A General Framework For Quantum Splines

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projectile 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 HamiltoMan 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.