Two-setting multisite Bell inequalities for loophole-free tests with up to 50$\%$ loss

We consider Bell experiments with $N$ spatially separated qubits where loss is present and restrict ourselves to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid, and Drummond [E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure $W={\prod }_{k=1}^N{\eta }_k$ of overall detection efficiency, where ${\eta }_k$ is the efficiency at site $k$. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided ${\prod }_{k=1}^N{\eta }_k>2^{(2-N)}$, which implies a symmetric threshold efficiency of $\eta \to 50\%$, as $N\to \infty$. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below $0.5$, provided $N>3$.

Figure 1

The LHV and quantum predictions for $S$, for a given value of $W_2$ which is a measure of efficiency. The blue (gray) dashed curve is the Holder inequality bound, which provides an upper limit to the LHV prediction for $S$. The blue (gray) dotted line at $S=2$ is the CHSH inequality bound, which also provides an upper limit to the LHV prediction for $S$. The open blue squares give a scattering of actual predictions for LHV theories. The observation of $S$ greater than either the Holder cure or the CHSH curve confirms the failure of all LHV theories. The dashed black line gives the quantum prediction for the special case of a maximally entangled Bell state.

Figure 2

The LHV and quantum predictions for $M_3$, versus efficiency. Panel (a) plots $M_3$, for a specified value of $W_N$. The blue (gray) dashed curve is the Holder inequality bound, which gives an upper limit to the LHV predictions. The blue (gray) dotted line is the upper LHV bound given by the MABK inequality. The open blue squares are a scattering of actual LHV predictions. The dashed black line is the quantum prediction for the maximally entangled GHZ state. Panel (b) plots the predictions versus the actual efficiency at each site, for the case of symmetric detection efficiencies where ${\eta }_k=\eta$, $\forall k$.

Figure 3

The LHV and quantum predictions for $A{r}_4$ versus efficiency. The curves are as defined in Figs. 1 and 2. Panel (a) gives the predictions for $A{r}_4$, for a given value of $W_N$. Panel (b) gives the predictions for the special symmetric case, where ${\eta }_k=\eta$, $\forall k$. The blue (gray) curves are the Holder and MABK inequality bounds to the LHV predictions. The dashed black line is the quantum prediction for a GHZ state.

Figure 4

The LHV and quantum predictions for $A{r}_6$ versus efficiency. The curves are as defined in Figs. 1, 2, and 3. Panel (a) gives the predictions for $A{r}_6$ for a given value of $W_N$. Panel (b) gives the predictions for the special symmetric case where ${\eta }_k=\eta$, $\forall k$. The blue (gray) curves are the Holder and MABK inequality upper bounds to the LHV predictions. The dashed black line is the quantum prediction for a GHZ state. The Holder region is clearly apparent and in this region, we see that the Holder curve is a close fit to the actual LHV predictions, indicated by the blue squares. The quantum prediction violates the LHV one, for $\eta$ approaching the limit of $0.5$, in this region.