Geometric interpretation and experimental test of Leggett inequalities with nonmaximally entangled states

Leggett inequality states that nonlocal hidden-variable models might still be incompatible with the predictions of quantum physics. However, its theoretical and experimental demonstration is only in the scenario of 2-dimensional maximally entangled systems. An open question remains as to whether the Leggett inequality can be violated by nonmaximally entangled states. Here, we answer this question both in theory and experiment. Specifically, from the point of view of geometry, we theoretically map the problem of maximizing the correlation measure in the Leggett inequality to maximizing the sum of an ellipse's diameter and semi-diameter axes, accordingly, demonstrating that the violation of the Leggett inequality requires a more robust entanglement than that of Bell's theory. Experimentally, by leveraging the controllable photonic orbital angular momentum entanglement, we demonstrate the violation of Leggett-type inequalities by more than 8.7 and 4.5 standard deviations under concurrence C = 0.95 and 0.9, respectively. Our observations indicate that, the requirement for quantum correlation should be increased to exclude a particular class of non-local hidden variable theories that abide by Leggett's model, providing insights into the boundaries of quantum correlation and the limitations imposed by non-local hidden variables.