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An estimation of the controllability time for single-input systems on compact Lie Groups

Nowadays, quantum control appears to be one of the most challenging applications of control theory.

Indeed in many processes in modern technology it is necessary to induce a transition between energy levels of a quantum mechanical system, that usually is a molecule (in photochemistry or in laser spectroscopy) or a spin system (e.g. in nuclear magnetic resonance). In the experiments, excitation and ionization are induced by means of a sequence of external pulses (e.g. lasers or magnetic fields). Usually the transfer should be as fast as possible in order to minimize the effects of relaxation and decoherence that are always present. In most of the cases, the description of such processes translates into a multilinear control system on the unitary group SU(n) or on the group of unitary transformations on a complex infinite dimensional Hilbert space. One of the most promising application is in quantum information processing for the realization of quantum gates (namely the quantum counterpart of the elementary operations AND, OR, NOT).

This paper contains a crucial estimate for the minimum time necessary to induce a transition in a quantum mechanical system. More precisely, for a generic single-input control system on a compact semi-simple Lie group (and in particular on SU(n)), an upper and a lower bound on the minimum time necessary to steer (with unbounded controls) the identity to any other point of the group are provided. The most important byproduct of this paper is that the ideas presented generalize to infinite dimension, allowing to get controllability results for the bilinear Schroedinger equation (as a PDE) by means of Galerkin approximations, as obtained in a recent paper by one of the authors et al.