The standard fair sampling assumption is not necessary to test local realism

TL;DR

This paper demonstrates that the common fair sampling assumption in Bell tests is unnecessary, showing conditions under which local realism can be tested without it, and explores implications for Tsirelson's bound and Bell inequalities.

Contribution

It proves that detection efficiency can depend on measurement settings if it factorizes appropriately, removing the need for the fair sampling assumption in Bell tests.

Findings

Detection efficiency can depend on measurement settings if it factorizes.

Tsirelson's bound must be satisfied under certain conditions.

Possible violation of Tsirelson's bound while CHSH inequality holds for unentangled states.

Abstract

Almost all Bell-inequality experiments to date have used postselection, and therefore relied on the fair sampling assumption for their interpretation. The standard form of the fair sampling assumption is that the loss is independent of the measurement settings, so the ensemble of detected systems provides a fair statistical sample of the total ensemble. This is often assumed to be needed to interpret Bell inequality experiments as ruling out hidden-variable theories. Here we show that it is not necessary; the loss can depend on measurement settings, provided the detection efficiency factorises as a function of the measurement settings and any hidden variable. This condition implies that Tsirelson's bound must be satisfied for entangled states. On the other hand, we show that it is possible for Tsirelson's bound to be violated while the CHSH-Bell inequality still holds for unentangled states, and present an experimentally feasible example.