Determining quantum coherence with minimal resources

We characterize minimal measurement setups for validating the quantum coherence of an unknown quantum state. We show that for a d-level system, the optimal strategy consists of measuring d orthonormal bases such that each measured basis is mutually unbiased with respect to the reference basis, and together with the reference basis they form an informationally complete set of measurements. We prove that, in general, any strategy capable of validating quantum coherence allows one to evaluate also the exact value of coherence. We then give an explicit construction of the optimal measurements for arbitrary dimensions, and we derive a reconstruction formula for the off-diagonal terms. We also demonstrate that the same measurement setup is optimal for the modified task of verifying if the coherence is above or below a given threshold value. Finally, we show that the certification of entanglement of bipartite maximally correlated states is intimately connected with the certification of coherence.