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We introduce a geometric framework for the study of Bell nonlocality inHilbert space, aiming to provide intrinsic measures based solely on theproperties of quantum states for a given Bell inequality. Recognizing thatnonlocality is inherently measurement-dependent, our approach focuses on fixinga specific Bell scenario -- that is, a chosen number of measurements andoutcomes per party, along with a specific Bell inequality -- and, for eachquantum state, maximizing the violation over all possible measurement choiceswithin the given scenario. We then characterize the geometry of thecorresponding set of local states, defined as those states for which noviolation occurs for any choice of the measurements. Within this framework, wedefine geometric measures of nonlocality based on the distance of a givenquantum state from the set of local states, using quantities such as the tracedistance, the Hilbert-Schmidt distance, and the relative entropy. We firstestablish the general formalism, emphasizing the challenges posed by theNP-hardness of characterizing local sets for arbitrary Bell scenarios. We thenspecialize to the case of the Clauser-Horne-Shimony-Holt (CHSH) inequality intwo-qubit systems, where the local set is fully characterized and an explicitgeometric analysis is possible. In this setting, we derive geometric measuresof nonlocality for Bell-diagonal and Werner states, showing that the localstate closest to a Werner state is a Werner state, and analogously forBell-diagonal states. For each quantum state, our results offer a metric-basedcharacterization of Bell nonlocality that is independent of specificexperimental implementations and naturally extend to different Bell scenarioswhenever the corresponding set of local states is accessible.

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