Converting quantum coherence to genuine multipartite entanglement and nonlocality

We study the relations between quantum coherence and quantum nonlocality, genuine quantum entanglement, and genuine quantum nonlocality. We show that the coherence of a qubit state can be converted to the nonlocality of two-qubit states via incoherent operations. The results are also generalized to the qudit case. Furthermore, rigorous relations between the quantum coherence of a single-partite state and the genuine multipartite quantum entanglement, as well as the genuine three-qubit quantum nonlocality, are established.

Figure 1

One to one mapping between quantum coherence and quantum nonlocality under incoherent transformations. For initial incoherent states, under incoherent operations the resulting states admit local hidden variable (LHV) models while, for coherent states, the resulting states admit no local hidden variable (NLHV) models.

Figure 2

Surfaces of $\langle \mathcal{T}\rangle ,\langle \mathcal{NS}\rangle ,\mathcal{C}\left(\mathcal{T}\right)$, and $\mathcal{C}\left(\mathcal{NS}\right)$ with respect to state ${\rho }^s$. $\langle \mathcal{T}\rangle$ and $\langle \mathcal{NS}\rangle$ are the corresponding values of the state $\mathcal{U}({\rho }^s\otimes |0\rangle {\langle 0|}^A\otimes |0\rangle \langle 0{|}^B),\mathcal{C}\left(\mathcal{T}\right)\left[\mathcal{C}\right(\mathcal{NS}\left)\right]$ is the coherence of ${\rho }^s$ such that $\mathcal{U}({\rho }^s\otimes |0\rangle {\langle 0|}^A\otimes |0\rangle \langle 0{|}^B)$ violate the $\mathcal{T}$ ($\mathcal{NS}$) inequality, and $a+ib={\rho }_{01}^s$.