A unified framework for Bell inequalities from continuous-variable contextuality

TL;DR

The paper presents a dimension-agnostic framework that recasts Bell non-locality as contextuality in continuous-variable and hybrid systems, using the contextual fraction $CF(e)$ to quantify nonlocality and to derive an optimal Bell inequality from experimental data. It employs a convergent binning relaxation of an infinite linear program to compute $CF(e)$ and the corresponding Bell functional, enabling practical analysis of CV, DV-CV, and multimode experiments. The results demonstrate both recovery of known CV violations and new instances where CV nonlocality is not CHSH-reducible, including a CGLMP-equivalent CV inequality and high contextual fractions in multimode and GKP-based encodings. The work provides a versatile toolkit for analyzing CV nonlocality in near-term experiments, while raising open questions about the fundamental nature of CV nonlocality and the scalability of the approach.

Abstract

Although the original EPR paradox was formulated in terms of position and momentum, most studies of these phenomena have focused on measurement scenarios with only a discrete number of possible measurement outcomes. Here, we present a framework for studying non-locality that is agnostic to the dimension of the physical systems involved, allowing us to probe purely continuous-variable, discrete-variable, or hybrid non-locality. Our approach allows us to find the optimal Bell inequality for any given measurement scenario and quantifies the amount of non-locality that is present in measurement statistics. This formalism unifies the existing literature on continuous-variable non-locality and allows us to identify new states in which Bell non-locality can be probed through homodyne detection. Notably, we find the first example of continuous-variable non-locality that cannot be mapped to a CHSH Bell inequality. Moreover, we provide several examples of simple hybrid DV-CV entangled states that could lead to near-term violation of Bell inequalities.

Figures (14)

• Figure 1: Top left: description of a general Bell scenario for two parties. Each party performs a measurement among different available settings $a_i$ ($b_i$), and obtain measurement outcomes $o_{a_i}$($o_{b_i}$). Top right: continuous-variable Bell experiment. Each party performs a homodyne detection measurement, setting different quadrature measurements by tuning the phases $\varphi_{A}$($\varphi_{B}$) of their local oscillators. Bottom left: The data obtained from this kind of experiment usually comes in the form of histograms of commuting variables, or contexts. The marginals of contexts sharing variables have to be consistent with each other. The set of all histograms provides an empirical model. Bottom right: We test each empirical model for Bell nonlocality (contextuality), and as a byproduct obtain the optimal Bell inequality for the given empirical model. The plots represent the distribution of values for the filters$\beta_{ij}$ that parametrize the Bell inequality. For the specific example shown, the optimal Bell inequality is equivalent to sign binning followed by CHSH.
• Figure 2: Contextual fraction associated to the measurement statistics of two photon subtracted states from the family described in equation \ref{['eq:two_photon_subtracted_family1']}. In all cases, a search was performed over different configurations of the homodyne detector settings to obtain the maximum contextual fraction possible. For all the non-zero values, they correspond to measuring $\varphi_1\in \{0,\pi/2\}$ and $\varphi_2\in\{-\pi/4,\pi/4\}$.
• Figure 3: On the left the empirical distribution corresponding to the homodyne statistics obtained by measuring settings $\varphi_A\in \{0,\pi/2\}$ and $\varphi_B\in\{-\pi/4,\pi/4\}$, on a two photon subtracted state, from the family described in equation \ref{['eq:two_photon_subtracted_family1']}. This corresponds to the case $r_1=-r_2=0.68$, $\theta=\pi/2$. On the right: the filter functions, $\beta_C$, that define the optimal Bell inequality for the empirical model considered.
• Figure 4: On the left the empirical distribution corresponding to the homodyne statistics obtained by measuring settings $\varphi_A\in \{0,\pi/2\}$ and $\varphi_B\in\{0,\pi/2\}$, on the state described jointly by equations \ref{['eq:Fred_state_general']} and \ref{['eq:Fred_state_basis']}. On the right: the filter functions, $\beta_C$, that define the optimal Bell inequality for the empirical model considered.
• Figure 5: Probability densities for the $q$ quadrature of the GKP code words used to approximate the state in \ref{['eq:entangled_GKP']}.

Theorems & Definitions (2)

• Definition 1
• Definition 2