  We establish a general operational one-to-one mapping between coherence
measures and entanglement measures: Any entanglement measure of bipartite pure
states is the minimum of a suitable coherence measure over product bases. Any
coherence measure of pure states, with extension to mixed states by convex
roof, is the maximum entanglement generated by incoherent operations acting on
the system and an incoherent ancilla. Remarkably, the generalized CNOT gate is
the universal optimal incoherent operation. In this way, all convex-roof
coherence measures, including the coherence of formation, are endowed with
(additional) operational interpretations. By virtue of this connection, many
results on entanglement can be translated to the coherence setting, and vice
versa. As applications, we provide tight observable lower bounds for
generalized entanglement concurrence and coherence concurrence, which enable
experimentalists to quantify entanglement and coherence of the maximal
dimension in real experiments.
