Quantum coherence: A fundamental resource for establishing genuine multipartite correlations

We establish the profound equivalence between the coherence measures defined via the convex roof method and two classes of genuine multipartite entanglement (GME) under certain conditions. Initially we construct two distinct classes of measures for genuine multipartite entanglement utilizing real symmetric concave functions and the convex roof technique. We then demonstrate that all coherence measures for any qudit states, defined through the convex roof approach, are identical to our two classes of GME measures of the states combined with an incoherent ancilla under an incoherent unitary operation. This relationship implies that genuine multipartite entanglement can be generated from the coherence incoherent in an initial state through the incoherent unitary operations. Furthermore, we conduct a preliminary exploration into the interplay between coherence and genuine quantum correlations such as genuine multipartite steering and genuine multipartite nonlocality. We show that the coherence exists in a qubit state if and only if genuine multipartite nonlocality is present for the special three-qubit $X$ states, obtained by performing unitary incoherent operations on the tensor product of the qubit state and the incoherent ancilla. Moreover, under the above process, we prove that the coherence of the qubit state exists if and only if the corresponding special $X$ state is genuinely multipartite steerable.