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

TL;DR

This work develops a universal geometric framework to quantify Bell nonlocality by measuring the distance from a quantum state to the set of local states across all Bell scenarios. It defines multiple contractive distance-based and entropic measures, proves structural results that the closest local state to Werner/isotropic states remains within the same family and, for two-qubit Bell-diagonal states, is Bell-diagonal, enabling explicit calculations. The authors provide exact CHSH nonlocality formulas for Werner and Bell-diagonal two-qubit states and extend the analysis to isotropic and Werner states in higher dimensions via CGLMP, offering both analytical and numerical methods, as well as lower bounds in generic scenarios. These results establish a consistent Hilbert-space perspective on nonlocality, with potential applications to quantum many-body systems and future work on higher-dimensional and continuous-variable extensions.

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 (CHSH) inequality in two-qubit systems and the Collins-Gisin-Linden-Massar-Popescu (CGLMP) 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

Figures (4)

• Figure 1: Normalized geometric measures of CHSH nonlocality for Werner states as a function of the Werner parameter $w$ in the interval $\frac{1}{\sqrt{2}} \leq w \leq 1$. Dashed straight line: Trace and Hilbert-Schmidt measures $\widetilde{\mathcal{M}}_\mathrm{Tr} = \widetilde{\mathcal{M}}_{\mathrm{HS}}$. Dashed-dotted curve: Hellinger and Bures measure $\widetilde{\mathcal{M}}_\mathrm{He} = \widetilde{\mathcal{M}}_\mathrm{Bu}$. Solid curve: Relative entropy measure $\widetilde{\mathcal{M}}_{\mathrm{Re}}$. Dashed-dotted curve: Hellinger measure $\widetilde{\mathcal{M}}_\mathrm{He}$. All plotted quantities are dimensionless.
• Figure 2: Normalized measures of CHSH nonlocality for the convex combination of two Bell states as functions of the convex combination parameter $p$ in the interval $p \geq \frac{1}{2}$. The same behavior holds in the symmetric case $p \leq \frac{1}{2}$, with the maximum achieved in $p=0$. Black dashed curve: Trace measure $\widetilde{\mathcal{M}}_\mathrm{Tr}$. Black solid curve: Relative entropy measure $\widetilde{\mathcal{M}}_{\mathrm{Re}}$. Grey dashed curve: Hellinger measure $\widetilde{\mathcal{M}}_\mathrm{He}$. Grey solid curve: Hilbert-Schmidt measure $\widetilde{\mathcal{M}}_\mathrm{HS}$. All plotted quantities are dimensionless.
• Figure 3: Normalized Hilbert-Schmidt measure $\widetilde{\mathcal{M}}_\mathrm{HS}$ of CHSH nonlocality for Bell-diagonal states as a function of the two free parameters $e_{1}$ and $e_{2}$, with $e_{4}=0$, $e_{3}=1-e_{1}-e_{2}$, and normalization coefficient $\mathcal{M}_\mathrm{HS}^{max}$. The maximum $\widetilde{\mathcal{M}}_\mathrm{HS}=1$ is reached in the three cases $(e_{1},e_{2},e_{3})=(1,0,0)$; $(e_{1},e_{2},e_{3})=(0,1,0)$; and $(e_{1},e_{2},e_{3})=(0,0,1)$. Each case corresponds to a different Bell state. The central region of minima is composed by local states, i.e. states with $\widetilde{\mathcal{M}}_\mathrm{HS}=0$. All plotted quantities are dimensionless.
• Figure 4: Semitransparent tetrahedron: set of Bell-diagonal states. The set is parameterized by the matrix elements $(a_1, a_2, a_3)$ in Eq. (\ref{['BD_matrix']}). Solid figure inside the tetrahedron: set of local states. Solid grey line: set of Werner states.