Not All Probability Density Functions Are Tomograms

This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true p...

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Bibliographic Details
Published in:	Entropy (Basel, Switzerland) Vol. 26; no. 3; p. 176
Main Authors:	Markovich, Liubov A, Urbanetz, Justus, Man'ko, Vladimir I
Format:	Journal Article
Language:	English
Published:	Switzerland MDPI AG 20-02-2024 MDPI
Subjects:	Analysis Atoms characteristic function Distribution (Probability theory) Fourier transforms Hilbert space Nonparametric statistics Operators (mathematics) Probability Probability density functions Probability distribution probability distribution function Probability theory quantum state reconstruction Random variables symplectic tomogram Wave functions probability distribution function characteristic function symplectic tomogram quantum state reconstruction
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Summary:	This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and nonparametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum".
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23
ISSN:	1099-4300 1099-4300
DOI:	10.3390/e26030176