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Kluwer Academic Publishers eBooks

Quantum-Tomography Method in Information Processing

Margarita A. Man’ko

Published Feb 24, 2006• 5 citations
Publisher

Abstract

Analogy of time-dependent analytic signal to the wave function is used to apply in signal analysis the new tomographic approach developed recently in quantum mechanics and quantum optics. The tomographic probability and analytic signal are shown to be connected by the integral transform with a kernel related to the fractional Fourier transform.

DOI
10.1007/0-306-47114-0_23

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Analogy of time-dependent analytic signal to the wave function is used to apply in signal analysis the new tomographic approach developed recently in quantum mechanics and quantum optics. The tomographic probability and analytic signal are shown to be connected by the integral transform with a kernel related to the fractional Fourier transform.
The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.
Abstract We study an application of the quantum tomography framework for the time-frequency analysis of modulated signals. In particular, we calculate optical tomographic representations and Wigner–Ville distributions for signals with amplitude and frequency modulations. We also consider time-frequency entropic relations for modulated signals, which are naturally associated with the Fourier analysis. A numerical toolbox for calculating optical time-frequency tomograms based on pseudo Wigner–Ville distributions for modulated signals is provided.
A method of quantum-state tomography for quantum-information processing is described. The method is based on the use of the Fourier-transform technique and involves detection of all the diagonal elements of the density matrix in a one-dimensional experiment and all the off-diagonal elements by a two-dimensional experiment. The method is efficient for a large number of qubits (> 5).(>~5).(> 5). The proposed method is outlined using a two-qubit system and demonstrated using simulations by tomographing arbitrary complex density matrices of two- and four-qubit systems.