Operational one-to-one mapping between coherence and entanglement measures

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.

Figure 1

Operational one-to-one mapping between coherence monotones and entanglement monotones. Any entanglement monotone $E_f$ for $f\in {\mathcal{F}}_{\mathrm{sc}}$ between B and A is the minimum of $C_f$ over local unitary transformations. Any coherent monotone $C_f$ on the system B is the maximum of $E_f$ generated by incoherent operations acting on the system B and an incoherent ancilla A. The generalized cnot gate is the universal optimal incoherent operation.