Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1908.06793 · Lobachevskii Journal of Mathematics

On Definition of Quantum Tomography via the Sobolev Embedding Theorem

Grigori Gennadievich Amosov, Ya. A. Korennoy

Published Oct 1, 2019• 2 citations
arXivHTMLPDF

Abstract

We obtain sufficient conditions on kernels of quantum states under which Wigner functions, optical quantum tomograms and linking their formulas are correctly defined. Our approach is based upon the Sobolev Embedding theorem. The transition probability formula and the fractional Fourier transform are discussed in this framework.
DOI
10.1134/s1995080219100044

Related papers

Quantum-Tomography Method in Information Processing

Margarita A. Man’ko

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.

Kluwer Academic Publishers eBooks•Feb 24, 2006•5 citations

Wigner function and the probability representation of quantum states

Margarita A. Man’ko, Vladimir I. Man’ko

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.

arXiv:1403.3189•EPJ Web of Conferences•Jan 1, 2014•1 citation

Quantum Tomography Approach in Signal Analysis

Margarita A. Man’ko

Some properties of the fractional Fourier transform, which is used in information processing, are presented in connection with the tomography transform of optical signals. Relation of the Green function of the quantum harmonic oscillator to the fractional Fourier transform is elucidated.

arXiv:quant-ph/9906010•Jun 2, 1999•3 citations

Relation between optical Fresnel transformation and quantum tomography in two-mode entangled case

Hong-Yi Fan, Liyun Hu

Similar in spirit to the preceding work 9 Fan, H-Y and Hu, L-Y. 2009. Opt. Commun., 282: 3734–3736. [Crossref], [Web of Science ®] , [Google Scholar], where the relation between optical Fresnel transformation and quantum tomography is revealed, we explore this kind of relationship in a two-mode entangled case. We show that under the two-mode Fresnel transformation the entangled state density becomes density operator , which is just the Radon transform of the two-mode Wigner operator in entangled form, i.e. where F 2 is a two-mode Fresnel operator in quantum optics, and s , r are the complex-value expressions of (A,B,C,D). So the probability distribution is the tomography (Radon transform of the two-mode Wigner function), and gives a tomogram. Similarly, we find where is the conjugated state to .

arXiv:0911.1596•Journal of Modern Optics•Jan 10, 2011•1 citation

Wigner functions and spin tomograms for qubit states

Петер Адам, V. A. Andreev, Iulia Ghiu +3 more

We establish the relation of the spin tomogram to the Wigner function on a discrete phase space of qubits. We use the quantizers and dequantizers of the spin tomographic star-product scheme for qubits to derive the expression for the kernel connecting Wigner symbols on the discrete phase space with the tomographic symbols.

arXiv:1312.6848•Dec 24, 2013•0 citations

Quantum tomography

J. C. Garrison, R. Y. Chiao

This chapter begins with a review of classical tomography, which reconstructs hidden structures inside an object by successive transmissions of an X-ray beam at different angles and lateral displacements. By interpreting the Wigner distribution as an analogue of the density distribution in a physical object, the mathematical methods of classical tomography — the Radon transform, the Fourier slice theorem, and the inverse Radon transform — are adapted to perform a high resolution determination of a quantum state of light. In optical homodyne tomography, the classical transmission angle is replaced by the adjustable phase of a local oscillator, and the successive lateral displacements are replaced by a series of measurements of the quadrature operator defined by the local oscillator phase. Experimental data is presented showing various properties of the Wigner distribution for a coherent state.

Oxford University Press eBooks•Jun 5, 2008•0 citations
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.
Similar in spirit to the preceding work 9 Fan, H-Y and Hu, L-Y. 2009. Opt. Commun., 282: 3734–3736. [Crossref], [Web of Science ®] , [Google Scholar], where the relation between optical Fresnel transformation and quantum tomography is revealed, we explore this kind of relationship in a two-mode entangled case. We show that under the two-mode Fresnel transformation the entangled state density becomes density operator , which is just the Radon transform of the two-mode Wigner operator in entangled form, i.e. where F 2 is a two-mode Fresnel operator in quantum optics, and s , r are the complex-value expressions of (A,B,C,D). So the probability distribution is the tomography (Radon transform of the two-mode Wigner function), and gives a tomogram. Similarly, we find where is the conjugated state to .
This chapter begins with a review of classical tomography, which reconstructs hidden structures inside an object by successive transmissions of an X-ray beam at different angles and lateral displacements. By interpreting the Wigner distribution as an analogue of the density distribution in a physical object, the mathematical methods of classical tomography — the Radon transform, the Fourier slice theorem, and the inverse Radon transform — are adapted to perform a high resolution determination of a quantum state of light. In optical homodyne tomography, the classical transmission angle is replaced by the adjustable phase of a local oscillator, and the successive lateral displacements are replaced by a series of measurements of the quadrature operator defined by the local oscillator phase. Experimental data is presented showing various properties of the Wigner distribution for a coherent state.