Quantum speed limit and optimal control

The Heisenberg uncertainty principle, $\Delta E\Delta t\geq \hbar$, implies that a system cannot pass through distinguishable, i.e. orthogonal, states within arbitrarily short time. In the case of a time-independent Hamiltonian, the presence of this ultimate bound has been well established and summarized in the concept of a maximum allowed velocity, called \emph{quantum speed limit} (QSL). On other hand for a time-dependent Hamiltonian the problem started to be addressed only very recently and is still open. Optimal control theory offers a valuable tool to explore this issue: we test its performance in two paradigmatic cases, Landau-Zener model and transfer of information along a chain of coupled spins, and show that the results are compatible with the ultimate limits enabled by quantum mechanics.