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 offiltersand define the fair sampling assumption as a property of these filters, equivalent to the definition introduced in Ref. \cite{Berry10}. When this assumption is fulfilled, the post-selected data is reproduced by an ideal experiment where lossless devices measure afilteredstate which can be obtained from theactualstate 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.
Highlights
Measurement devices in many quantum experiments, and notably in quantum optics, have a finite non-unit efficiency: such devices may refuse to provide an outcome for the desired measurement, and produce a “no-click” event instead [2]
Performing Bell tests devoid of the detection loophole is still challenging and, as such, most experiments at present still rely on fair sampling. This naturally raises the following question: if assessing fair sampling requires a detailed description of the way the measurement device works, what remains of the device-independent framework in scenarios based on Bell tests relying upon the fair sampling assumption? The aim of our work is to provide a concrete answer to this question
We argue that protocols involving lossy devices and post-selection are secure if fair sampling holds unconditionally, and in particular according to the positive operator-valued measure (POVM) description possessed by the adversary
Summary
Measurement devices in many quantum experiments, and notably in quantum optics, have a finite non-unit efficiency: such devices may refuse to provide an outcome for the desired measurement, and produce a “no-click” event instead [2]. Input and on the quantum state, see panel (b) of Figure 1 Under this assumption the sampling can be regarded as “fair”, since the post-selected data can be seen as being generated from an ideal quantum experiment where lossless devices measure a filtered state. We provide a prescription for the minimal information that in an actual quantum experiment should be directly assessed in order to verify that a device (approximately) satisfies fair sampling Assuming that this minimal information is publicly available and trusted, the certification of the device can be safely completed using a standard device-independent protocol with post-selected data. Supplementary material and further considerations are presented in the Appendices
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