Extending the fair sampling assumption using causal diagrams

Valentin Gebhart and Augusto Smerzi

QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy

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Abstract

Discarding undesirable measurement results in Bell experiments opens the detection loophole that prevents a conclusive demonstration of nonlocality. As closing the detection loophole represents a major technical challenge for many practical Bell experiments, it is customary to assume the so-called fair sampling assumption (FSA) that, in its original form, states that the collectively postselected statistics are a fair sample of the ideal statistics. Here, we analyze the FSA from the viewpoint of causal inference: We derive a causal structure that must be present in any causal model that faithfully encapsulates the FSA. This provides an easy, intuitive, and unifying approach that includes different accepted forms of the FSA and underlines what is really assumed when using the FSA. We then show that the FSA can not only be applied in scenarios with non-ideal detectors or transmission losses, but also in ideal experiments where only parts of the correlations are postselected, e.g., when the particles' destinations are in a superposition state. Finally, we demonstrate that the FSA is also applicable in multipartite scenarios that test for (genuine) multipartite nonlocality.

One of the main obstacles in the demonstration of Bell nonlocality is the requirement for highly efficient detectors. This challenging difficulty is commonly avoided by assuming that the possible local-realistic explanations of the observed statistics are restricted, which is known as the fair sampling assumption (FSA). In this work, we derive a necessary structure in the causal diagrams of the local-hidden variable models, that must be present to faithfully encapsulate the FSA. This structure highlights what one really assumes when assuming the FSA, and can be used to compare different forms of the FSA found in the literature. Finally, we show that the causal-diagram FSA can also be applied in Bell experiments where the particle's destinations are random, or in multipartite experiments testing for genuine multipartite nonlocality.

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Cited by

[1] Valentin Gebhart and Augusto Smerzi, "Coincidence postselection for genuine multipartite nonlocality: Causal diagrams and threshold efficiencies", Physical Review A 106 6, 062202 (2022).

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