Kinematical bounds on evolution and optimization of mixed quantum states

Upper and lower bounds are established for time-dependent ensemble averages of observables of driven quantum systems in mixed states. They limit controllability of observables independently of the control fields. Narrower bounds are established when the observable is a projector onto a pure quantum state or subspace. They are optimal in the sense of being kinematically achievable. Calculations on nonlinear optimal control of a four-level model indicate that these kinematical bounds are dynamically achievable asymptotically with increasing control pulse strength.