It is well-known that local measurements performed on shared quantum systems can lead to nonlocal correlations, certified by a Bell inequality violation [1]. These correlations could not be obtained by any classical mechanism. Although this conclusion follows straightforwardly from the mathematical structure of quantum theory, proving it experimentally is a hard task. In fact, even if nonlocal correlations are observed, some experimental conditions have to be satisfied in order to guarantee that no classical mechanism can reproduce the obtained data. In this brief note we will comment on the most common mismatches between the theoretical description and experimental realisation of Bell tests, the so-called loopholes (for a more complete discussion see [2]).

In many Bell experiments, specially the ones involving photons, the measurements performed do not always return a conclusive result, due to detection inefficiencies. For instance, when making photon-polarisation measurements one usually observes not only the two outcomes corresponding to orthogonal polarisations but also a third event, corresponding to a no-detection. The most general way of treating this “no-click” event is by considering it as an additional measurement outcome. Nonlocal correlations can then be certified if there is no classical model reproducing the complete statistics observed. It turns out that if the no-click outcome occurs with a probability higher than a certain threshold one can not guarantee nonlocality. This threshold was obtained for several inequalities and situations. For instance, in the case two observers choose two measurement choices with ideally two outputs, one can show using the CHSH inequality that their detectors should click with probability higher than 82.8% if they share a maximally entangled state or 66.7% if partially entangled pure states are considered. These requirements can be lowered in other situations, for example when different measurements have different detection efficiencies (see e.g [3, 4, 5] and the I_{3322} inequality, if higher dimensional systems are used (see the I^{4}_{4422} inequality [6]) or more parties are considered (eg. through the Mermin or PVB inequalities [7, 8]).

Another important requirement in a Bell test is the space-like separation among the measurement events. In fact, if the devices of the observers can communicate during the measurement rounds one can not exclude the possibility that the correlations observed are simply due to an agreement set by this communication. In practice one usually avoids this possibility by separating the observers sufficiently to guarantees that any communication leading to nonlocal correlations should travel faster that light.

Although the detection and locality loopholes are regarded as the most serious practical difficulties in Bell tests there exist other loopholes to take into account. Related to the previous loophole is the assumption that the observers can choose their measurement settings freely, i.e. independently of earlier events. A failure of this assumption is called the measurement-independence loophole and was recently investigated in [9, 10] (see also highlight on measurement settings independence). Another possible problem is the memory-loophole, that takes place when the classical strategy used to simulate the nonlocal correlations is updated in each round of the experiment. A way of overcoming this problem has been proposed in [11]. Finally, in practice one only has access to finite statistics. Thus any conclusion about a Bell inequality violation has to be statistical. A way of treating this fact can be found in [12].

Daniel Cavalcanti

published online on 4 Feb 2014

[1] J. S. Bell, *On the Einstein Podolsky Rosen paradox*, Physics 1, **195** (1964).

[2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, Stephanie Wehner, *Bell nonlocality*, arXiv:1303.2849.

[3] N. Brunner, N. Gisin, V. Scarani, and C. Simon, *Detection Loophole in Asymmetric Bell Experiments*, Phys. Rev. Lett. **98**, 220403 (2007).

[4] A. Cabello and J. A. Larsson, *Minimum Detection Efficiency for a Loophole-Free Atom-Photon Bell Experiment*, Phys. Rev. Lett. **98**, 220402 (2007).

[5] G. Garbarino, *Minimum detection efficiencies for a loophole-free observable-asymmetric Bell-type test*, Phys. Rev. A **81**, 032106 (2010).

[6] T. Vertési, S. Pironio, N. Brunner, *Closing the detection loophole in Bell experiments using qudits*, Phys. Rev. Lett. **104**, 060401 (2010).

[7] Cabello, A., D. Rodriguez, and I. Villanueva, *Necessary and Sufficient Detection Efficiency for the Mermin Inequalities*, Rev. Lett. **101**, 120402 (2008).

[8] Pal, K., T. Vertesi, and N. Brunner, *Closing the detection loophole in multipartite Bell tests using Greenberger-Horne-Zeilinger states*, Phys. Rev. A **86**, 062111 (2012).

[9] M. Hall, *Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement Independence*, Phys. Rev. Lett. **105**, 250404 (2010).

[10] J. Barrett and N. Gisin, *How Much Measurement Independence Is Needed to Demonstrate Nonlocality?*, Phys. Rev. Lett. **106**, 100406 (2011).

[11] J. Barrett, D. Collins, L. Hardy, A. Kent, S. Popescu, *Quantum nonlocality, Bell inequalities and the memory loophole*, Phys. Rev. A **66**, 042111 (2002).

[12] R. Gill, *Statistics, Causality and Bell's Theorem*, arXiv:1207.5103.