Quantum tomography and the quantum Radon transform

Alberto Ibort; Alberto López-Yela

doi:10.3934/ipi.2021021

Article Contents

2021, Volume 15, Issue 5: 893-928. Doi: 10.3934/ipi.2021021

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Quantum tomography and the quantum Radon transform

Alberto Ibort^1, and
Alberto López-Yela^2,

1.
Instituto de Ciencias Matemáticas (CSIC - UAM - UC3M - UCM) ICMAT and Depto. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain
2.
Dpto. de Teoría de la señal y comunicaciones, Univ. Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain

Received: July 2019

Revised: February 2021

Early access: March 2021

Published: October 2021

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Abstract

A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of $ C^* $-algebras is presented. Given a $ C^* $-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on $ C^* $-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.

The abstract theory is realized by using dynamical systems, that is, groups represented on $ C^* $-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Left: Section of a body irradiated with gamma radiation and the measure of the tomogram $ \mathcal{W}(X, \theta) $. Right: Scheme of quantum state tomography. The quantum tomogram $ \mathcal{W}_\rho (X, \theta) $ obtained by homodyne measuring the quadrature operator $ \mathbf{X}_\theta = X -\textbf{Q}\cos\theta-\textbf{P}\sin\theta $

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Figure 2. Tomographic problem

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Figure 3. Tomographic map $ U $

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Figure 4. Sampling diagram

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Figure 5. Positive Transform diagram

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Figure 6. Bloch's sphere representing states of a particle with spin $ 1/2 $

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