Quantum Speed Limit for Change of Basis

Quantum speed limits provide ultimate bounds on the time required to transform one quantum state into another. Here, we extend the notion of quantum speed limits to collections of quantum states, investigating the time for converting a basis of states into an unbiased one. We provide tight bounds for systems of dimension smaller than 5, and general bounds for multi-qubit systems and Hilbert space dimension $d$. For two-qubit systems, we show that the fastest transformation implements two Hadamards and a swap of the qubits simultaneously. We further prove that for qutrit systems the evolution time depends on the particular type of the unbiased basis. We also investigate speed limits for coherence generation, providing the minimal time to establish a certain amount of coherence with a unitary evolution.