Quantum speed limits for change of basis

Quantum speed limits provide ultimate bounds on the time required to transform one quantum state into another. Here, we introduce a novel notion of quantum speed limits for collections of quantum states, investigating the time for converting a basis of states into an unbiased one as well as basis permutation. Establishing an unbiased basis, we provide tight bounds for the systems of dimension smaller than 5, and general bounds for multi-qubit systems and the 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. Permuting a basis, we obtain the exact expression for the Hilbert space of dimension d. We also investigate speed limits for coherence generation, providing the minimal time to establish a certain amount of coherence with a unitary evolution.