⚛️ quantum physics

Extending Bell's Theorem: Nonlocality via Measurement Dependence

This paper demonstrates that by imposing a no-signalling condition, a version of Bell's theorem can be established without assuming Measurement Independence, thereby showing that certain violations of this assumption are testable in principle and associated with signalling.

Original authors: G. Bacciagaluppi, R. Hermens, G. Leegwater

Published 2026-02-13

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: G. Bacciagaluppi, R. Hermens, G. Leegwater

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

The Big Picture: The Cosmic Coin Flip

Imagine Alice and Bob are two friends on opposite sides of the world. They each have a magic coin. Every time they flip their coins, the results are perfectly synchronized. If Alice gets "Heads," Bob gets "Heads." If Alice gets "Tails," Bob gets "Tails." This happens even if they flip the coins at the exact same time, faster than a signal could travel between them.

In physics, this is called quantum entanglement. It's spooky because it seems to break the rule that nothing can travel faster than light.

For decades, physicists have tried to explain this "spooky action" using a hidden script (called Hidden Variables) that the coins carry with them from the start. The famous Bell's Theorem proved that if this script exists, it must obey three rules:

Locality: Alice's coin can't instantly affect Bob's coin.
Realism: The coins have a definite state before they are flipped.
Measurement Independence (MI): The script the coins carry is totally independent of how Alice and Bob decide to flip them. (i.e., The coins don't "know" or "care" if Alice chooses to flip them with her left hand or right hand).

Experiments have shown that the universe violates Bell's inequalities. This means at least one of those three rules is broken. Most physicists assume the "Locality" rule is broken (the coins talk to each other instantly).

This paper asks a different question: What if the "Locality" rule is actually fine, but the Measurement Independence rule is broken? What if the coins do know how Alice and Bob are going to flip them?

The Core Idea: The "Conspiracy" vs. The "Signal"

Usually, when people suggest that the coins know the flip settings, they call it a "conspiracy." Imagine a scenario where the universe is rigged so that every time Alice decides to use her left hand, the universe secretly arranges for the coins to have a specific script that matches that choice. It feels like a cosmic joke where everything is pre-planned to trick us.

The authors of this paper say: "Wait a minute. Let's not just call it a conspiracy. Let's see if we can actually use this connection to send a message."

They introduce a concept called "Signalling in Principle."

The Analogy: The Magic Dice Shop

Imagine Charlie runs a shop that sells these magic dice to Alice and Bob.

Standard Quantum Mechanics: Charlie sells dice that are perfectly random. No matter how Alice or Bob decide to roll them, the dice don't care.
The "Conspiracy" View: Charlie secretly knows what Alice and Bob will do, and he only gives them dice that match. But he does this so perfectly that Alice and Bob can't tell the difference. It looks random to them.
The "Signalling" View (This Paper): The authors say, "What if Charlie has a special batch of dice? These dice are 'out of equilibrium.' They are rigged in a way that if Alice chooses a specific setting, the dice actually tilt the odds for Bob."

If Charlie can prepare these special "out-of-equilibrium" batches of dice, then Alice could change her setting, and Bob would immediately see a change in his results. Alice could send a message to Bob.

The Main Discovery: Three Types of "Spooky" Connections

The paper argues that there are actually three different ways the universe can be "spooky," and we can tell them apart by checking if a signal can be sent.

Type 1: The Silent Correlation (Outcome Independence Violated)
- Analogy: Alice and Bob have two coins that are magically linked. When Alice flips hers, Bob's flips the same way.
- Can they signal? No. Even though they are linked, Alice can't force her coin to be "Heads" to tell Bob a message. The link is there, but it's silent. This is what standard Quantum Mechanics looks like.
Type 2: The Instant Shout (Parameter Independence Violated)
- Analogy: Alice and Bob are connected by a super-fast, invisible telephone wire. If Alice changes her setting, she instantly shouts to Bob, changing his result.
- Can they signal? Yes. But there's a catch: To send a message, Alice has to speak before Bob listens. It requires a specific order of time (Alice acts, then Bob reacts). This is "Action at a Distance."
Type 3: The Timeless Whisper (Measurement Independence Violated)
- Analogy: This is the new idea. Imagine the dice are rigged so that the choice of how to flip them is baked into the dice's history.
- Can they signal? Yes. And here is the kicker: They don't need a specific time order. Alice doesn't have to act "before" Bob. Because the connection is based on a shared history or a global constraint (like a pre-written script that depends on the future), the signal can work in a way that feels like it's going "backwards and forwards" in time.
- The Catch: To make this work, Charlie (the source) must be able to prepare these special "non-quantum" batches of dice. If he can't, the signal remains theoretical.

The "Schulman Model": A Concrete Example

To prove this isn't just math, the authors look at a specific model called the Schulman Model.

Imagine a particle that has to make a "choice" of how to rotate between two measurements.
In this model, the particle's rotation depends on the future measurement settings.
Usually, this looks like a perfect quantum system. But the authors show that if you have a source that can produce a "biased" version of this system (where the particles aren't perfectly balanced), Alice and Bob could use that bias to send a secret message.

Why Does This Matter? (Experimental Metaphysics)

The authors call this "Experimental Metaphysics." They are saying:

"We don't just need to know that the universe is weird. We need to know how it is weird."

If we find a way to prepare these special "non-quantum" ensembles (the rigged dice), we could test the universe:

If we can send a signal without a time order, we know the universe violates Measurement Independence.
If we can only send a signal with a time order, we know it violates Locality (Action at a Distance).

The Bottom Line

This paper extends Bell's Theorem. It says:

Don't just dismiss "Measurement Independence" violations as conspiracies.
Check if they allow for "Signalling in Principle."
If they do, it's a new, distinct form of non-locality. It's "milder" than instant action-at-a-distance (because it might fit with relativity in a weird, zig-zag way) but "stronger" than silent correlations (because it allows communication).

In short: The universe might not be breaking the speed limit by shouting across the room (Action at a Distance). Instead, it might be that the room itself is built in a way where the walls "know" what you're going to say before you say it (Measurement Dependence). And if we can find the right tools, we might be able to use that knowledge to whisper a secret message across the room.

1. Problem Statement

Standard derivations of Bell's inequalities rely on three core assumptions:

Outcome Independence (OI): The outcome of one measurement is independent of the outcome of the other, given the hidden variables.
Parameter Independence (PI): The outcome of one measurement is independent of the setting (context) of the distant measurement, given the hidden variables.
Measurement Independence (MI): The distribution of hidden variables ( $\lambda$ ) at the source is statistically independent of the future measurement settings chosen by Alice and Bob ( $I, J$ ).

While experimental violations of Bell inequalities are widely accepted as proof of nonlocality (violations of OI or PI), some theoretical approaches attempt to preserve locality by rejecting Measurement Independence (often termed "superdeterminism" or "retrocausality"). These models suggest that the hidden variables and measurement settings share a common cause or are correlated via retrocausal influences.

The Core Problem:
If MI is violated, Bell inequalities can be violated without requiring action-at-a-distance (violations of PI). However, critics often dismiss such models as "conspiratorial" or untestable because the correlation between settings and hidden variables is assumed to be unobservable. The authors ask: Can violations of MI be distinguished from violations of PI or OI in a way that makes them operationally testable? Specifically, can a violation of MI lead to "signalling in principle" (the ability to transmit information) even if PI and OI hold?

2. Methodology

The authors employ a rigorous mathematical framework based on hidden variable models for bipartite spin-1/2 systems. Their methodology involves:

Operational Definition of Signalling: They define "signalling in principle" not by assuming a "sub-quantum demon" (an agent with direct access to hidden variables), but by the existence of non-quantum ensembles. These are ensembles where the distribution of hidden variables deviates from the standard quantum equilibrium (Born rule) but is dynamically allowed by the theory.
The Equiprobability Theorem: They utilize and extend theorems (originally by Squires, Hardy, Brown, and Colbeck-Renner) stating that if a model satisfies PI and MI and reproduces quantum correlations for maximally entangled states, the marginal probabilities for individual spin measurements must be equiprobable (i.e., $\langle A \rangle = 0$ ).
Chain of Correlations: They construct proofs involving chains of spin measurements at small angular intervals to bound the deviation from equiprobability.
Sub-Ensembles: They distinguish between theoretical sub-distributions and operationally preparable sub-ensembles. Their results hinge on the assumption that a source (Charlie) can reliably prepare these non-quantum sub-ensembles.

3. Key Contributions

A. Formalizing "Signalling in Principle" for MI Violations

The paper establishes that if a hidden variable model violates MI (while satisfying OI and PI), and if the model allows for the preparation of non-quantum ensembles (ensembles where hidden variables are not distributed according to the quantum state), then signalling is possible in principle.

Unlike violations of PI (action-at-a-distance), which require a preferred time ordering for signalling, violations of MI allow signalling without a preferred time order.
This makes the violation of MI operationally distinguishable from the violation of PI.

B. Extension of Bell's Theorem

The authors prove a generalized version of Bell's theorem (Theorems 2 and 2') that does not assume MI.

Theorem 2: If an ensemble violates Bell inequalities (specifically the CHSH inequality) and satisfies OI, and if the correlations along a chain of spin directions are sufficiently high, then if Charlie can prepare appropriate sub-ensembles, at least one party (Alice or Bob) can signal.
This implies that No-Signalling (the prohibition of faster-than-light communication) is a sufficient condition to derive Bell inequalities, even without assuming MI. If a theory respects No-Signalling, it cannot violate Bell inequalities unless it violates MI and prevents the preparation of non-quantum ensembles.

C. Analysis of the Schulman/Wharton Model

The paper analyzes the "Schulman model" (generalized by Wharton), a retrocausal hidden variable model that violates MI to reproduce quantum mechanics.

They demonstrate that while the standard quantum predictions of this model satisfy MI (in the sense of reproducing the Born rule), the model's underlying dynamics allow for non-quantum distributions (e.g., ensembles where hidden spins are aligned in specific ways).
If such non-quantum ensembles can be prepared, the Schulman model would allow for signalling, thereby making the violation of MI testable.

4. Key Results

Theorem 1 & 1': If a hidden variable model satisfies PI and OI but reproduces quantum joint probabilities, and if the marginal expectation value $\langle A \rangle \neq 0$ (violation of equiprobability), then signalling is possible.
Theorem 2 & 2': Experimental violation of Bell inequalities implies the existence of sub-ensembles that allow signalling, provided:
- OI holds.
- Correlations along a chain of spin directions are high (bounded by $\gamma$ ).
- The Bell violation is large enough ( $\epsilon \geq 4n\gamma$ ).
- Crucially: The source can prepare these specific non-quantum sub-ensembles.
Distinction of Nonlocality: The paper categorizes nonlocality into three operationally distinct types:
- Correlations without signalling: Standard quantum mechanics (violates OI/PI in a way that prevents signalling).
- Signalling with preferred time order: Violations of PI (Action-at-a-distance, e.g., Pilot-wave theory).
- Signalling with no preferred time order: Violations of MI (Retrocausality/Measurement Dependence).

5. Significance and Implications

Testability of Superdeterminism: The paper challenges the view that "superdeterministic" or retrocausal models are inherently untestable. It argues that if such theories allow for non-equilibrium (non-quantum) ensembles, they predict observable signalling, making them falsifiable in principle.
Experimental Metaphysics: The work extends Shimony's program of "experimental metaphysics." It suggests that by testing for the absence of signalling in specific non-quantum ensembles, one can constrain the space of viable hidden variable theories, even those that violate MI.
Refutation of "Conspiracy" Loophole: While standard "conspiracy" models (where settings are correlated with hidden variables via common causes) are ruled out by "Big Bell" and "Cosmic Bell" experiments, this paper shows that systematic, law-like violations of MI (e.g., retrocausality) are not ruled out by those experiments alone. However, they are ruled out if one assumes the No-Signalling principle holds universally.
Bohr and Everett: The authors briefly discuss how interpretations like Bohr's (contextual) or Everett's (many-worlds) might formally violate MI but do not allow signalling because the hidden variables are not causally dependent on settings in a way that allows control. This highlights that the mechanism of the MI violation (causal vs. formal) determines testability.

Conclusion

The paper provides a robust theoretical framework showing that Measurement Independence is not just a philosophical assumption but a physical constraint with operational consequences. If a theory violates MI to explain Bell violations, it must either:

Allow for the preparation of non-quantum ensembles, leading to signalling in principle (and thus being testable/falsifiable).
Or, strictly forbid such ensembles, which implies a specific structure to the theory (e.g., global constraints or specific boundary conditions) that prevents the extraction of information.

By linking the violation of MI to the possibility of signalling, the authors extend Bell's theorem to cover a broader class of local hidden variable theories, offering a new pathway to experimentally distinguish between different forms of quantum nonlocality.