  Active control of quantum systems enables diverse applications ranging from
quantum computation to manipulation of molecular processes. Maximum speeds and
related bounds have been identified from uncertainty principles and related
inequalities, but such bounds utilize only coarse system information, and
loosen significantly in the presence of constraints and complex interaction
dynamics. We show that an integral-equation-based formulation of conservation
laws in quantum dynamics leads to a systematic framework for identifying
fundamental limits to any quantum control scenario. We demonstrate the utility
of our bounds in three scenarios -- three-level driving, decoherence
suppression, and maximum-fidelity gate implementations -- and show that in each
case our bounds are tight or nearly so. Global bounds complement
local-optimization-based designs, illuminating performance levels that may be
possible as well as those that cannot be surpassed.
