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TMF, 2018, Volume 197, Number 2, Pages 328–342 (Mi tmf9562)  

Unnormalized tomograms and quasidistributions of quantum states

V. I. Man'koab, L. A. Markovichbcde

a Lebedev Physical Institute, RAS, Moscow, Russia
b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Oblast, Russia
c Kharkevich Institute for Information Transmission Problems, Moscow, Russia
d Trapeznikov Institute of Control Sciences, Moscow, Russia
e International Center for Quantum Optics and Quantum Technologies (the Russian Quantum Center), Moscow, Russia

Abstract: We consider tomograms and quasidistributions, such as the Wigner functions, the Glauber–Sudarshan $P$-functions, and the Husimi $Q$-functions, that violate the standard normalization condition for probability distribution fucntions. We introduce special conditions for the Wigner function to determine the tomogram with the Radon transform and study three different examples of states like the de Broglie plane wave, the Moschinsky shutter problem, and the stationary state of a charged particle in a uniform constant electric field. We show that their tomograms and quasidistribution functions expressed in terms of the Dirac delta function, the Airy function, and Fresnel integrals violate the standard normalization condition and the density matrix of the state therefore cannot always be reconstructed. We propose a method that allows circumventing this problem using a special tomogram in the limit form.

Keywords: quantum tomography, quasidistribution, normalization condition, plane wave, Moschinsky shutter, particle in an electric field.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
The research of L. A. Markovich in Secs. 3 and 4 was supported in part by a grant from the Russian Science Foundation (Project No. 16-08-01285_a).


DOI: https://doi.org/10.4213/tmf9562

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English version:
Theoretical and Mathematical Physics, 2018, 197:2, 1677–1689

Bibliographic databases:

Document Type: Article
PACS: 03.65.-w
MSC: 81Q99
Received: 11.03.2018

Citation: V. I. Man'ko, L. A. Markovich, “Unnormalized tomograms and quasidistributions of quantum states”, TMF, 197:2 (2018), 328–342; Theoret. and Math. Phys., 197:2 (2018), 1677–1689

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