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Colliders are not Testing Locality via Bell's Inequality nor Providing an Unconditional Proof of Entanglement

The paper argues that collider experiments cannot provide unconditional proofs of entanglement or test locality via Bell's inequalities because the measurement of commuting final-state momenta allows for the construction of a local hidden variable theory that reproduces the observed data while satisfying Bell's inequality.

Original authors: Steven A. Abel, Herbi K. Dreiner, Rhitaja Sengupta, Lorenzo Ubaldi

Published 2026-08-13

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Steven A. Abel, Herbi K. Dreiner, Rhitaja Sengupta, Lorenzo Ubaldi

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

Imagine you are trying to figure out if the universe is a giant, interconnected web where things happen instantly across vast distances, or if it's more like a collection of independent actors who only know what's right in front of them. This is the heart of a famous debate in physics between "locality" (things only affect their immediate neighbors) and "quantum entanglement" (particles can be linked in a spooky way that defies distance). To test this, scientists use a mathematical rule called Bell's Inequality. Think of it like a strict traffic law: if the universe follows the rules of local, independent actors, the traffic flow must obey this law. If the traffic breaks the law, it proves the universe is using that spooky, interconnected quantum magic. For decades, experiments with light particles (photons) have broken this law, suggesting the universe is indeed quantum and non-local. But recently, physicists at giant particle smashers called colliders (like the Large Hadron Collider) have been trying to run these same tests using heavy particles like top quarks and tau leptons, hoping to see if the quantum magic holds up at super-high energies.

Now, here comes a new paper that acts like a skeptical detective arriving at the crime scene. The authors, a team of physicists, argue that the recent attempts to test Bell's Inequality at these colliders are actually looking at the wrong clues. They claim that no matter how hard you try, you cannot prove the universe is non-local or that particles are entangled just by measuring the speeds and directions of the debris flying out of a collision. Why? Because the tools they are using—the measuring sticks of momentum—can be perfectly mimicked by a "local" explanation. It's like trying to prove a magician is using telepathy by watching them shuffle a deck of cards, only to realize that a simple, local trick (like a hidden compartment) could explain the shuffle just as well. The paper suggests that the "spooky" correlations seen in collider data might just be a clever illusion created by the way we measure things, not a fundamental feature of reality.

The Paper's Big Reveal: The "Momentum" Trap

The paper, titled "Colliders are not Testing Locality via Bell's Inequality nor Providing an Unconditional Proof of Entanglement," delivers a rather deflating message for those hoping to see quantum weirdness confirmed at the highest energies. The authors argue that every proposal to test Bell's Inequality at colliders—whether it involves top quarks, tau leptons, or Higgs bosons—fails to provide a true test. They claim that the data collected from these machines can always be explained by a "Local Hidden Variable Theory" (LHVT).

To understand this, imagine a pair of twins separated at birth. In a "quantum" world, they might share a secret code that lets them instantly coordinate their actions no matter how far apart they are. In a "local" world, they might have been given a matching instruction manual (the hidden variable) before they left home. If you see them doing the same thing, you can't tell if it's magic or just the manual. The authors say that in collider experiments, we are only measuring the momentum (the speed and direction) of the particles after they fly apart. Because momentum is a "commuting" quantity (a fancy way of saying you can measure all its parts at once without them messing each other up), the math of the collision naturally produces a pattern that looks like a matching instruction manual.

The authors use a clever construction, originally proposed by a physicist named Kasday and adapted for colliders, to show that you can build a "local" model that perfectly copies the data. It's like building a fake magic trick that uses a hidden compartment instead of telepathy. Since the data fits this "local" model perfectly, you can't claim you've proven the "telepathy" (entanglement) exists. The paper states that the differential cross-section (the mathematical description of how often particles fly out at certain angles) is the local hidden variable theory. Therefore, the experiment is circular: you assume quantum mechanics to interpret the data, and then use that data to "prove" quantum mechanics.

Why the "Spooky" Tests Fail

The paper walks through several specific scenarios where scientists have tried to find entanglement, and explains why each one falls flat.

1. The Tau Lepton Trap (LEP and LHC)
First, they revisit an old experiment at the LEP collider involving tau leptons decaying into pions. They show that the angle between the pions follows a pattern that satisfies Bell's Inequality. This means the data can be explained by a local theory. Even though the tau particles are entangled in the Standard Model, the measurement of the pions' momenta can be described by a local script. The authors argue that extracting the "spin" information (the quantum part) from the "momentum" data requires assuming quantum mechanics in the first place, which makes the test invalid.

2. The Top Quark Showdown (LHC)
This is a hot topic. The ATLAS and CMS experiments at the LHC have claimed to see evidence of entanglement in top quark pairs. They measure the angles of the leptons produced when the top quarks decay. The authors argue that the distribution of these angles is just a function of the momenta, which commute. Just like with the taus, you can build a local model that fits this distribution perfectly.
The paper points out a specific logical loop: To claim entanglement, scientists calculate a number called $D$ from the data and compare it to a theoretical limit. But to get that number from the data, they must assume the Standard Model (which is a quantum theory) is correct. If you assume the theory to get the number, and then use the number to prove the theory, you aren't testing anything. The authors show that their local model produces the same $D$ value as the experiment, proving that a theory without entanglement can explain the results just as well.

3. The Higgs Boson Hurdle
The paper also looks at the Higgs boson decaying into pairs of W or Z bosons. These are spin-1 particles, which have three possible spin states, making Bell's Inequality (designed for two-state systems) tricky to apply directly. Even if scientists try to filter the data to only look at "transverse" (side-to-side) spins, the authors argue this filtering acts like a "cut" on the data.
This leads to a "detection loophole." Imagine you are counting people entering a room, but you only count those wearing red hats. If you find a pattern, it might be because of the red hats, not because of magic. Similarly, cutting the data to isolate specific spins introduces artificial correlations that can fake a violation of Bell's Inequality. The authors show that even with these cuts, the underlying data can still be described by a local, separable theory.

The "Cut" That Breaks the Rules

One of the most vivid parts of the paper is the explanation of how "momentum cuts" can create fake violations of Bell's Inequality. The authors describe a scenario where you try to isolate specific particle behaviors by rejecting data that doesn't fit a certain angle (a "cut").
They explain that rejecting data is effectively adding a third outcome: "no click" or "rejected." In a true Bell test, you need to account for all possibilities. If you throw away the "rejected" events, you might accidentally create a pattern that looks like a violation of the rules, even if the underlying physics is perfectly local. It's like a magician who only shows you the tricks that work and hides the ones that fail, making the act look impossible. The paper argues that in collider physics, these cuts are often necessary to isolate the signal, but they ruin the purity of the Bell test, allowing local theories to mimic quantum violations.

The Final Verdict

The authors conclude with a "no-go theorem." They state that as long as we are only measuring the final momenta of particles at colliders, we cannot test for locality or prove entanglement unconditionally. The data can always be described by a local, separable theory where the particles are just following a pre-written script (the hidden variables) rather than communicating instantly.

They are not saying entanglement doesn't exist; they are saying that collider experiments, as currently designed and analyzed, cannot prove it. The "entanglement" seen in the data is consistent with the Standard Model, but it is also consistent with a local, non-entangled explanation. To truly test Bell's Inequality, you would need to measure non-commuting properties (like different spin components) directly, which is impossible for free particles in a collider environment.

In short, the paper tells us that the colliders are not the magic mirrors we thought they were. They are showing us a reflection that looks quantum, but it's a reflection that a local theory can also produce. Until we find a way to measure the "spooky" parts without relying on assumptions that already include the "spooky" parts, the question of whether the universe is truly non-local at these high energies remains unanswered by these experiments. The authors urge caution against claiming a "proof" of entanglement when the data could just as easily be explained by a local, classical story.

Technical Summary: Colliders are not Testing Locality via Bell's Inequality nor Providing an Unconditional Proof of Entanglement

Problem Statement
Recent literature has proposed using high-energy collider experiments, particularly at the Large Hadron Collider (LHC) and LEP, to test the principle of locality via Bell's inequalities and to provide unconditional evidence of quantum entanglement. Proposed processes include top-quark pair production ( $t\bar{t}$ ), tau-lepton pair production ( $\tau^+\tau^-$ ) from $Z^0$ or Higgs decays, and Higgs boson decays to vector bosons ( $H \to VV^*$ ). These proposals typically rely on measuring the angular correlations of final-state particle momenta (e.g., leptons or pions) to infer underlying spin correlations. The central problem addressed by this paper is whether these momentum-based measurements can genuinely distinguish between quantum mechanical entanglement and local hidden variable theories (LHVTs), or if they inherently suffer from logical circularity and methodological flaws.

Methodology
The authors employ a theoretical construction originally proposed by Kasday (1971) and previously adapted to collider contexts by Abel, Dittmar, and Dreiner (1992). The core of their methodology rests on the observation that collider experiments measure the 4-momenta of final-state particles. Crucially, all momentum components commute.

LHVT Construction: The authors demonstrate that because the measured observables (momenta) commute, the differential cross-section itself can be interpreted as a Local Hidden Variable Theory (LHVT). In this framework, the hidden variables are identified directly with the unit momentum vectors of the decay products.
Numerical Validation: Using MadGraph5 aMC@NLO, the authors perform numerical simulations for specific benchmark processes at the LHC (14 TeV) and LEP. They generate the full differential cross-sections for:
- $e^+e^- \to Z^0 \to \tau^+\tau^- \to \pi^+\pi^-\nu\bar{\nu}$
- $pp \to H \to \tau^+\tau^- \to \pi^+\pi^-\nu\bar{\nu}$
- $pp \to t\bar{t} \to (b\ell^+\nu)(\bar{b}\ell^-\bar{\nu})$
- $pp \to H \to ZZ^* \to \ell^+\ell^-\ell'^+\ell'^-$ and $H \to WW^* \to \ell\nu\ell'\bar{\nu}$
Inequality Testing: They normalize these differential cross-sections to form correlation functions $P(\vec{a}, \vec{b})$ and test them against the Bell inequality and the CHSH inequality.
Analysis of Cuts: The authors investigate the effect of phase-space cuts (e.g., isolating transverse polarizations or specific invariant mass regions) on these inequalities, analyzing how data rejection (the "detection loophole") can artificially induce violations.

Key Contributions and Results

The LHVT Equivalence: The paper establishes that for any process where the final state involves commuting observables (momenta), the normalized differential cross-section constitutes an LHVT. By construction, this LHVT is local and generates separable correlations. Consequently, it necessarily satisfies Bell's and CHSH inequalities for all angles.
Numerical Confirmation: The authors' numerical simulations for $t\bar{t}$ , $\tau^+\tau^-$ , and Higgs decay channels confirm that the resulting correlation functions satisfy the CHSH inequality. For instance, in the $t\bar{t}$ near-threshold region, the fitted correlation function $P_{QM} = \frac{1}{2}(1 + 0.54 \cos \theta_{\ell\ell})$ satisfies the inequality, contradicting claims in the literature that these specific kinematic regions violate locality.
The Circularity of Entanglement Tests: The paper argues that claims of observing entanglement (e.g., via the Peres-Horodecki criterion applied to the spin density matrix) are logically circular in this context. To extract the spin density matrix parameters ( $B^\pm, C$ ) from the measured angular distributions of momenta, one must assume the validity of Quantum Field Theory (specifically the Standard Model's analyzing power). If one assumes QM to derive the spin correlations, one cannot then use those correlations to test QM.
Separability of Momentum Correlations: The authors show that the measured momentum correlations can be fully described by a classical, local, and separable function. Therefore, no "unconditional" proof of entanglement is possible at colliders; one can only show that the Standard Model (which includes entanglement) describes the data, but a separable LHVT describes the data equally well.
The Role of Momentum Cuts: The paper demonstrates that applying momentum cuts to isolate specific spin states (e.g., transverse vector bosons) introduces a "detection loophole." By rejecting events that do not meet specific kinematic criteria, the effective probability distribution is altered. The authors show that such cuts can artificially create apparent violations of Bell's inequalities, even when the underlying process is fully described by a local theory. This is a manifestation of adding a third outcome (rejection) to a binary measurement problem.
Vector Boson Decays: For Higgs decays to massive vector bosons ( $H \to VV^*$ ), the authors note that Bell's inequality does not directly apply because massive spin-1 particles have three spin states, not two. While the CGLMP inequality exists for higher spins, it applies to spin measurements, not the angular correlations of momenta measured at colliders. Even when restricting to transversely polarized bosons (2 states), the resulting momentum correlations remain an LHVT satisfying Bell's inequality.

Significance and Claims
The paper claims to establish a "no-go theorem" for testing locality via Bell's inequalities at colliders. The authors argue that:

Locality cannot be tested: Because the measured observables (momenta) commute, a local hidden variable theory can always be constructed to reproduce the data exactly. Thus, a violation of Bell's inequality is impossible in these setups, and no test of locality can be performed.
Entanglement cannot be unconditionally proven: Since the data can be described by a separable LHVT without invoking quantum entanglement, collider experiments cannot provide unconditional observational evidence of entanglement. Any claim of entanglement relies on the prior assumption of the Standard Model's quantum mechanical framework.
Critique of Current Proposals: The paper asserts that recent proposals and measurements (e.g., by ATLAS and CMS) claiming to test locality or prove entanglement are flawed due to the circular logic of assuming QM to extract spin correlations and the failure to account for the fact that momentum correlations are inherently local and separable.

The authors conclude that while the Standard Model consistently describes the data with entangled states, the experimental setup at colliders is fundamentally incapable of distinguishing between these quantum predictions and a classical, local, separable explanation based solely on momentum measurements.

The Paper's Big Reveal: The "Momentum" Trap

Why the "Spooky" Tests Fail

The "Cut" That Breaks the Rules

The Final Verdict

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