Quantum Nonlocality with Arbitrary Limited Detection Efficiency

The demonstration and use of nonlocality, as defined by Bell’s theorem, rely strongly on dealing with nondetection events due to losses and detectors’inefficiencies. Otherwise, the so-called detection loophole could be exploited. The only way to avoid this is to have detection efficiencies that are above a certain threshold. We introduce the intermediate assumption of limited detection efficiency, that is, in each run of the experiment, the overall detection efficiency is lower bounded by ${\eta }_{\min }>0$. Hence, in an adversarial scenario, the adversaries have arbitrary large but not full control over the inefficiencies. We analyze the set of possible correlations that satisfy limited detection locality and show that they necessarily satisfy some linear Bell-like inequalities. We prove that quantum theory predicts the violation of one of these inequalities for all ${\eta }_{\min }>0$. Hence, nonlocality can be demonstrated with arbitrarily small limited detection efficiencies. We validate this assumption experimentally via a twin-photon implementation in which two users are provided with one photon each out of a partially entangled pair. We exploit on each side a passive switch followed by two measurement devices with fixed settings. Assuming the switches are not fully controlled by an adversary, nor by hypothetical local variables, we reveal the nonlocality of the established correlations despite a low overall detection efficiency.

Figure 1

Two boxes each receive a particle, emitted by a common source. They are given inputs $x$ and $y$, which we depict here as the setting of an active switch, and return outputs $a$ and $b$, respectively. There is the possibility for nondetection events, in which case the corresponding output variable takes the value $\varnothing$. Since these losses can be seen as happening inside the box, they can depend on the inputs $x$ and $y$, respectively. We analyze the limited detection local case of this scenario, meaning that a local hidden variable $\lambda$ not only fully describes the state of the particle, but can also influence whether or not a nondetection event occurs.

Figure 2

A source emits supposedly entangled photons towards two measurement stations. In each station, a passive switch routes the individual incoming photons to one out of two pairs of detectors, corresponding to either settings 0 or 1. The boxes are given inputs $x$ and $y$, which decide which pairs of detectors are switched on, and return outputs $a$ and $b$ depending on which detector fires. If the photon goes towards the inactive pair of detectors it is simply considered as lost and not registered. We compare this scenario to the scenario where a local hidden variable $\lambda$ that can influence the losses and settings tries to emulate our observed correlations.