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ISSN 1054-660X, Laser Physics, 2006, Vol. 16, No. 8, pp. 1264 ­ 1267.
© MAIK "Nauka / Interperiodica" (Russia), 2006. Original Text © Astro, Ltd., 2006.

QUANTUM INFORMATION AND COMPUTATION

Absolute Robustness of the First Principal Fourier State Component and Noise Suppression in Hilbert Space
Yu. I. Bogdanova and S. P. Kulikb
a

Institute of Physics and Technology, Russian Academy of Sciences, Nakhimovskii pr. 34, Moscow, 117218 Russia b Faculty of Physics, Moscow State University, Vorob'evy gory 1, str. 2, Moscow, 119992 Russia
e-mail: skulik@qopt.phys.msu.su Received March 28, 2006

Abstract--The effect of the absolute robustness of the first principal Fourier-state component is developed upon rounding off the density matrix elements. A protocol for extracting the pure state using the Fourier transform and auxiliary qubits is proposed. PACS numbers: 03.65.Wj, 03.65.Yz, 03.67.­a DOI: 10.1134/S1054660X06080159

1. ABSOLUTE ROBUSTNESS OF THE FIRST PRINCIPAL COMPONENT OF THE FOURIER STATES The effect of uncontrolled errors and noise on the states of the system is inherent in data transfer via quantum communication channels and quantum calculations [1]. In connection with this, a search for robust algorithms for data storage, processing, and transfer is a topical problem of quantum information theory. It is demonstrated in the processing of the experimental data and the numerical mathematical simulation in the problems of the quantum tomography of optical states that the first principal component of an extremely rough and inaccurate experimental density matrix may serve as a relatively good approximation of the theoretically expected pure state vector [2, 3]. In this work, we consider the results of the analytical and numerical analysis of the most interesting scenario, in which the initial (noiseless) states represent arbitrary pure Fourier harmonics. The case when noise is added to the initial state is also investigated. The roundoff procedure given below is a particular case of such noise adding. In this case, the first principal components appear absolutely robust upon an arbitrary rounding off of the density matrix elements (i.e., rounding off to any nonzero number of digits in the binary, decimal, or any other representation). For such a roundoff, the density matrix exhibits nonzero and even negative eigenvalues and the principal eigenvalue is varied. However, the corresponding eigenfunction (the first principal component) remains unchanged and exactly equals the initial pure Fourier harmonics. The Fourier state that corresponds to the k th harmonic in s-dimensional space is described with the

state vector whose components are represented as 2 c j = exp i ----- kj , s j = 1, 2, ..., s . (1)

The total number of Fourier states in the s-dimensional space given by expression (1) with k = 1, ..., s is s. State vector (1) and its density matrix obey the following normalization condition: c c = Tr ( cc ) = s .
+ +

(2)

Note that, in this expression, the trace of the density matrix is normalized to the dimension of space (s) rather than unity. This deviation from the conventional approach makes it possible to avoid the degeneration of the density matrix to the zero matrix upon rounding off in spaces with a relatively high dimension s. Below, we illustrate the absolute robustness of the Fourier harmonic using a specific example corresponding to k = 3 and s = 5. In this case, state vector (1) represents the following column: ~ n= ­0.8090 ­ 0.5878 i 0.3090 + 0.9511 i 0.3090 ­ 0.9511 i ­0.8090 + 0.5878 i 1.0000 ­ 0.0000 i .

The corresponding exact density matrix is written as (we present four digits in spite of the fact that 16 digits are taken into account)

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ABSOLUTE ROBUSTNESS OF THE FIRST PRINCIPAL FOURIER STATE COMPONENT

1265

=

1 ­0.8090 ­ 0.5878 i 0.3090 + 0.9511 i 0.3090 ­ 0.9511 i ­0.8090 + 0.5878 i

­0.8090 + 0.5878 i 1 ­0.8090 ­ 0.5878 i 0.3090 + 0.9511 i 0.3090 ­ 0.9511 i

0.3090 ­ 0.9511 i ­0.8090 + 0.5878 i 1 ­0.8090 ­ 0.5878 i 0.3090 + 0.9511 i

0.3090 + 0.9511 i 0.3090 ­ 0.9511 i ­0.8090 + 0.5878 i 1 ­0.8090 ­ 0.5878 i

­0.8090 ­ 0.5878 i 0.3090 + 0.9511 i 0.3090 ­ 0.9511 i ­0.8090 + 0.5878 i 1

.

A unique nonzero eigenvalue of this density matrix is s = 5. The corresponding eigenvector represents the vector of the state under consideration. The rounding off accurate to the l th binary fraction digit is represented as round ( 2 ) ~ = -------------------------- . l 2
l

(3)

Here, the function round denotes rounding off to the nearest integer. Note that this function independently acts upon the real and imaginary parts of a complex number. In the case of rounding off, for example, in the decimal system, we should substitute 10 for 2 in formula (3). Consider the maximum rough rounding off of the density matrix under consideration. If we use zero binary digits (l = 0), 0 or ±1 should be substituted for the real and imaginary parts of each element of the density matrix. Thus, the rounding off yields ~= 1 ­1 ­ i 0+i 0­i ­1 + i ­1 + i 1 ­1 ­ i 0+i 0­i 0­i ­1 + i 1 ­1 ­ i 0+i 0+i 0­i ­1 + i 1 ­1 ­ i ­1 ­ i 0+i 0­i ­1 + i 1 .

which cannot be adequately rounded off to either zero or unity. In this case, the rounding off is related to a spontaneous symmetry violation: the algorithm choosing between zero and unity resembles the situation of Buridan's ass. Eventually, the absolute robustness is preserved in this case regardless of the choice between zero and unity provided that the rounding off of different elements of the density matrix is correlated. The general features of the rounded-off density matrix are as follows: (i) the sum of eigenvalues equals s and (ii) the maximum eigenvalue is no less than s at any roundoff. Thus, negative eigenvalues emerge. In general, the reason for the absolute robustness of the Fourier states is as follows. In accordance with definition (1), a shift of the number of the component leads to multiplying by a constant phase factor written as c
j+1

= c j exp ( i ) ,

(4)

2 where = ----- k is a constant shift angle or parameter. s Hereafter, the summation of subscripts is performed with respect to the modulus of s, so that the first subscript follows the s th subscript. The following expression for the elements of the density matrix is derived from formula (4):
k + 1, j + 1

= k, j .

(5)

The eigenvalues of this matrix are ­1.0000; ­0.4596; ­0.3446; 1.1085; 5.6957. Note that the first principal component, which corresponds to the maximum eigenvalue (5.6957), determines the eigenvector, which is exactly equal to the initial state vector. This is interpreted as the absolute robustness of the first principal component of the Fourier state density matrix. The absolute robustness of the first principal component of the rounded-off density matrix illustrated with the above simple example is a general feature. It is valid for all of the Fourier states in the Hilbert space with an arbitrary dimension s upon the rounding off to any number of digits in any system. A single exception corresponds to the rounding off accurate to zero digits (l = 0) in spaces whose dimension s is a multiple of three (s = 3, 6, 9, ...). In this case, we arrive at such values as cos(/3) = sin(/6) = 0.5,
LASER PHYSICS Vol. 16 No. 8 2006

This equality is satisfied for any (arbitrarily rough) rounding off of the density matrix. Thus, the condition for the absolute robustness lies in the fact that equality (5) is satisfied upon the rounding off. Indeed, assume that the product of a rough density ~ matrix and Fourier state vector c represents the vector a: ~ (6) kj c j = a k . Repeated subscripts in formula (6) and the formulas below imply summation. With allowance for expressions (4) and (5), we have ~ ~ a k + 1 = k + 1, j c j = k, j ­ 1 c j ­ 1 exp ( i ) (7) = a k exp ( i ) . Expression (7) shows that vector a satisfies expression (4) as well as vector c does. This means that the vectors a and c coincide accurate to a constant factor. In other words, the Fourier states remain the eigenvectors of the rounded-off density matrix.


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BOGDANOV, KULIK

Apparently, in a real problem, it does not suffice to reconstruct arbitrary Fourier states: one should also reconstruct an arbitrary linear combination of these states. In this case, a quantum state can be reproduced at an arbitrarily high accuracy in the presence of any noise due to the application of additional auxiliary (control) qubits. The corresponding protocol is presented in the next section. 2. EXTRACTION OF A PURE STATE USING THE FOURIER TRANSFORM AND AUXILIARY QUBITS First, we briefly characterize the mathematical model of noise. We assume Gaussian errors. In the Hilbert space, the action of noise leads to the fact that an arbitrary (k th) element of the statistical ensemble is described with the random state vector ~ n
(k)

the remaining s ­ 1 eigenvectors. All of these eigenvectors correspond to a single eigenvalue = ------------- . s + s (12)

In the analysis of the experimental data, it is expedient to estimate effective noise power in the Hilbert space with formula (11), which yields s ( 1 ­ max ) = ------------------------- . s max ­ 1 (13)

To estimate the weight wpure of the pure component in the mixture, we can use the following formula: w
pure

s max ­ 1 1 = ----------- = -------------------- . s­1 1+

(14)

1 = --------------- c 1+

(0)

+ ----------- 1+

(k)

(8)

rather than an ideal state vector c(0). Here, (k) is a random Gaussian complex unity vector in s-dimensional Hilbert space. Its mean value equals zero, and the covariance matrix is represented as 1 i * = -- ij , j s i, j = 1, ..., s . (9)

Apparently, the noise model under consideration is a phenomenological model. However, it can be realized in real physical systems. For example, a relationship between the noise power in the Hilbert space and temperature can be established for a quantum ensemble of two-level atoms in a thermostat. This model is not directly related to the problem under study and needs to be analyzed elsewhere. Note that the motivation for this work is related to the real experimental data regarding the quantum tomography of optical three-level systems. We assume that L1 information and L2 auxiliary (control) qubits correspond to one register. Let an arbitrary 2 -dimensional quantum state and zeros be recorded in the information and control qubits, respectively. We assume that the data under study are stored in the momentum representation.1 We also assume that all of the unitary transformations and calculations are performed in momentum space. Let noise with power be added to the signal under study in momentum space due to the aforementioned calculations. After the transformations and calculations in momentum space, we return to the original coordinate space with the inverse Fourier transform. The measurements are performed with all of the control qubits. With 1 the probability wpure = ----------- , all of them are found in 1+ the zero state. With the probability 1 ­ wpure = ----------- , 1+ some (at least one but most likely half) of the control qubits will contain errors. If the control measurements do not yield errors, with a relatively high probability, a state that is close to the desired ideal state is stored in
1
L
1

After averaging with respect to the representatives of the statistical ensemble, the density matrix of the quantum state under consideration is written as 1 ij = ----------- 1+
(0) ij

ij + ----------- ---- , 1+ s

i, j = 1, ..., s .

(10)

Density matrix (10) shows that the averaging of incoherent noise in the Hilbert space results in a mix1 ture of a pure state with the weight ----------- and white 1+ noise (completely incoherent state) with the weight ----------- . The density matrix of the latter is proportional to 1+ the unity matrix. We can easily find explicit eigenvalues and eigenvectors of density matrix (10). The first principal component is identical to the ideal state vector. The corresponding eigenvalue is given by
max

s+ = ------------- . s + s

(11)

An arbitrary set of basis vectors in the (s ­ 1)-dimensional Hilbert space that represents an orthogonal complement to the exact state vector may serve as a set of

The original coordinate representation is transformed into the momentum representation using the Fourier transform of the register containing L1 + L2 qubits. LASER PHYSICS Vol. 16 No. 8 2006


ABSOLUTE ROBUSTNESS OF THE FIRST PRINCIPAL FOURIER STATE COMPONENT

1267

L1 information qubits. The fidelity2 of this state to the original state can be estimated as F 1 ­ ------ . L2 2 (15)

This formula and formula (14) for the estimation of the weight are proven by the numerical calculations. A simple qualitative interpretation is possible for formula (15). Indeed, the total noise with power is uniformly distributed in 2 -dimensional Hilbert space, whereas the desired signal is concentrated in a space with lower dimension ( 2 ). Only a fraction of 1 2 the total noise represented as -------------- = ------ corresponds L1 + L2 L2 2 2 to the desired subspace. This corresponds to the above formula. A state with a relatively high purity can be reached due to a decrease in the coherence, whose probability is estimated as 1 ­ wpure = ----------- . This quantity corre1+ sponds to the fraction of the representatives that fail the
2
L
1

measurements of the control qubits. The numerical calculations show that such representatives do not contain valuable information. The advantage of the above approach lies in the fact that the correctness of the transformations and the closeness of the state contained therein to an exact ideal vector in Hilbert space that represents, in general, a complicated superposition of 2 basis states is ensured with a relatively high accuracy given by expression (15) in the absence of measurements involving the information qubits. ACKNOWLEDGMENTS This work was supported by the Federal Agency on Science and Innovations of the Russian Federation (contract no. 2006-RI-19.0/001/593) and the Russian Foundation for Basic Research (project no. 06-0216769). REFERENCES
1. M. A. Nelsen and I. L. Chuang, Quantum and Quantum Information (Cambridge Cambridge, 2000). 2. Yu. I. Bogdanov, M. V. Chekhova, S. P. Phys. Rev. Lett. 93, 230 503 (2004). 3. Yu. I. Bogdanov, M. V. Chekhova, S. P. quant-ph/0411192 (2004). Computation Univ. Press, Kulik, et al., Kulik, et al.,
L
1

L1 + L

2

L

1

Fidelity F is defined as the square of the modulus of the scalar product of the ideal (exact) and real states. The closeness of this parameter to unity characterizes the quality of the prepared states.

LASER PHYSICS

Vol. 16

No. 8

2006