Abstract
We demonstrate an experimental test of the Clauser–Horne– Shimony–Holt (CHSH) Bell inequality which seemingly exhibits correlations beyond the limits imposed by quantum mechanics. Inspired by the idea of Fourier synthesis, we design analysers that measure specific superpositions of orbital angular momentum (OAM) states, such that when one analyser is rotated with respect to the other, the resulting coincidence curves are similar to a square-wave. Calculating the CHSH Bell parameter, S, from these curves result to values beyond the Tsirelson bound of . We obtain S = 3.99 ± 0.02, implying almost perfect nonlocal Popescu–Rohrlich correlations. The ‘super-quantum’ values of S is only possible in our experiment because our experiment, subtly, does not comply with fair-sampling. The way our Bell test fails fair-sampling is not immediately obvious and requires knowledge of the states being measured. Our experiment highlights the caution needed in Bell-type experiments based on measurements within high-dimensional state spaces such as that of OAM, especially in the advent of device-independent quantum protocols.
Highlights
We limit ourselves to the Clauser-Horne-Shimony-Holt (CHSH) inequality and variants thereof, which is the most tested version of the Bell inequality [15]
The Bell parameter, S, in the CHSH inequality is derived from the correlation function, E, of observables OA and OB as a function of the orientations of the analysers
The inequality is violated by entangled state√s in quantum mechanics, which predicts that the maximum achievable value of S is 2 2, the Tsirelson bound [11]
Summary
We limit ourselves to the Clauser-Horne-Shimony-Holt (CHSH) inequality and variants thereof, which is the most tested version of the Bell inequality [15]. A subtle choice of measurement states in order to obtain a desired correlation curve, as in our case, can lead to explicit violation of the fair-sampling assumption. For (2) and (1) to be considered equivalent, the detected coincidences (i.e. the four count rates), are assumed to be a fair sample of all the photons emitted by the source.
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