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

TL;DR

This paper investigates how post-selection under the fair sampling assumption influences device-independent quantum certifications, showing that under certain conditions, the filtered data reliably reflects the true quantum state and supports secure quantum protocols.

Contribution

It provides a detailed analysis of post-selection effects in device-independent tests, defining fair sampling as a property of filters and demonstrating the conditions under which certified conclusions remain valid.

Findings

Post-selected data can be modeled as an ideal, lossless measurement of a filtered state.

Fair sampling assumption ensures the validity of device-independent conclusions under certain conditions.

Results hold with small deviations from perfect fair sampling.

Abstract

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 [Berry et. al., PRA 81(1), 012109 (2010)]. 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.