject of this work. The logarithmic likelihood (logarithm of the likelihood function) is, except for an insignificant constant, ln L =

ln L ln L = ----------- ----------- , c j c* s

(19)


i

where lnL is given by Eq. (13). We will average in Eq. (19) on the assumption that the event detection obeys the Poisson law. Therefore, I =

( k i ln ( i t i ) ­ i t i ) .

(13)
sj

We also introduce the matrices with elements defined by the following formulas: I
sj


i

t i i --- -------i -------- . i c j cs *

(20)

=


i

t i X * X ij , is

(14)

J

sj

=


i

k ---i X * X ij , i is

s, j = 1, 2, 3 .

After going, in accordance with Eq. (7), from the event-generation frequencies (intensities) to the amplitudes of the processes, the expression for the Fisher information matrix is greatly simplified and takes the form of Eq. (14). If the exposure times for all processes are the same (t = ti = const), then I = t( X X ).
+

(15)

The matrix I is determined from the experimental protocol and, thus, is known a priori (before the experiment). It coincides with so-called Fisher information matrix (see below). On the contrary, the matrix J is determined by the experimental values of ki and by the unknown event-generation intensities i. In terms of these matrices, the condition for the extremum of function (13) can be written as Ic = Jc ,
JETP LETTERS Vol. 78 No. 6 2003

(21)

Thus, the Fisher information matrix, in fact, is determined by the observation time for the statistical ensemble. In other words, time plays the role of the most fundamental measure of information.
2

(16)

This theory is valid not only for an X matrix of the particular form (8) but also in the general case.


356 Table 2 Number No. angle, deg of events

BOGDANOV et al.

State vector k


=1



9

Theory c1, c2, c3 0.3094, 0.6248 + 0.1921i, 0.5713 + 0.3880i 0.4702, 0.6649 + 0.2368i, 0.4105 + 0.3349i 0.5518, 0.6548 + 0.2566i, 0.3289 + 0.3045i 0.6310, 0.6248 + 0.2744i, 0.2497 + 0.2717i 0.7053, 0.5758 + 0.2901i, 0.1754 + 0.2368i 0.7724, 0.5094 + 0.3036i, 0.1083 + 0.2002i

Experiment c1, c2, c3 0.3549, 0.5620 + 0.2684i, 0.5467 + 0.4328i 0.4961, 0.6444 + 0.2486i, 0.4017 + 0.3399i 0.5234, 0.6637 + 0.2446i, 0.3687 + 0.2997i 0.6466, 0.6184 + 0.2897i, 0.2521 + 0.2281i 0.7266, 0.5890 + 0.1900i, 0.2224 + 0.1992i 0.7953, 0.5107 + 0.2519i, 0.1217 + 0.1685i

F

2

1

17.5

2012

0.9946

15.925*

2

22.5

1997

0.9990

3.6482

3

25

6119

0.9979

16.384*

4

27.5

1281

0.9976

7.4987

5

30

2245

0.9921

26.977*

6

32.5

2753

0.9967

11.239

4. Analysis of experimental data. The examples of reconstruction of the qutrit states by the maximum likelihood method are given in Table 2. In the next-to-last column, the values of the fidelity parameter F defined as F = c c
pure state

c

calc

c exp

2

(22)

and indicating, in our case, the measure of correspondence between the theoretical and experimental state vectors are presented. The asterisk (*) denotes the experimental 2 values that exceed the critical value 11.345 at a confidence level of 99%. For these experiments, one can state with a guarantee of 99% accuracy that the uncertainties in setting the measurement parameters and their instabilities are statistically significant. In other words, a comparison of the reconstruction results with the fundamental statistical level of accuracy can serve as a basis for some problems such as the setup adjustment, operation stability control, revelation of foreign interference in the system, etc. Thus, for a small sample size, statistical errors prevail, whereas for large sample sizes, the setting errors and the instability of protocol parameters dominate. In our case, the observation time was such that both these types of errors played a significant part. For some experiments, the 2 values were lower than the critical level, and for some other experiments, these values were higher than the critical level. The larger the sam-

ple size for which the setup operation errors and instabilities are as yet insignificant, the higher the quality of the experiment.3 The process of quantum information accumulation is described by Eqs. (14) and (21); as the measurement time increases, the quantum states of the greater and greater number of ensemble representatives break and information about the object of interest progressively increases. Accordingly, the statistical error becomes more and more small. Therefore, only the errors of the first (statistical) type bear a fundamental quantum character. The errors of the second (setting error) type are, fundamentally, classical, because they are caused by the researcher's incomplete knowledge; i.e., a more exact information exists, in principle, but it is inaccessible to the experimenter. The state-preparation procedure considered in Section 1 assumes that only pure states of the form (2) are fed into the measuring unit of the setup. The same conclusion can be drawn from the data analysis: a comparison of the results of reconstructing quantum state in the approximation of a pure ensemble with the results of the separation of a mixture into two components (socalled quasi-Bayes algorithm [14]) indicates that the estimator for a pure state vector is very close to the estimator for the major density-matrix component.
3

The corresponding number of experiments can be called coherence volume. JETP LETTERS Vol. 78 No. 6 2003


STATISTICAL RECONSTRUCTION OF THE QUANTUM STATES

357

Conclusions. The procedure of measuring the quantum state of a three-level optical system formed by a frequency- and spatially degenerate biphoton field has been considered in this work. The method of statistical estimation of the quantum state through solving the likelihood equation and examining the statistical properties of the resulting estimators has been developed. Based on the experimental data (fourth-order field moments) and the root method of estimating quantum states, the initial wave function has been reconstructed for biphoton qutrits. The inaccuracy in setting the measurement parameters are analyzed for the cases where their instability is statistically significant. This work was supported in part by the Russian Foundation for Basic Research (project nos. 02-0216843 and 03-02-16444) and INTAS (grant no. 212201). REFERENCES
1. A. V. Burlakov and D. N. Klyshko, Pis'ma Zh. èksp. Teor. Fiz. 69, 795 (1999) [JETP Lett. 69, 839 (1999)]. 2. H. Bechmann-Pasquinucci and W. Tittel, Phys. Rev. A 61, 062 308 (2000). 3. H. Bechmann-Pasquinucci and A. Peres, Phys. Rev. Lett. 85, 3313 (2000). 4. M. V. Chekhova, S. P. Kulik, G. A. Maslennikov, and A. A. Zhukov, quant-ph/0305115. 5. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980), p. 256. 6. A. V. Belinsky and D. N. Klyshko, Laser Phys. 4, 663 (1994).

7. A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, et al., Phys. Rev. A 60, 4209 (1999). 8. A. V. Burlakov and M. V. Chekhova, Pis'ma Zh. èksp. Teor. Fiz. 75, 505 (2002) [JETP Lett. 75, 432 (2002)]. 9. A. V. Burlakov, L. A. Krivitskioe, S. P. Kulik, et al., Opt. Spektrosk. 94, 744 (2003) [Opt. Spectrosc. 94, 684 (2003)]. 10. L. A. Krivitskioe, S. P. Kulik, A. N. Penin, and M. V. Chekhova, Zh. èksp. Teor. Fiz. 124 (4), 943 (2003) [JETP 97, 846 (2003)]. 11. D. N. Klyshko, Zh. èksp. Teor. Fiz. 111, 1955 (1997) [JETP 84, 1065 (1997)]. 12. Yu. I. Bogdanov, Fundamental Problem of Statistical Data Analysis: Root Approach (Mosk. Inst. èlektron. Tekh., Moscow, 2002); physics/0211109 (2002). 13. Yu. I. Bogdanov, Quantum Mechanical View of Mathematical Statistics, in Progress in Quantum Physics Research (Nova Science, New York, 2003); quantph/0303013 (2003). 14. Yu. I. Bogdanov, Root Estimator of Quantum States, in Progress in Quantum Physics Research (Nova Science, New York, 2003); quant-ph/0303014 (2003). 15. H. Cramer, Mathematical Methods of Statistics (Princeton Univ. Press, Princeton, N.J., 1946; Mir, Moscow, 1975). 16. M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, 4th ed. (Griffin, London, 1977; Nauka, Moscow, 1973). 17. Optical Materials for Infrared Engineering (Mir, Moscow, 1965).

Translated by V. Sakun

JETP LETTERS

Vol. 78

No. 6

2003