Towards a Quantum Sampling Theory: The Case of Finite Groups

Garcı́a, Antonio G.; Hernández-Medina, Miguel Á.; Ibort, Alberto

doi:10.1007/978-3-030-24748-5_11

Springer Proceedings in Physics

2019

DOI: 10.1007/978-3-030-24748-5_11

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Towards a Quantum Sampling Theory: The Case of Finite Groups

Antonio G. Garcı́a

Miguel Á. Hernández-Medina²,

Alberto Ibort

Abstract: Nyquist-Shannon sampling theorem, instrumental in classical telecommunication technologies, is extended to quantum systems supporting a unitary representation of a finite group G. Two main ideas from the classical theory having natural counterparts in the quantum setting: frames and invariant subspaces, provide the mathematical background for the theory. The main ingredients of classical sampling theorems are discussed and their quantum counterparts are thoroughly analyzed in this simple situation. A few examp… Show more

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2021

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Cited by 1 publication

References 28 publications

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Quantum tomography and the quantum Radon transform

Ibort¹,

López-Yela²

2021

IPI

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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebras is presented. Given a <inline-formula><tex-math id="M2">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on <inline-formula><tex-math id="M3">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.</p><p style='text-indent:20px;'>The abstract theory is realized by using dynamical systems, that is, groups represented on <inline-formula><tex-math id="M4">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-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.</p>

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Quantum tomography and the quantum Radon transform

Ibort¹,

López-Yela²

2021

IPI

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Towards a Quantum Sampling Theory: The Case of Finite Groups

Cited by 1 publication

References 28 publications

Quantum tomography and the quantum Radon transform

Quantum tomography and the quantum Radon transform

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