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A general framework for quantum splines

CIDMA – Center for Research and Development in Mathematics and Applications, Higher Institute of Accounting and Administration, University of Aveiro, Aveiro 3810-500, Portugal

E-mail Address: abrunheiroligia@ua.pt

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M. Camarinha

Centre for Mathematics of University of Coimbra, Department of Mathematics, University of Coimbra, Coimbra 3001-454, Portugal

E-mail Address: mmlsc@mat.uc.pt

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J. Clemente-Gallardo

Department of Theoretical Physics, Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Edificio I+D, Mariano Esquillor s/n, Zaragoza 50018, Spain

E-mail Address: jesus.clementegallardo@bifi.es

Corresponding author.

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J. C. Cuchí

Department of Agricultural and Forest Engineering, University of Lleida, Av. Alcalde Rovira Roure 191, Lleida 25198, Spain

E-mail Address: cuchi@eagrof.udl.cat

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, and

P. Santos

Centre for Mathematics of University of Coimbra, Department of Mathematics, University of Coimbra, Coimbra 3001-454, Portugal

Department of Physics and Mathematics, Polytechnic Institute of Coimbra, ISEC, Rua Pedro Nunes – Quinta da Nora, Coimbra 3030-199, Portugal

E-mail Address: patricia@isec.pt

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https://doi.org/10.1142/S0219887818501475Cited by:3 (Source: Crossref)

Abstract

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on $𝔲^{*} (n)$ which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.

Keywords:

AMSC: 81Q93, 65D07, 81Q70

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