Quantum distance to uncontrollability and quantum speed limits

Distance to uncontrollability is a crucial concept in classical control theory. Here, we introduce quantum distance to uncontrollability as a measure of how close a universal quantum system is to a nonuniversal one. This allows us to provide a quantitative version of the quantum speed limit, decomposing the bound into geometric and dynamical components. We consider several physical examples including globally controlled solid state qubits, scrambling of quantum information, and a cross-Kerr system, showing that the quantum distance to uncontrollability provides a precise meaning to spectral crowding, weak interactions, and other bottlenecks to universality. We suggest that this measure should be taken into consideration in the design of quantum technology.