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Device-independent certifications employ Bell tests to guarantee the properfunctioning of an apparatus from the sole knowledge of observed measurementstatistics, i.e. without assumptions on the internal functioning of thedevices. When these Bell tests are implemented with devices having too lowefficiency, one has to post-select the events that lead to successfuldetections and thus rely on a fair sampling assumption. The question that weaddress in this paper is what remains of a device-independent certificationunder fair sampling. We provide an intuitive description of post-selections interms of filters and define the fair sampling assumption as a property of thesefilters, equivalent to the definition introduced in [Berry et. al., PRA 81(1),012109 (2010)]. When this assumption is fulfilled, the post-selected data isreproduced by an ideal experiment where lossless devices measure a filteredstate which can be obtained from the actual state via local probabilistic maps.Trusted conclusions can thus be obtained on the quantum properties of thisfiltered state and the corresponding measurement statistics can reliably beused, e.g., for randomness generation or quantum key distribution. We alsoexplore a stronger notion of fair sampling leading to the conclusion that thepost-selected data is a fair representation of the data that would be obtainedwith lossless detections. Furthermore, we show that our conclusions hold incases of small deviations from exact fair sampling. Finally, we describe setupspreviously or potentially used in Bell-type experiments under fair sampling andidentify the underlying device-specific assumptions.

How post-selection affects device-independent claims under the fair sampling assumption