Connecting Quantum Contextuality and Nonlocality

Abstract

Quantum theory departs from classical physics in its treatment of correlations, most prominently through the phenomena of contextuality and nonlocality. Once regarded primarily as foundational curiosities, these effects are now understood as key operational resources for quantum computation, communication, and simulation. Although traditionally investigated in distinct settings, recent theoretical and experimental advances have revealed deep conceptual, mathematical, and operational connections between them. This review presents a unified perspective on these developments based on sheaf-theoretic and graph-theoretic frameworks, which provide theory-independent characterizations of statistical correlations. These approaches clarify the structural relationship between contextuality and nonlocality, facilitate the formulation of experimentally testable inequalities, and guide implementations in realistic physical platforms, with particular emphasis on photonic systems. By bridging abstract theoretical structures and concrete experimental realizations, this review sheds light on the nonclassical foundations of quantum correlations and their emerging role in quantum technologies.

Figures (5)

• Figure 1: Interpreting "$>$" as "appears visually closer," the Penrose triangle gives rise to locally consistent orderings that cannot be obtained as the restrictions of any global section mansfield2013mathematical. A global section would correspond to a strict total order on $\{A,B,C\}$, which is incompatible with these visual assignments.
• Figure 2: The exclusivity graph of the CHSH scenario, ${\cal G}_{\rm CHSH}$cabello2014graph.
• Figure 3: Scheme of the experiment by Liu et al.liu2016nonlocality. A source emits pairs of entangled particles. One particle encodes two qubits (qubits 1 and 2) and is sent to Alice's laboratory, where three sequential measurements are performed $C$, $A$, and $B$). The other particle, encoding qubits 3 and 4, is sent to Bob's laboratory, where a single measurement is performed ($B'$).
• Figure 4: Scheme of the experiment by Xue et al.xue2023synchronous, illustrating the synchronous test of Bell nonlocality and state-dependent contextuality. Polarization-entangled photon pairs are generated via spontaneous parametric down-conversion. One photon of each pair encodes a qubit in its polarization degree of freedom and is measured by Alice using standard polarization analysis, implementing the CHSH test with Bob. The other photon encodes a qutrit in its combined polarization and spatial degrees of freedom and is measured by Bob through a sequence of compatible measurements realized with cascaded Mach-Zehnder interferometers, enabling the test of the KCBS inequality. The experimental architecture allows both inequalities to be evaluated within the same operational scenario.
• Figure 5: Scheme of the experiment by Sheng et al.sheng2025orbital. The inset (a) shows the original two-photon OAM spectrum of limited spiral bandwidth before entanglement concentration, while (b) shows the maximally entangled OAM spectrum after concentration.

Theorems & Definitions (11)

• Definition 2.1
• Definition 2.2
• Example 2.1
• Definition 2.3
• Example 2.2
• Proposition 2.1
• Proposition 2.2
• Example 2.3
• Proposition 2.3
• Example 3.1