  We present a genuine coherence measure based on a quasi-relative entropy as a
difference between quasi-entropies of the dephased and the original states. The
measure satisfies non-negativity and monotonicity under genuine incoherent
operations (GIO). It is strongly monotone under GIO in two- and
three-dimensions, or for pure states in any dimension, making it a genuine
coherence monotone. We provide a bound on the error term in the monotonicity
relation in terms of the trace distance between the original and the dephased
states. Moreover, the lower bound on the coherence measure can also be
calculated in terms of this trace distance.
