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Since quantum coherence is an undoubted characteristic trait of quantumphysics, the quantification and application of quantum coherence has been oneof the long-standing central topics in quantum information science. Within theframework of a resource theory of quantum coherence proposed recently, a\textit{fiducial basis} should be pre-selected for characterizing the quantumcoherence in specific circumstances, namely, the quantum coherence is a\textit{basis-dependent} quantity. Therefore, a natural question is raised:what are the maximum and minimum coherences contained in a certain quantumstate with respect to a generic basis? While the minimum case is trivial, it isnot so intuitive to verify in which basis the quantum coherence is maximal.Based on the coherence measure of relative entropy, we indicate the particularbasis in which the quantum coherence is maximal for a given state, where theFourier matrix (or more generally, \textit{complex Hadamard matrices}) plays acritical role in determining the basis. Intriguingly, though we can prove thatthe basis associated with the Fourier matrix is a stationary point foroptimizing the $l_1$ norm of coherence, numerical simulation shows that it isnot a global optimal choice.

Maximal coherence in generic basis