Examinando por Autor "Ibort, Alberto"

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A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups
(American Institute of Physics, 2017-10-11) Ibort, Alberto; López Yela, Alberto; Moro, Julio
A numerical algorithm that computes the decomposition of any finite-dimensional unitary reducible representation of a compact Lie group is presented. The algorithm, which does not rely on an algebraic insight into the group structure, is inspired by quantum mechanical notions. After generating two adapted states (these objects will be conveniently defined in Definition II.1) and after appropriate algebraic manipulations, the algorithm returns the block matrix structure of the representation in terms of its irreducible components. It also provides an adapted orthonormal basis. The algorithm can be used to compute the Clebsch–Gordan coefficients of the tensor product of irreducible representations of a given compact Lie group. The performance of the algorithm is tested on various examples: the decomposition of the regular representation of two finite groups and the computation of Clebsch–Gordan coefficients of two examples of tensor products of representations of SU(2).
Quantum tomography and the quantum Radon transform
(American Institute of Mathematical Sciences, 2021-10) Ibort, Alberto; López Yela, Alberto
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 -algebras is presented. Given a -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 -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 -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.