An Estimate of Quantum Evolution Speed.

M. Andrecut, Mazhar Ali

David Velasco Villamizar, A. C. S. Leal, R. Auccaise +1 more

Finding the solutions of the equations that describe the dynamics of a given physical system is crucial in order to obtain important information about its evolution. However, by using estimation theory, it is possible to obtain, under certain limitations, some information on its dynamics. The quantum-speed-limit (QSL) theory was originally used to estimate the shortest time in which a Hamiltonian drives an initial state to a final one for a given fidelity. Using the QSL theory in a slightly different way, we are able to estimate the running time of a given quantum process. For that purpose, we impose the saturation of the Anandan-Aharonov bound in a rotating frame of reference where the state of the system travels slower than in the original frame (laboratory frame). Through this procedure it is possible to estimate the actual evolution time in the laboratory frame of reference with good accuracy when compared to previous methods. Our method is tested successfully to predict the time spent in the evolution of nuclear spins 1/2 and 3/2 in NMR systems. We find that the estimated time according to our method is better than previous approaches by up to four orders of magnitude. One disadvantage of our method is that we need to solve a number of transcendental equations, which increases with the system dimension and parameter discretization used to solve such equations numerically.

David Velasco Villamizar, Eduardo I. Duzzioni, Adriane Consuelo Leal Auccaise +1 more

Finding the solutions of the equations that describe the dynamics of a given physical system is crucial in order to obtain important information about its evolution. However, by using estimation theory, it is possible to obtain, under certain limitations, some information on its dynamics. The quantum-speed-limit (QSL) theory was originally used to estimate the shortest time in which a Hamiltonian drives an initial state to a final one for a given fidelity. Using the QSL theory in a slightly different way, we are able to estimate the running time of a given quantum process. For that purpose, we impose the saturation of the Anandan-Aharonov bound in a rotating frame of reference where the state of the system travels slower than in the original frame (laboratory frame). Through this procedure it is possible to estimate the actual evolution time in the laboratory frame of reference with good accuracy when compared to previous methods. Our method is tested successfully to predict the time spent in the evolution of nuclear spins 1/2 and 3/2 in NMR systems. We find that the estimated time according to our method is better than previous approaches by up to four orders of magnitude. One disadvantage of our method is that we need to solve a number of transcendental equations, which increases with the system dimension and parameter discretization used to solve such equations numerically.

Sergio A Albeverio, Alexander Konstantinovich Motovilov

Abstract By a quantum speed limit one usually understands an estimate on how fast a quantum system can evolve between two distinguishable states. The most known quantum speed limit is given in the form of the celebrated Mandelstam–Tamm inequality that bounds the speed of the evolution of a state in terms of its energy dispersion. In contrast to the basic Mandelstam–Tamm inequality, we are concerned not with a single state but with a (possibly infinite-dimensional) subspace which is subject to the Schrödinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on the speed of such a subspace evolution. These bounds may be viewed as further generalizations of the Mandelstam–Tamm inequality. Our study includes the case of unbounded Hamiltonians.