Necessary and sufficient criterion for extremal quantum correlations in the simplest Bell scenario

In the study of quantum nonlocality, one obstacle is that the analytical criterion for identifying the boundaries between quantum and postquantum correlations has not yet been given, even in the simplest Bell scenario. We propose a plausible, analytical, necessary and sufficient condition ensuring that a nonlocal quantum correlation in the simplest scenario is an extremal boundary point. Our extremality condition amounts to certifying an information-theoretical quantity; the probability of guessing a measurement outcome of a distant party optimized using any quantum instrument. We show that this quantity can be upper and lower bounded from any correlation in a device-independent way, and we use numerical calculations to confirm that coincidence of the upper and lower bounds appears to be necessary and sufficient for the extremality.

Figure 1

The relation between the RHS and LHS of Eq. (4) for 20 000 randomly chosen nonlocal realizations $\left(C_{\mathrm{CHSH}}>2\right)$, when $\langle A_x{B}_y\rangle$ are not scaled (black dots) and are scaled by $D_x^B$ or $D_y^A$ (red dots). The fraction of nonlocal realizations among all realizations is roughly 0.6%. In the inset, both $[{\prod }_{xy}sin({\varphi }_x^B-{\theta }_y^B),C_{\mathrm{CHSH}}]$ and $[{\prod }_{xy}sin({\varphi }_y^A-{\theta }_x^A),C_{\mathrm{CHSH}}]$ are plotted for the same realizations (black dots). The blue dots show the results, where $V_{11}$ is randomly selected from $[-0.05,0.05]$, and $V_x^A,V_y^B$ are from $[-0.0125,0.0125]$ (the others are from $[-1,1])$, to effectively pick up rare cases.

Figure 2

For each of 40 000 randomly constructed realizations, ${\lambda }_{\max }$ is obtained using the NPA hierarchy, and $(\mathrm{\Delta },{\lambda }_{\max })$ is plotted. The inset shows the distribution of $C_{\mathrm{CHSH}}$ and $\mathrm{\Delta }$, which indicates that $C_{\mathrm{CHSH}}\ge 2$ when $\mathrm{\Delta }=0$.

Figure 3

In the correlation space $\alpha \mathbf{PR}+\beta {\mathbf{P}}_{\mathbf{L}}+(1-\alpha -\beta )\mathbf{I}$, the quantum boundary is the red straight line as proved in [18]. Here, $\mathbf{PR}$ is the postquantum correlation produced by the Popescu-Rohrlich box [4], ${\mathbf{P}}_{\mathbf{L}}$ is a local deterministic correlation with $\langle A_x\rangle =\langle B_y\rangle =1$, and $\mathbf{T}$ is a correlation attaining the Tsirelson bound. Along this boundary, ${\left(D_{xy}^B\right)}^2\equiv {\langle A_x\rangle }^2+S_{xy}^{+}$ and ${\left(D_{xy}^A\right)}^2\equiv {\langle B_y\rangle }^2+S_{xy}^{+}$ are plotted. Since $D_x^B=D_y^A=1$ at both edges of the boundary, the correct device-independent upper bounds are equal to 1 over the entirety of the boundary.