Time-optimal quantum transformations with bounded bandwidth

In this paper, we derive sharp lower bounds, also known as quantum speed limits, for the time it takes to transform a system into state such that an observable assumes its lowest average value. We assume is initially in incoherent relative and evolves according von Neumann equation with Hamiltonian whose bandwidth uniformly bounded. The transformation depends intricately on observable's initial state's eigenvalue spectrum constellation of associated eigenspaces. problem finding limits consequently divides different cases requiring strategies. large number cases, simultaneously develop method break down complex manageable ones. derivations involve both combinatorial differential geometric techniques. study multipartite systems show allowing correlations between parts can up time. final section, use obtain upper bounds power which energy be extracted from batteries.