Device-independent certifications employ Bell tests to guarantee the proper functioning of an apparatus from the sole knowledge of observed measurement statistics, i.e. without assumptions on the internal functioning of the devices. When these Bell tests are implemented with devices having too low efficiency, one has to post-select the events that lead to successful detections and thus rely on a fair sampling assumption. The question that we address in this paper is what remains of a device-independent certification under fair sampling. We provide an intuitive description of post-selections in terms of $filters$ and define the fair sampling assumption as a property of these filters, equivalent to the definition introduced in Ref. . When this assumption is fulfilled, the post-selected data is reproduced by an ideal experiment where lossless devices measure a $filtered$ state which can be obtained from the $actual$ state via local probabilistic maps. Trusted conclusions can thus be obtained on the quantum properties of this filtered state and the corresponding measurement statistics can reliably be used, e.g., for randomness generation or quantum key distribution. We also explore a stronger notion of fair sampling leading to the conclusion that the post-selected data is a fair representation of the data that would be obtained with lossless detections. Furthermore, we show that our conclusions hold in cases of small deviations from exact fair sampling. Finally, we describe setups previously or potentially used in Bell-type experiments under fair sampling and identify the underlying device-specific assumptions.
Device-independence is yet another surprising feature of quantum mechanics. Sometimes it is possible to guarantee that a set of devices are operating according to their intended specifications without ever needing to ``look inside'' to discover how exactly do they work. The core idea is that a violation --by a sufficiently large margin-- of a Bell inequality is enough to certify that entanglement must be present inside the devices that are used in the Bell test. These Bell tests can then be incorporated in more involved protocol that allow to perform practically useful tasks, such as the production of certified truly random numbers, or secure distribution of keys using only untrusted quantum devices.
There is however a difficult technological challenge ahead, namely, for many of these protocols to work the Bell inequality has to be violated by a large margin --the experimentally observed value of the Bell operator has to be much higher than the classically reachable value. This puts an extra burden in performing these protocols, to the extent that, for example, as of today there has been no experimental demonstration of a fully device-independent quantum key distribution protocol. The main obstacle to these goals is to reach a sufficiently high photon preparation and detector efficiency. The situation is similar to the status of experimental Bell tests in the past, when low efficiency was one of the main limiting factors to go fully loophole-free. Nonetheless, many Bell experiments with photons yielding a large Bell violation have been performed prior to 2015, by using a so-called ``fair sampling'' assumption. The idea is that, according to this assumption, the non-detection events are locally random and do not depend on the quantum state of the incoming photon, therefore post-selection of the cases where the detectors click results in a unbiased sample from the original data.
The question that we address in our work, then, is whether it is possible to use the fair sampling assumption also in the new context of device-independent certifications to allow the implementation of experiments that would otherwise be impossible with the currently existing technology. The question is delicate, as one has to clearly distinguish between the properties of the devices that are assumed to be known, in order to satisfy fair sampling, from the properties that we want to certify in a device-independent fashion. In our work we show that such a distinction is possible and, also, that only a partial physical characterization of the device is required to be confident that fair sampling holds. We introduce a entirely quantum and general formulation of fair-sampling in which the proof of security and correctness is particularly simple and appealing, stemming from an equivalence between the experiment ``in the lab'' and an ideal quantum experiment using lossless photon sources and detectors. Moreover, we perform a careful analysis of what happens in the case in which the assumption is only approximately satisfied, which will always happen in any real-world situation. We believe that our study of approximate fair sampling could be particularly useful for the analysis of upcoming experiments, since this aspect has been neglected in several recent proof-of-principle demonstrations of device-independent protocols that have assumed that fair sampling holds exactly.
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