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arXiv:2104.09721

Tomography in Loop Quantum Cosmology

Jasel Berra–Montiel, Alberto Molgado

Published Apr 20, 2021• 0 citations
arXivHTMLPDF

Abstract

We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position and momentum, we efficiently interpret the properties of such tomograms, being consequent with Heisenberg's uncertainty principle. We also obtain, by means of the dual tomographic symbols, the expectation value for the volume operator, which coincides with the values reported in the literature. We expect that our findings result interesting as the introduced tomographic representation may be further benefited from the well-developed measure techniques in the areas of Quantum optics and Quantum information theory.

DOI
10.48550/arxiv.2104.09721

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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.
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.
The review of star-product formalism providing the possibility to describe quantum states and quantum observables by means of the functions called symbols of operators which are obtained by means of bijective maps of the operators acting in Hilbert space onto these functions is presented. Examples of the Wigner-Weyl symbols (like Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified with the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators which are needed to construct the bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relation of the structure constants of the associative star-product of the operator symbols to the quantizer-dequantizer operators is reviewed.