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Journal of Mathematical Physics

Integral transforms between tomogram and quasi-probability functions based on quantizer-dequantizer operators formalism

Vladimir I. Man’ko, Liubov A. Markovich

Published Oct 1, 2020• 9 citations
Publisher

Abstract

An application of a quantizer–dequantizer method as a unifying description for representations of states in quantum mechanics is considered. Well-known quasi-distributions and tomograms are rewritten in terms of the dequantizer and quantizer operators. Using this description of the tomographic probability function and its symbol, we construct the invertible integral transforms between the tomogram and the quasi-probability distributions such as Wigner, Kirkwood–Rihaczek, Choi–Williams, P- and Q-functions, and others.

DOI
10.1063/5.0019203

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An application of a quantizer–dequantizer method as a unifying description for representations of states in quantum mechanics is considered. Well-known quasi-distributions and tomograms are rewritten in terms of the dequantizer and quantizer operators. Using this description of the tomographic probability function and its symbol, we construct the invertible integral transforms between the tomogram and the quasi-probability distributions such as Wigner, Kirkwood–Rihaczek, Choi–Williams, P- and Q-functions, and others.
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Abstract This paper presents a review of star-product formalism. This formalism provides a description for quantum states and observables by means of the functions called’ symbols of operators’. Those functions are obtained via bijective maps of the operators acting in Hilbert space. Examples of the Wigner-Weyl symbols (Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified for the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators required for construction of bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relationship between structure constants of associative star-product of operator symbols and quantizer-dequantizer operators is reviewed.
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 A review of the approach where the states of quantum systems are identified with fair probability distributions is presented. The quantizer–dequantizer operators used to construct the invertible map of the density operators onto the probability distributions are applied to obtain the kinetic equations for probability distributions identified with the quantum system states. For qubit states, the von Neumann evolution equation for the density operator is explicitly given in the form of kinetic equation for the probability distribution. Simplest tomographic probability distributions describing the states of multimode quantum oscillator are constructed.
We present a new formulation of conventional quantum mechanics, in which the notion of a quantum state is identified via a fair probability distribution of the position measured in a reference frame of the phase space with rotated axes. In this formulation, the quantum evolution equation as well as the equation for finding energy levels are expressed as linear equations for the probability distributions that determine the quantum states. We also give the integral transforms relating the probability distribution (called the tomographic-probability distribution or the state tomogram) to the density matrix and the Wigner function and discuss their connection with the Radon transform. Qudit states are considered and the invertible map of the state density operators onto the probability vectors is discussed. The tomographic entropies and entropic uncertainty relations are reviewed. We demonstrate the uncertainty relations for the position and momentum and the entropic uncertainty relations in the tomographic-probability representation, which is suitable for an experimental check of the uncertainty relations.