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ISSN 0030-400X, Optics and Spectroscopy, 2007, Vol. 103, No. 1, pp. 107­115. © Pleiades Publishing, Ltd., 2007. Original Russian Text © Yu.I. Bogdanov, R.F. Galeev, S.P. Kulik, E.V. Moreva, 2007, published in Optika i Spektroskopiya, 2007, Vol. 103, No. 1, pp. 112­120.

SINGLE-PHOTON DETECTION AND RECONSTRUCTION (TOMOGRAPHY) OF OPTICAL-FIELD STATES

Mathematical Modeling of the Accuracy Characteristics in Problems of Precision Quantum Tomography of Biphoton States
Yu. I. Bogdanova, R. F. Galeevb, S. P. Kulikb, and E. V. Morevac
a

Institute of Physics and Technology, Russian Academy of Sciences, Moscow, 117218 Russia b Faculty of Physics, Moscow State University, Moscow, 119992 Russia c Moscow Engineering Physics Institute (State University), Moscow, 115409 Russia
Received October 12, 2006

Abstract--A model that approximately takes into account instrumental errors in problems of precision reconstruction of quantum states is considered. The model is based on the notion of coherence volume, which characterizes the quality of the experimental and technological realization of the measurement protocol of a quantum state. Various sources of instrumental errors that affect the reconstruction accuracy of quantum states are mathematically modeled. It is shown that, for precision tomography of quantum states, one should take into account random coincidences that occur because the operating time of the photon detection system is finite. Among other possible sources of instrumental errors of the measurement protocol, one should take into account errors of the rotation angles and optical thicknesses of polarization plates, a possible drift in the parameters of the pump laser, and so on. PACS numbers: 42.50.-p DOI: 10.1134/S0030400X0707017X

1. INTRODUCTION The development of technology of such multidimensional quantum systems as qudits (quantum dits), which are an alternative to more traditional two-level systems, qubits, promise significant advantages in problems of quantum informatics and cryptography. In particular, the use of qudits allows one to demonstrate a stronger violation of the Bell inequalities compared to systems based on qubits, which is of fundamental importance for quantum physics and quantum theory of information [1]. Much applied interest in qudit systems also comes from their possible use in quantum cryptography in key distribution problems [2]. Multilevel quantum systems prove to be more robust against unavoidable noise in communication channels. At the same time, the technology of preparation and measurement of qudits seems to be much more technically complicated than in the case of qubits. Efficient application of multilevel quantum systems in quantum cryptographic protocols requires secure complete high-precision control over quantum states. The possibility of such a high-precision control over three- and four-level single-mode polarized quantum states of biphotons was experimentally demonstrated for the first time in [3­7]. Quantum communication systems should ensure the technology of preparation, transformation, and measurement of signals of a specific quantum-mechanical nature. The quantum signal determined by the vector of state in an abstract Hilbert space differs in principle

from signals of the classical nature. An important distinctive feature of quantum signals compared to classical ones is the fundamental necessity of statistical description of their behavior. Technologically, the quantum state is determined by the procedure of its preparation. Such a procedure is termed the protocol. The protocol for the preparation of the quantum state specifies the quantum statistical ensemble that corresponds to it. This ensemble determines the potential possibility of generating an arbitrarily large number of representatives that are close to each other (ideally, identical). The measurement of an individual quantum object leads to a change in its quantum state (the wave function reduction); however, the experimenter deals not with a single object but with a large set of representatives of the quantum statistical ensemble. This particular feature leads to the situation that the traditional measurement procedures are necessarily replaced by measuring­computational algorithms, with the help of which experimentally obtained statistical data are subjected to a special mathematical processing aiming at reconstructing the parameters of the quantum state under study [8]. The complete high-precision control over a quantum state requires taking into account statistical fluctuations and instrumental errors that arise in the course of realization of measuring­computational algorithms. The statistical fluctuations are connected with the fundamental quantum nature of the states under investiga-

107


108 NPBS IF 1 V

BOGDANOV et al.

The vector of the measured four-level quantum state, ququart, is | = c 1 |2 H, 0 V + c 2 |1 H, 1 V + c 3 |1 V , 1 H + c 4 |0 H, 2 V . (1)

1, 2

PP1 D1 QWP1 HWP1 QWP2 HWP2

V

PP2 D2 CC

Fig. 1. Modeled scheme for the measurement of biphoton states. Frequency modes 1 and 2 are separated with a nonpolarizing beam splitter NPBS, pass through quarter-wave plates QWP1 and QWP2, half-wave plates HWP1 and HWP2, and polarization filters PP1 and PP2. An interference filter IF in the upper arm of the scheme transmits only photons of mode 1. The counts are registered with detectors D1 and D2 and are fed to a coincidence circuit (CC).

tion and arise because the number of representatives of the quantum ensemble subjected to measurement is finite. As the observation time increases, the measurement process destroys the quantum states of an increasingly greater number of the ensemble representatives. Correspondingly, more and more exact information on the quantum state is extracted (the level of statistical fluctuations in the estimation of the vector of the quantum state becomes increasingly smaller). However, the accuracy level of the control over a quantum state cannot be arbitrarily high because of the occurrence of unavoidable technological restrictions and instrumental errors related to them (such as errors of the angle setting and errors of the parameters of polarization instruments, noise in the photon detection system, unstable operation of the pump laser, etc.). The objective of this paper is to systematically study the effect of various instrumental errors on the quantum state reconstruction accuracy. The study is performed by using as an example the protocol for the measurement of four-level polarization quantum states of biphotons that we experimentally realized in [7]. The theoretical consideration performed and the results of the mathematical modeling make it possible to determine technological requirements to the parameters of an experimental setup that are necessary for the secure high-precision control over the quantum state. The results of this study are important for the efficient practical realization of the elements and protocols of qudit-based quantum informatics. 2. MEASUREMENT PROTOCOL Pure polarization states arising due to spontaneous parametric down-conversion of light [9] are considered.

It is assumed the biphoton, ququart, is formed by two frequency-distinguished photons, which (conventionally) correspond to signal and idler modes. Under steady-state conditions, the energy of each pumping photon ( p) equals the sum of the energies of the "signal" ( s) and "idler" ( i) photons; therefore, p = s + i. The notation |2H, 0V refers to the state in which both photons are horizontally polarized; the component |1H, 1V describes the state where the signal photon is horizontally polarized, while the idler photon is vertically polarized; conversely, the designation stands for the case where the signal photon is vertically polarized, while the idler photon is horizontally polarized; and, finally, |1V, 1H is the state in which both photons are vertically polarized. The procedure of preparation of these photons was described in detail in our recent paper [7]. There are two protocols for the measurement of the state of ququarts, biphotons. According to the first protocol, a ququart is subjected to specified polarization transformations as a whole object. The transformations are performed with the help of phase plates that independently act on the polarization state of each of the photons from the pair. Then, the biphoton is split into two photons with the help of a beam splitter. The polarization states of each of the photons are projected onto fixed states and are registered with two detectors whose outputs are connected to a coincidence circuit. According to the second protocol, the initial state of a fourlevel system, ququart, is split into two independent modes, and each of the photons forming this biphoton is separately subjected to polarization transformations. In both protocols, the registered event is the coincidence of counts of the two photodetectors. Figure 1 shows the schematic of the corresponding experimental setup that was used in [7] and whose functioning is numerically modeled in this study. For this purpose, the accuracy of the method for reconstruction of quantum states was successively modeled as a function of the parameters of the protocol: the orientation angles and thicknesses of quarter-wave ( /4 ) ( /4 ) ( /2 ) (1, 2, h 1 , h 2 ) and half-wave (1, 2, h 1 , h 2 ) plates, respectively; the effect of the instability (drift) of the pumping laser radiation power on the accuracy of quantum tomography results was considered; and the algorithm of statistical quantum state reconstruction was refined taking into account accidental coincidences (see Section 3). The quantum tomography protocol considered was reduced to the analysis of statistical data obtained upon measurement of 16 quantum processes. The amplitude
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( /2 )


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Setting parameters of the angles and of the quarter-wave and half-wave plates, respectively in signal (s) and idler (i) channels; M ( = 1, 2, ..., 16) is the amplitude of the corresponding measured process 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Setting parameters s 0 0 0 0 0 0 45° 45° 45° 45° 0 0 0 0 0 0
s

Amplitude of the process
i

i 0 0 0 0 0 0 0 0 0 ­45° ­45° ­45° ­ 45° ­90° ­90° ­90°

M c1/2 c2/2 c4/2 c3/2 1/2 2 (c1 ­ ic3) 1/2 2 (c2 ­ ic4) 1/2 2 (c2 ­ c4) 1/2 2 (c1 ­ c3) 1/4[(c1 ­ c3) + i(c2 ­ ic4)] 1/4[c1 + c2 ­ c3 ­ c4] 1/4[(c1 + c2) ­ i(c3 + c4)] 1/2 2 (c1 + c2) 1/2 2 (c3 + c4) 1/2 2 (c3 ­ ic4) 1/2 2 (c1 ­ ic2) 1/4[(c1 ­ c4) ­ i(c2 + c3)]

45° 45° 0 0 22.5° 22.5° 22.5° 22.5° 22.5° 22.5° 22.5° 45° 0 0 45° 22.5°

­ 45° 0 0 ­ 45° ­ 45° 0 0 ­ 45° ­22.5° ­22.5° ­22.5° ­22.5° ­22.5° ­22.5° ­22.5° ­22.5°

of each of such quantum processes can be defined by the formula M j = 1/2 ( a 1 a 2 c 1 + a 1 b 2 c 2 + b 1 a 2 c 3 + b 1 b 2 c 4 ) , j = 1, 2, ..., 16 , where a 1 = ­ r /2 ( 1 ) t */4 ( 1 ) ­ t /2 ( 1 ) r /4 ( 1 ) , a 2 = ­ r /2 ( 2 ) t */4 ( 2 ) ­ t /2 ( 2 ) r /4 ( 2 ) , b 1 = ­ r /2 ( 1 ) r */4 ( 1 ) + t /2 ( 1 ) t /4 ( 1 ) , b 2 = ­ r /2 ( 2 ) r */4 ( 2 ) + t /2 ( 2 ) t /4 ( 2 ) . Here, the amplitude transmission tj and reflection rj coefficients are used, which are defined as functions of the orientation angles j by the following formulas [7]: t j = cos j + i sin j cos 2 j , r j = i sin j sin 2 j , j = ( no j ­ ne j ) h / . (3) (2)

Relation (2) determines the linear dependence between the components of the initial vector of state and the amplitude of the measured quantum process. Using the instrumental matrix X, the relation between the amplitudes of the processes Mj ( j = 1, 2, ..., 16) and the components of the state vector cr (r = 1, 2, 3, 4) can be written in the form of the following matrix equality: Mj =


r=1

4

X jr c r ,

j = 1, 2, ..., 16.

(4)

The row in the instrumental matrix X corresponding to the jth quantum process has the form X j = 1/2 ( a 1 a 2 a 1 b 2 b 1 a 2 b 1 b 2 ) , (5)

where the values of the complex parameters a1, a2, b1, and b2 are different for different rows of the measurement protocol. The setting parameters of the protocol under consideration are summarized in the table. The squared modulus of the amplitude of a quantum process determines the generation intensity of corresponding events, j = M* M j. j (6)

Here, j is the optical thickness of the plate; h its geometrical thickness; and (noj ­ nej) is the difference between the refractive indices of the ordinary and extraordinary rays in the crystal at the wavelength . For an ideal half-wave plate, it is necessary to set j = /2, whereas, for an ideal quarter-wave plate, j = /4.
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The event generation intensity j is the main quantity that can be measured (j is measured in units of frequency (Hz)). The number of events that occur within any given time interval satisfies the Poisson distribution with the average value j tj, where tj is the exposure time for the jth process. In accordance with the statistical interpretation of Bohr's principle of complementarity, the quantities j determine the intensities of corresponding complementary Poisson processes [8]. The state vector in relation (4) is not assumed to be a priori normalized to some quantity (for example, to unity). The norm of the vector c is obtained as a result of the reconstruction of the quantum state carriers important information on the total intensity of all processes considered in the experiment (the corresponding normalization condition is presented in the next section). 3. THE QUANTUM STATE RECONSTRUCTION ALGORITHM TAKING INTO ACCOUNT ACCIDENTAL COINCIDENCES A significant source of instrumental errors is connected with the finite efficiency and finite operating time of detectors. The efficiency of photon detectors is finite and, as a rule, small compared to unity; therefore, only a part of the generated biphotons leads to the appearance of events in the coincidence circuit. The coincidence is detected only if different photons forming a biphoton are incident into different arms of the setup and are simultaneously registered by the corresponding detectors. In other cases where either both photons are incident into the same arm or only one of the detectors operates (because of the finite quantum efficiency) coincidences do not occur. In real experiments, the detection efficiency of single photons exceeds by many times the coincidence intensity. Taking into account the finite operating time of the photon detection system, this circumstance leads to the appearance of accidental coincidences. Indeed, if two events that are registered by the two detectors are separated by a time interval shorter than the characteristic system operating time , the registration of these events is fixed as a coincidence. Let (1) and (2) be the detection intensities of random events (the number of events per unit time) for the first and second detectors, respectively. Then, the intensity of accidental coincidences can be calculated by the formula
(0)

^ (1) ^ (2) cidences (0) can approximately be considered to be set a priori (up to possible variations in the parameter ). During the exposure time tj in the jth quantum process, the number of accidental coincidences can be esti(0) (0) ^ (1) ^ (2) mated as k j t j j t j j j . The number of true coincidences, determined by correlations between photons, can be estimated as the difference between the total number of coincidences kj detected during the fixed exposure time and the number of accidental coin(0) cidences k j . However, this estimation is very rough. In particular, under such an approach, physically meaningless negative values of the number of true coincidences can arise because of statistical fluctuations of the number of events. A more exact approach that is free of inherent contradictions is based on the maximum likelihood method. The algorithm considered is a direct generalization of the algorithm described in [3, 5] to the case of the occurrence of accidental coincidences. The coincidence intensity j in the jth quantum process is the sum of true coincidences j, caused by biphotons (the useful signal), and of accidental coinci(0) dences j , j = j + j .
(0)

(8)

The intensity of true coincidences j is determined by the amplitude of the quantum process Mj, j = M j M*, j where Mj = (9)


s

X js c s .

According to (7), the intensity of accidental coinci(0) dences j is Therefore, j =
(0) j

= j j .

(1)

(2)

(10)

= .

(1)

(2)

(7)


r, s

(0) X * X js c * c s + j . jr r

(11)

We will estimate the intensities (1) and (2) via the ^ (1) ^ (2) corresponding operating frequencies and of the detectors. The operating time is estimated experimentally assuming that (1) and (2) are independent from each other. Then the intensity of accidental coin-

As estimated parameters, we will consider the components of the state vector c. Within the framework of the maximum likelihood method, the estimates of parameters are obtained from the extremum condition for the so-called likelihood function [3, 5].
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MATHEMATICAL MODELING OF THE ACCURACY CHARACTERISTICS Number of observations 160 120 80 40 0 0.2 40 0 1.0 0.9994 Fh () 80 Number of observations 120 (b)

111

0.4

0.6

0.8

0.9996

0.9998

Logarithmic likelihood ­88 ­92 ­96 ­ 100 ­ 104 0.96 0.98 1.00 1.02 1.04 /0 (c)

1.0000 Fh

Fig. 2. Comparison of the results of statistical reconstruction of quantum states (a) without and (b) with taking into account accidental coincidences. In the first case, the average information fidelity F h = 0.761 and in the second case, F h = 0.99989. Each histogram represents the result of 500 numerical experiments with the sample size of about 80 000 events each. The number of accidental coincidences was approximately equal to 50% of the total number of events. Panel (c) shows the estimation of the operating time of the detection system.

In our case, the likelihood function is defined by the product of Poisson probabilities, L=

The likelihood equation automatically yields the following normalization condition:


j

( jt j) ---------------- exp ( ­ j t j ) , k j!
j

k

(12)


j

j k j -------------------) = (0 j + j


j

( jt j ).

(16)

where kj is the number of coincidences observed in the jth quantum process within the exposure time tj. Completely similar to [3, 5], we will obtain the following likelihood equation for the estimation of the state vector: Ic = Jc , (13) where I and J are the so-called Hermitian and empirical Fisher information matrices, respectively. In our case, the elements of these matrices are determined by the following formulas: I
rs

=


j

t j X * X js , jr

(14)

J

rs

=


j

kj ----- X * X js . j jr
Vol. 103 No. 1

(15)

j Here, -------------------) is the estimate for the fraction of non(0 j + j accidental coincidences in the jth quantum process. The normalization condition obtained means that the total number of nonaccidental events detected in all processes (the left-hand side) is equal to the sum of products of the generation intensities (registration) of nonaccidental events and the exposure time. The algorithm described is well applicable even in the case where the intensity of accidental coincidences is comparable or exceeds the intensity of the useful signal. This is illustrated in Figs. 2a and 2b for the case in which the number of accidental extraneous coincidences is approximately equal to the number of useful events. Upon reconstruction of an arbitrarily chosen (random) state vector, the information fidelity level (see Section 4) in the series of 500 numerical experiments was, on the average, close to unity ( F h = 0.99989) if

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accidental coincidences were taken into account, and the information fidelity was very poor (on the average, F h = 0.761) if such coincidences were not considered. Note that, if the operating time of a system is a priori unknown, it can be coincided as an adjustable parameter in the maximal likelihood method. Figure 2c shows that, in the numerical calculations, the maximally likely parameter proves to be very close to the unknown true parameter 0 (the ratio /0 proves to be close to unity). 4. THE THEORY OF STATISTICAL FLUCTUATIONS The estimate of the quantum state of a normally operating quantum information system should fluctuate within the limits predicted by the statistical theory. Comparison of the results of reconstruction of quantum states with a fundamental statistical accuracy level can form the basis for the solution of such problems as setup adjustment, system stability operation control, detection of external intervention into the operation of the system, and so on. The model described below specifies that ideal accuracy level of quantum state reconstruction that can be activated if instrumental errors of the protocol will be negligibly small. The theory of statistical fluctuations was described in more detail in [4]. The unnormalized state vector c estimated based on the maximal likelihood method differs from the exact state vector c(0) by a small random quantity c = c(0) ­ c. It is convenient to transform the complex fluctuation vector c into a real vector of the double length. We will explicitly express the real and imaginary parts of the (1) (2) fluctuation vector cj = c j ­ i c j and pass from the complex vector c to the real vector x: x = c c c c
(1) 1

where the matrix H is the so-called complete information matrix. It has the following block structure: H = Re ( I + K ) ­ Im ( I + K ) Im ( I ­ K ) Re ( I ­ K ) . (19)

Here, along with the Hermitian Fisher information matrix I (14), the symmetric Fisher information matrix K is introduced, whose elements are defined by the following expression: K
sj

=




t --------- X s X j . 2 M

(20)

c 1 c2 c = : cs

:
(1) s (2) 1

:
(2) s

.

The matrix H is real and symmetric. Its dimension is two times greater than the dimension of the matrices I and K. For ququarts, I and K are 4 â 4 matrices; therefore, the dimension of H is 8 â 8. We will formulate the characteristic condition of the completeness of measurements. It is reduced to the requirement that one and only one eigenvalue of the complete information matrix H should be zero, whereas the remaining eigenvalues must be strictly positive. The eigenvector whose eigenvalue is zero defines the direction of gauge fluctuations (such fluctuations are physically meaningless). The eigenvectors corresponding to the remaining eigenvalues specify the directions of principal fluctuations of the state vector in the Hilbert space. The knowledge of quantitative laws in which statistical fluctuations are obeyed makes it possible to estimate the distributions of various physical characteristics. The most important information criterion, which determines the general level of all possible statistical fluctuations in a quantum information system, is the -squared criterion. Analogously to (17), we will consider the transformation of the complex state vector into a real vector of the double length, x= : (1) cs . (2) c1 : (2) cs c
(1) 1

(17)

c=

c1 c2 : cs

(21)

In particular, for ququarts (s = 4), this transformation makes it possible to pass from four-component complex vectors to eight-component real vectors. In the new representation, the fluctuation of the logarithmic likelihood function can be represented in the form ln L = ­ H sj s j = ­ | H | , (18)

It can be shown that the information contained in the state vector is equal to the doubled total number of observations in all quantum processes, | H | = 2 n , where n = (22)





k .
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The -squared criterion can be written in the form invariant with respect to the scale of the state vector (recall that we consider the unnormalized state vector), | H | ( 2 s ­ 1 ) ----------------------- ------------------------ . 4n | H |
2

Coherence volume 106

(23)

10

5

This expression describes the distribution of relative information fluctuations. It shows that, as the number of observations increases, the relative information uncertainty of the quantum state decreases as 1/n. According to (23), the average value of relative information fluctuations is given by | H | 2 s ­ 1 ----------------------- ------------- . | H | 4n (24)

3 2 10
4

1

10

3

0

1

2

3 4 5 Instrumental error level

As a measure of correspondence between the theoretical state vector and its estimate, we will introduce the characteristic that is termed the information fidelity: | H | F H = 1 ­ ----------------------- . | H | (25)

Fig. 3. Dependence of the coherence volume on the level of various instrumental errors: (1) the inaccuracies (in degrees) in setting of the angles of the protocol, (2) the imperfections (in degrees) of half- and quarter-wave plates, and (3) the drift (in percent) in the pump laser radiation power.

Correspondingly, the quantity 1 ­ FH can be termed the information loss. Recall that the conventional definition of the fidelity F between the theoretical c0 and reconstructed c state vectors is given by the formula c c0 F = --------------------------- . + + ( c c ) ( c0 c0 )
+ 2

(26)

The convenience of the information fidelity FH consists in its simpler statistical properties compared to the conventional fidelity F. For systems where statistical fluctuations dominate, the information fidelity FH is the random quantity, based on the square distribution, F
H

problems of reconstruction of quantum states. The model proposed is based on the notion of the coherence volume, which characterizes the quality of the experimental realization of the protocol for the measurement of a quantum state. Numerical experiments show that the effect of instrumental errors can approximately be taken into account if some smaller number neff (the effective number of observations) is taken in the statistical model instead of the real number of physical observations n. Numerical calculations show that the effective number of observations can be estimated by the formula 1/ n
ef

(2s ­ 1) = 1 ­ ------------------------ , 4n

2

= 1/ n + 1/ n 0 .

(28)

(27)

where 2(2s ­ 1) is the random quantity having the square distribution with the 2s ­ 1 degrees of freedom. In an ideal quantum information system, as the number of observations increases, the information fidelity asymptotically tends to unity, and, correspondingly, the information loss tends to zero. The noise caused by instrumental errors, which is complementary to statistical fluctuations, leads to a decrease in the informational fidelity level compared to the theoretical level given by (27). This phenomenon is numerically studied in the next section. 5. EFFECTIVE SAMPLE SIZE AND COHERENCE VOLUME In this section, we will describe the model that can approximately take into account instrumental errors in
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We will term the parameter n0 in this formula the coherence volume. If n n0 (small sample size), then neff n. In this case, the influence of instrumental errors is negligibly small compared to the effect of statistical fluctuations. In the opposite case, n n0 (large sample size), and neff n0. In this case, instrumental errors play a decisive role. The level of instrumental errors determines the value of the coherence volume n0. The lower the level of instrumental errors, the greater the achievable coherence volume n0. Regardless of however large the number of observations can be, the reconstruction accuracy will not exceed the level determined by the coherence volume. Figure 3 shows how rapidly the coherence volume decreases with increasing level of various instrumental errors. The inaccuracies in setting of the angles of the protocol and the imperfections of half- and quarterwave plates are given in the figure in degrees, while the


114 Observation number 100 80 60 40 20 0 0.9996 0.9998 ()

BOGDANOV et al. Observation number 200 160 120 80 40 0 1.0000 0.9994 0.9996 0.9998 1.0000 Fh (b)

Fig. 4. Comparison of the results of 500 numerical experiments with 20000 events in each (histograms) with the predictions of the ideal theory (curves): (a) instrumental errors are insignificant (the inaccuracies in setting of the angles of the protocol and the imperfections of the half- and quarter-wave plates are about 0.1°, the pump laser power drift is of the order of 0.1%, and the level of accidental coincidences is 1%) and (b) instrumental errors are significant (the inaccuracies in setting of the angles of the protocol and the imperfections of the half- and quarter-wave plates are about 1°, the pump laser power drift is of the order of 1%, and the level of accidental coincidences is 10%); neff 8800 ± 200, n0 15 700 ± 700.

drift in the pump laser radiation power is shown in percent. The inaccuracies in setting of the angles of the protocol and the imperfections of the half- and quarterwave plates were modeled by means of a uniform distribution. The entire range (in degrees) of the uniformly distributed error was specified (a one-degree error was modeled with the help of a random quantity uniformly distributed over the interval [­0.5; 0.5]). The drift, i.e., the change in the radiation power of the pump laser linear in time, was specified in percent with respect to the initial value. The drift was a determinate (nonrandom) but unknown quantity. In other words, the drift affected the data but the reconstruction algorithm did not take into account this influence. By a one-percent drift was meant a one-percent change in the pump laser power during the entire experiment (within the total time interval necessary for the measurement of all of the 16 processes). The most interesting practical case is where all the types of instrumental errors considered above act simultaneously. Figure 4 compares the results of the numerical experiments with the ideal theory and gives the idea of the errors that can be considered to be small or large. It is seen that the instrumental errors were insignificant and the calculation results were close to the predictions of the ideal theory if the inaccuracies in setting of the angles of the protocol and the imperfections of the half- and quarter-wave plates were about 0.1°, the pump laser power drift was of the order of 0.1%, and the level of accidental coincidences was 1%. As these errors were increased by an order of magnitude, they became significant, and the calculation results appreciably changed compared to the ideal theory. There were 500 experiments, each consisting of 20 thousands of observations. Note that, if the sample size in each experiment was increased by an order of magnitude (from 20 thousand to 200 thousand), then

the previously insignificant instrumental errors (presented above) would become significant. CONCLUSIONS The main results of this study are formulated as follows. (i) A method of mathematical modeling that approximately takes into account instrumental errors in problems of precision reconstruction of quantum states was developed. (ii) The ideal statistical model of the reconstruction accuracy of quantum states was generalized to the case where the instrumental errors of the measurement protocol should be taken into account. It was shown that the parameters introduced in the study--the effective number of observations and the coherence volume-- determine the level of the quantum state reconstruction accuracy from the experimental data. (iii) The method for taking into account accidental coincidences that should be considered in real experiments on precision tomography of quantum states was developed. The effects of errors in setting of the rotation angles and optical thicknesses of polarization plates were numerically studied, and the possible drift of the pump laser power was taken into account. (iv) The mathematical model developed and the obtained results of the numerical modeling make it possible to specify the technological requirements to the parameters of the experimental setup, the satisfaction of which ensures secure control of the quantum state with a given accuracy. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research (project no. 06-02-16769) and
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Translated by V. Rogovooe

OPTICS AND SPECTROSCOPY

Vol. 103

No. 1

2007