Geometric measures of CHSH quantum nonlocality: characterization, quantification, and comparison by distances and operations

Abstract: We introduce a geometric framework for the study of Bell nonlocality in Hilbert space, aiming to provide intrinsic measures based solely on the properties of quantum states for a given Bell inequality. Recognizing that nonlocality is inherently measurement-dependent, our approach focuses on fixing a specific Bell scenario -- that is, a chosen number of measurements and outcomes per party, along with a specific Bell inequality -- and, for each quantum state, maximizing the violation over all possible measurement choices within the given scenario. We then characterize the geometry of the corresponding set of local states, defined as those states for which no violation occurs for any choice of the measurements. Within this framework, we define geometric measures of nonlocality based on the distance of a given quantum state from the set of local states, using quantities such as the trace distance, the Hilbert-Schmidt distance, and the relative entropy. We first establish the general formalism, emphasizing the challenges posed by the NP-hardness of characterizing local sets for arbitrary Bell scenarios. We then specialize to the case of the Clauser-Horne-Shimony-Holt (CHSH) inequality in two-qubit systems, where the local set is fully characterized and an explicit geometric analysis is possible. In this setting, we derive geometric measures of nonlocality for Bell-diagonal and Werner states, showing that the local state closest to a Werner state is a Werner state, and analogously for Bell-diagonal states. For each quantum state, our results offer a metric-based characterization of Bell nonlocality that is independent of specific experimental implementations and naturally extend to different Bell scenarios whenever the corresponding set of local states is accessible.