Abstract
Reasoning about Bell nonlocality from the correlations observed in post-selected data is always a matter of concern. This is because conditioning on the outcomes is a source of non-causal correlations, known as aselection bias, rising doubts whether the conclusion concerns the actual causal process or maybe it is just an effect of processing the data. Yet, even in the idealised case without detection inefficiencies, post-selection is an integral part of experimental designs, not least because it is a part of the entanglement generation process itself. In this paper we discuss a broad class of scenarios with post-selection on multiple spatially distributed outcomes. A simple criterion is worked out, called theall-but-oneprinciple, showing when the conclusions about nonlocality from breaking Bell inequalities with post-selected data remain in force. Generality of this result, attained by adopting the high-level diagrammatic tools of causal inference, provides safe grounds for systematic reasoning based on the standard form of multipartite Bell inequalities in a wide array of entanglement generation schemes, without worrying about the dangers of selection bias. In particular, it can be applied to post-selection defined by single-particle events in each detection chanel when the number of particles in the system is conserved.
Highlights
The study of experimental correlations provides a window into the underlying causal mechanisms, even when their exact nature remains obscured. In his seminal works [1], John Bell showed that seemingly innocuous assumptions about the causal structure of realistic models leave a mark on the observed statistics
Because entanglement is not a property generated on demand, every Bell experiment must resort to post-selection
This opens the doors to the selection bias introducing non-causal correlations into the data, and threatening the conclusions expected to be drawn from the experiment
Summary
The study of experimental correlations provides a window into the underlying causal mechanisms, even when their exact nature remains obscured. The generic structure of events is richer than that required for the intended Bell inequality, and post-selection aims at retaining only those experimental trials, based on some well-defined criterion, which are potentially interesting for the violation of the desired inequality This poses an issue regarding the legitimacy of the conclusion about Bell nonlocality in such scenarios, since conditioning is often a source of non-causal correlations. It is interesting to ask about general conditions when post-selection, due to entanglement generation, does not compromise the conclusion of nonlocality from the violation of some given Bell inequality This problem has been discussed only for some particular scenarios for two and three parties, and the analysis typically involved the entire pattern of experimental outcomes present in a given experiment [35,36,37]. We give a full poof of this criterion preceded with a brief discussion of the selection bias and Bell nonlocality under postselection
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