Experimental test of the chained CHSH inequality in a two-dimensional orbital angular momentum system

The chained CHSH inequality is of paramount importance in quantum information science, demonstrating superiority over the CHSH inequality in some quantum information tasks. Hitherto, experimental verification was rarely considered to verify the chained CHSH inequality, as previous experimental efforts primarily focused on the CHSH inequality. Here, by employing two-photon orbital angular momentum (OAM) entanglement, we experimentally demonstrate the chained CHSH across multiple settings. In theory, we demonstrate a geometrical connection of optimal measurement vectors and establish that the optimal measurement vectors reside in a circle through the center of the Bloch sphere. This allows us to construct multiple sets of optimal measurement vectors on different circles. In the experiment, we typically choose circles in the XOY, XOZ, and YOZ planes to observe violations of the chained CHSH inequality. Our experimental results indicate that the maximum violation increases with setting k, underscoring the dependence of the maximum violation on the measurement angles.