Entropic Characterization of Quantum States with Maximal Evolution under Given Energy Constraints

A measure [Formula: see text] for the amount of dynamical evolution exhibited by a quantum system during a time interval [Formula: see text] is defined in terms of how distinguishable from each other are, on average, the states of the system at different times. We investigate some properties of the measure [Formula: see text] showing that, for increasing values of the interval’s duration, the measure quickly reaches an asymptotic value given by the linear entropy of the energy distribution associated with the system’s (pure) quantum state. This leads to the formulation of an entropic variational problem characterizing the quantum states that exhibit the largest amount of dynamical evolution under energy constraints given by the expectation value of the energy.