Closing the detection loophole in multipartite Bell tests using Greenberger-Horne-Zeilinger states

We investigate the problem of closing the detection loophole in multipartite Bell tests, and show that the required detection efficiencies can be significantly lowered compared to the bipartite case. In particular, we present Bell tests based on $n$-qubit Greenberger-Horne-Zeilinger states, which can tolerate efficiencies as low as $38\%$ for a reasonable number of parties and measurements. Even in the presence of a significant amount of noise, efficiencies below $50\%$ can be tolerated, which is encouraging given recent experimental progress. Finally, we give strong evidence that, for a sufficiently large number of parties and measurements, arbitrarily small efficiencies can be tolerated, even in the presence of an arbitrary large amount of noise.

Figure 1

Threshold efficiency ${\eta }^{*}$ required given an $n$-qubit GHZ state with visibility $v$. Each observer has $m$ measurement settings.

Figure 2

Threshold detector efficiency ${\eta }^{*}$ versus the visibility $v$ of the GHZ state, in the case $m=n$. This indicates that in the limit of large $n$ and $m$, ${\eta }^{*}$ can become arbitrarily small, even for small visibilities $v$.

Figure 3

The set of straight lines defining the region of allowed parameters $x$ and $y$. Each straight line corresponds to one condition as given in Eq. (9) (with equality). Hence all points $(x,y)$ located below all lines will satisfy all conditions (9). Hence for any such choice of parameters $x$ and $y$, the local bound of the Bell inequality will be smaller or equal to 0 as desired. Here we have $n=5$ parties. Only the relevant lines are shown.