Hierarchy of the nonlocal advantage of quantum coherence and Bell nonlocality

Quantum coherence and nonlocality capture the nature of quantumness from different aspects. For the two-qubit states with a diagonal correlation matrix, we prove strictly a hierarchy between the nonlocal advantage of quantum coherence (NAQC) and Bell nonlocality by showing geometrically that the NAQC created on one qubit by local measurement on another qubit captures quantum correlation which is stronger than Bell nonlocality. For general states, our numerical results present strong evidence that this hierarchy may still hold. So the NAQC states form a subset of the states that can exhibit Bell nonlocality. We further propose a measure of NAQC that can be used for a quantitative study of it in bipartite states.

Figure 1

The tetrahedron $\mathcal{T}$ and octahedron $\mathcal{O}$ associated with ${\rho }_{\mathrm{Bell}}$ (a), and the level surfaces of $M\left(\tilde{\rho }\right)=1$ (b), $C_{l_1}^{na}\left({\rho }_{\mathrm{Bell}}\right)=\sqrt{6}$ (c), and $C_{re}^{na}\left({\rho }_{\mathrm{Bell}}\right)=C_{re}^m$ (d). The regions of Bell nonlocal states $\tilde{\rho }$ and NAQC states ${\rho }_{\mathrm{Bell}}$ are those outside the level surfaces.

Figure 2

Geometric representation of the polyhedron $\mathcal{P}$ (the red lines) inside $\mathcal{T}$. Here, $O$ and $O^{\prime }$ are the coordinate origin and the midpoint of $BC$, respectively, while the khaki curve is the curve of constant BN $M\left(\tilde{\rho }\right)=1$ at the facet $ABC$.

Figure 3

Critical $v_0^c$ for which (a) $C_{l_1}^{na}=\sqrt{6}$ and (b) $C_{re}^{na}=C_{re}^m$ vs $w_0$ with $w_1=0$ and 1. The other parameters are $\vec{s}=(s_1,w_0{s}_1,w_1{s}_1)$ and $\vec{r}=(r_1,w_0{r}_1,w_1{r}_1)$, with $s_1$ and $r_1$ being given in Eq. (13), and $v_0^c$ is plotted in the region of $v_0<0$.

Figure 4

The $x$ dependence of ${\tilde{Q}}_{l_1}\left({\rho }_1\right)$. Here, we choose $x\in [0,0.5]$ as ${\tilde{Q}}_{l_1}\left({\rho }_1\right)$ is symmetric with respect to $x=0.5$ for ${\rho }_1$. The inset shows the $x$ dependence of $\mathrm{\Delta }={\tilde{Q}}_{l_1}\left({\rho }_1\right)-Q_{l_1}\left({\rho }_1\right)$.