⚛️ quantum physics

A unified framework for data-driven Bell inequalities

This paper introduces a unified, dimension-agnostic framework for data-driven Bell inequalities that enables the discovery of optimal inequalities across discrete, continuous, and hybrid systems, revealing new non-local states and the first continuous-variable example that cannot be mapped to binary inequalities.

Original authors: Carlos Ernesto Lopetegui-González, Gaël Massé, Enky Oudot, Uta Isabella Meyer, Federico Centrone, Frédéric Grosshans, Pierre-Emmanuel Emeriau, Ulysse Chabaud, Mattia Walschaers

Published 2026-10-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Carlos Ernesto Lopetegui-González, Gaël Massé, Enky Oudot, Uta Isabella Meyer, Federico Centrone, Frédéric Grosshans, Pierre-Emmanuel Emeriau, Ulysse Chabaud, Mattia Walschaers

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles can be linked in ways that seem to defy the logic of our everyday experience. This phenomenon, known as non-locality, suggests that two objects can remain connected across vast distances, with a measurement on one instantly influencing the other, regardless of the space between them. For decades, scientists have tested this idea using systems where particles have a limited number of possible states, like a coin that can only be heads or tails. However, the original puzzle that sparked this field of inquiry involved properties that exist on a smooth, continuous scale, such as the position or momentum of a particle, which can take on any value along an infinite line. While the theoretical foundation was laid for these continuous systems, proving that they truly exhibit this strange non-local behavior in a laboratory has remained a stubborn challenge. Most attempts to test them have relied on forcing the continuous data into a binary "yes or no" format, essentially squashing a smooth curve into a few sharp points to make it fit existing tests. This approach, while useful, may have been missing the true nature of the connection.

A team of researchers has now developed a new, flexible framework that allows them to probe non-locality without forcing the data into a pre-defined shape. Instead of starting with a specific test and trying to fit the results to it, they begin with the raw experimental data itself. Imagine a scientist collecting a massive amount of measurements from a quantum experiment, where the results are recorded as a continuous stream of numbers. The researchers take this data and feed it into a sophisticated computer program designed to find the best possible test for that specific set of numbers. The program works by gradually refining how it looks at the data, starting with broad categories and then slicing the data into finer and finer pieces. With each step, it searches for a mathematical rule that proves the data cannot be explained by any local, classical cause. If such a rule is found, it reveals the presence of non-locality and, crucially, tells the scientists exactly what that rule looks like. This method is agnostic to the type of system being studied, meaning it works equally well for systems with discrete states, continuous variables, or a mix of both.

When the team applied this method to continuous systems, they discovered something surprising. Despite the data coming from a smooth, continuous world, the most effective tests they found were inherently discrete. The optimal rules for detecting non-locality in these systems turned out to be simple, step-like functions that grouped the continuous data into distinct buckets, much like sorting a pile of marbles into separate jars. This finding suggests that the non-locality present in these continuous systems is not a fundamentally new type of connection, but rather one that can be fully captured by the same discrete logic used in simpler systems. The researchers also found that they could not find any example of a continuous system that required a complex, smooth rule to prove its non-locality; every successful test they uncovered could be mapped back to a simpler, discrete inequality. This challenges the long-held hope that continuous systems might offer a unique form of non-locality that discrete systems cannot replicate.

The study also identified specific quantum states that are robust enough to show this non-local behavior even when they are not perfect. Using a technique called bosonic encoding, which maps quantum information onto the vibrations of light, the team simulated entangled states that could be detected using standard homodyne detectors, a common tool in optical laboratories. They found that even states with significant imperfections, or "noise," still displayed strong non-locality. For instance, they identified a two-mode state where the connection between the particles was so strong that it could not be described by the standard binary tests used for decades. Instead, the data required a test designed for three-level systems, a more complex version of the usual coin-flip scenario. Furthermore, they showed that by combining a discrete system, like a qubit, with a continuous one, they could create hybrid states that exhibited significant non-locality with relatively simple preparations. These hybrid states are within reach of current experimental technology, suggesting that the next generation of quantum experiments could demonstrate these effects without needing the ultra-precise, ideal conditions previously thought necessary.

The implications of this work extend beyond just finding new examples of non-locality. By providing a tool that automatically generates the best possible test for any given set of experimental data, the researchers have offered a universal method for certifying quantum behavior. This is particularly important for the development of quantum technologies, where proving that a device is truly quantum is essential for security and reliability. The framework allows scientists to take the messy, real-world data from an experiment and determine with high confidence whether it contains non-local correlations, and exactly how strong those correlations are. While the search for these states proved difficult, requiring a vast exploration of possibilities, the team's success in finding robust examples using bosonic codes suggests that the key to unlocking non-locality in continuous systems may lie in how the information is encoded, rather than in the raw complexity of the system itself. The work leaves open the question of whether a truly continuous form of non-locality exists, but it strongly suggests that if it does, it is hiding in a place that current methods have not yet been able to reach.

Technical Summary: A Unified Framework for Data-Driven Bell Inequalities

Problem Statement
The original Einstein-Podolsky-Rosen (EPR) paradox was formulated using continuous variables (CV), yet the vast majority of experimental tests of Bell non-locality and the associated literature focus on discrete variables (DV). While homodyne detection offers a practical route for CV quantum information processing, demonstrating Bell non-locality in these systems remains a significant challenge. Existing approaches typically rely on predefined binning strategies that map continuous outcomes to binary values ( $\pm 1$ ) to apply standard inequalities like CHSH, or they utilize moment-based polynomial inequalities that require a large number of modes to violate local bounds. Furthermore, previous methods often start with a specific Bell inequality without guaranteeing its optimality for the specific measurement scenario or state at hand. There is a lack of a general framework that can derive the optimal Bell inequality directly from experimental data, agnostic to whether the underlying system is CV, DV, or hybrid.

Methodology
The authors propose a unified, data-driven framework to quantify non-locality and derive optimal Bell inequalities for any measurement scenario. The core of the approach relies on the equivalence between Bell non-locality and contextuality, specifically utilizing the contextual fraction (CF) as a quantifier. The CF represents the portion of experimental statistics that cannot be explained by a local hidden-variable model (LHVM).

Optimization Formulation: The problem is formulated as an infinite-dimensional linear optimization program (LP) to find the maximum weight of a local model within the empirical data. The dual of this program yields the optimal Bell inequality (parametrized by filter functions $\beta_C$ ) and the contextual fraction.
Relaxation via Binning: Since the infinite LP cannot be solved exactly, the authors introduce a convergent hierarchy of finite linear programs. Instead of using moment-based relaxations (which they found unfruitful for detecting CV non-locality in their numerical exploration), they employ fine-grained binning of the continuous outcome distributions.
- The continuous outcome space is discretized into $N_b$ bins.
- The hierarchy is constructed by doubling the number of bins at each level ( $N_b = 2^k$ ), ensuring that each level is a coarse-graining of the next.
- This guarantees that the computed contextual fraction converges monotonically to the true value from below, providing a rigorous lower bound on non-locality.
Hybrid Capability: The framework is agnostic to dimensionality. It can treat discrete degrees of freedom exactly while applying the binning hierarchy to continuous degrees of freedom, making it suitable for hybrid DV-CV systems.

Key Contributions

Unified Framework: A single methodology to derive optimal Bell inequalities for CV, DV, and hybrid systems, generalizing existing DV methods to the continuous domain.
Data-Driven Optimization: The framework does not assume a specific inequality (e.g., CHSH). Instead, it takes empirical data as input and outputs the specific Bell inequality that maximally violates the local bound for that data.
Convergent Hierarchy: A numerical relaxation scheme based on binning that provides guaranteed lower bounds on the contextual fraction, converging to the continuous optimum as bin resolution increases.

Key Results
The authors applied their framework to several experimentally motivated scenarios:

Insights into CV Non-locality:
- Failure of Moment-Based Approaches: Extensive numerical searches using moment-based semidefinite programming (SDP) hierarchies failed to find relevant Bell inequalities for CV systems within practical polynomial degrees, whereas binning-based approaches succeeded.
- Discrete Nature of Optimal Inequalities: In all CV examples where non-locality was detected, the resulting optimal Bell inequalities were found to be effectively discrete (piecewise constant functions), often mapping to standard binary inequalities like CHSH. This suggests that homodyne-based non-locality may possess an intrinsically discrete structure.
- Robustness of GKP States: Using bosonic codes (GKP states), the authors found that even low-quality states (with significant damping and few peaks) suffice to observe significant non-locality. For example, entangled GKP states with damping parameters corresponding to $\sim 8.5$ dB squeezing showed a contextual fraction approaching the Tsirelson bound ( $\approx 0.41$ ).
- Qutrit Example: Using qutrit-encoded GKP states, the authors identified a two-mode bosonic state where the optimal Bell inequality is not equivalent to CHSH but corresponds to the CGLMP inequality (a qutrit inequality). This is the first example of two-mode CV non-locality requiring a non-binary inequality.
- Three-Mode Systems: For three-mode GHZ-like states encoded with GKP codewords, the framework detected a contextual fraction of up to 60.5%, significantly exceeding the two-mode limit.
Hybrid DV-CV Systems:
- The framework was applied to hybrid states (e.g., a qubit entangled with a CV mode).
- Simple hybrid states, such as those involving squeezed cat states or GKP encodings, were found to exhibit significant non-locality (contextual fractions of 15.6% to 40%) under homodyne detection.
- These states are within the reach of near-term experiments on platforms like circuit QED and ion traps.

Significance and Claims
The paper claims to provide a general, versatile tool for the experimental certification of non-locality. Its primary significance lies in:

Unification: It bridges the gap between CV and DV non-locality studies, allowing for a coherent analysis of hybrid systems.
Optimality: By deriving the inequality from the data, it ensures that the observed violation is the maximum possible for the given experimental constraints, rather than being limited by a pre-chosen inequality.
Practicality: It identifies that robust non-locality can be observed in CV systems using relatively simple, low-quality states (e.g., cat states or imperfect GKP states) and standard homodyne detection, suggesting a path toward near-term experimental realization.
Fundamental Insight: The results raise the question of whether CV non-locality under homodyne detection is fundamentally discrete, as all optimal inequalities found were discrete in nature. The authors conjecture that adding more measurement settings per mode does not necessarily increase the contextual fraction, though this remains an open question.

The authors note that while the method is powerful, it faces computational scaling limitations with the number of modes due to the exponential growth of the binning space, currently restricting high-resolution analysis to two or three modes.

Technical Summary: A Unified Framework for Data-Driven Bell Inequalities

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