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

Abstract

We introduce a geometric framework for studying Bell nonlocality in Hilbert space, where, for a given quantum state, nonlocality is quantified by the distance between the state and the set of local states. This approach applies to any Bell inequality and any measurement scenario. Whenever the local set is characterized, the proposed nonlocality measure can be computed explicitly. As a general result, we prove that for any scenario in arbitrary dimension the closest local state to a Werner state is itself a Werner state, and analogously, the closest local state to an isotropic state is again isotropic. In the two-qubit case, we further show that the closest local state to a Bell-diagonal state is Bell-diagonal as well. These structural results are independent of the specific Bell inequality considered, thus revealing intrinsic geometric features of these families of states and providing significant simplifications for computing the proposed measures. For the Clauser–Horne–Shimony–Holt inequality in two-qubit systems and the Collins-Gisin-Linden-Massar-Popescu inequality for two qudits of arbitrary finite dimension, we derive explicit geometric measures of nonlocality for Bell-diagonal, Werner, and isotropic states using various distance metrics, including the trace, Hellinger, Hilbert–Schmidt distances, and relative entropy. Furthermore, we prove in all generality that for all scenarios in which the local set is not fully characterized, the geometric measures provide rigorous lower bounds on nonlocality.