How can the principles of quantum nonlocality and contextuality be experimentally demonstrated in high-dimensional quantum systems, such as qudits encoded in photonic orbital angular momentum states, and what implications do such experiments have for extending Bell's theorem to more complex Hilbert spaces?

Quantum nonlocality and contextuality are foundational aspects of quantum mechanics, demonstrating that quantum correlations cannot be explained by local hidden variable theories. Nonlocality manifests through violations of Bell inequalities, while contextuality reveals the dependence of measurement outcomes on the chosen measurement settings. High-dimensional quantum systems, such as qudits (d-dimensional quantum states), offer an expanded Hilbert space compared to qubits, providing richer resources for testing these phenomena.

Photonic systems, particularly those using the orbital angular momentum (OAM) of light, are ideal platforms for encoding qudits due to their high-dimensional state space and controllability. Recent theoretical advancements propose generalizations of Bell inequalities and contextuality tests for such systems. Investigating these phenomena in high-dimensional settings pushes the boundaries of quantum theory, enabling deeper insights into entanglement, the structure of quantum correlations, and their potential applications in quantum communication and computation.