⚛️ quantum physics

A Appropriate Probability Model for the Bell Experiment

This paper proposes an explicit probability model for the CHSH Bell experiment that, by treating quantum expectations as conditional on observable detector settings and avoiding assumptions of realism, reconciles experimental results with the Bell inequality while demonstrating that non-separability implies non-determinism or non-locality.

Original authors: Kees van Hee, Kees van Berkel, Jan de Graaf

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kees van Hee, Kees van Berkel, Jan de Graaf

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 Mystery: The "Spooky" Coin Flip

Imagine you have two magical coins. You give one to Alice in New York and one to Bob in London. These coins are "entangled," meaning they are magically linked.

Here is the weird part:

When Alice flips her coin, it lands on Heads or Tails completely at random.
The moment she looks at it, Bob's coin instantly decides what it will be, even though he is thousands of miles away.
If Alice chooses to flip her coin in a specific way (let's call it "Setting A"), and Bob chooses "Setting B," their results match a very specific, weird pattern predicted by quantum physics.

For decades, physicists have been arguing about this. They say: "This is impossible! If the coins are far apart, Alice's choice shouldn't affect Bob's coin instantly. This violates the rules of 'locality' (things only affect what's right next to them) and 'realism' (the coins must have a definite state before we look at them)."

This argument is called the Bell Experiment. The math says the coins should behave one way, but the experiments show they behave another way. This is the "Bell Contradiction."

The Authors' New Idea: The "Menu" vs. The "Order"

The authors of this paper, Kees van Hee, Kees van Berkel, and the late Jan de Graaf, say: "Stop arguing about the physics. The problem is with the math we use to describe it."

They argue that previous models made a silly mistake, like asking a restaurant chef a question that can't be answered.

The Mistake: The "Impossible Menu"

Imagine a restaurant where you can order a burger or a pizza, but never both at the same time.

The Old Model: The old math assumed that even if you only ordered a burger, the kitchen already had a specific, pre-determined pizza ready to go, and a specific burger, and a specific salad, all sitting on the counter waiting for you. It assumed that every possible order existed simultaneously, even the ones you didn't pick.
The Problem: In the real quantum world, the "pizza" and the "burger" (the two different detector settings) cannot exist at the same time. You can only pick one setting per experiment.

The old math tried to add up the results of the burger, the pizza, and the salad all together to prove a contradiction. But you can't add up things that don't exist simultaneously!

The New Model: The "Conditional Order"

The authors propose a new way to look at the math. They say:

We should only calculate the probability of the result given the specific setting we actually chose.
Instead of asking, "What would the result be if we chose A, B, C, and D all at once?" (which is impossible), we ask, "What is the result if we chose A and B?"

The Analogy:
Think of the Bell experiment like a weather forecast.

The Old Way: Trying to predict if it will rain, snow, and be sunny all at the same time to prove a contradiction.
The New Way: Asking, "If it is raining, what is the chance of a flood?" and "If it is sunny, what is the chance of a drought?"
When you look at the results this way, the "contradiction" disappears. The quantum predictions fit perfectly with the experiments. The "violation" of the rules was just a math error caused by asking the wrong question.

The Hidden Variable: The "Secret Recipe"

The paper also tackles the idea of Hidden Variables. This is the idea that maybe the coins aren't random at all; maybe they have a secret "recipe" (a hidden variable) that determines the outcome, and we just don't know it yet.

The authors extended their new model to include this secret recipe. They proved that:

If you try to use a secret recipe to explain the results, the math breaks down.
The "recipe" cannot be separated into two independent parts (one for Alice, one for Bob).
This means the universe is either not deterministic (the outcome is truly random, not pre-written) OR not local (Alice and Bob are connected in a way that defies distance), or both.

The Takeaway

What did they actually solve?
They didn't prove that quantum mechanics is "spooky" or that Einstein was wrong about reality. Instead, they showed that the mathematical tool used to prove the contradiction was flawed.

Before: "The math says the universe is broken because the coins behave strangely."
Now: "The math was broken because we tried to add up things that can't be added. When we fix the math to only look at what we actually measure, the universe makes perfect sense."

In a nutshell:
The Bell experiment isn't a paradox that breaks physics; it's a puzzle that was solved by realizing we were trying to solve it with the wrong set of rules. The universe is consistent; our old probability model just had a blind spot.

1. Problem Statement

The paper addresses the Bell contradiction, a long-standing paradox in quantum physics where the predictions of quantum mechanics (specifically for entangled particles) appear to violate the Bell-CHSH inequality.

The Conflict: Standard derivations of the Bell inequality assume that measurement outcomes for all possible detector settings exist simultaneously (Realism) and that one detector's setting does not influence the other's outcome (Locality).
The Paradox: Quantum mechanical calculations and experimental results (e.g., Aspect et al.) yield expectation values that sum to $2\sqrt{2}$ (Tsirelson's bound), violating the classical limit of 2.
The Authors' Hypothesis: The authors argue that the contradiction arises not from a failure of realism or locality, but from an implicit and incorrect probability model used in standard derivations. They posit that standard models incorrectly treat unmeasured detector settings as existing random variables, whereas the physical reality only involves two simultaneously observable settings per measurement.

2. Methodology

The authors employ a rigorous mathematical approach, contrasting standard probability theory with quantum mechanical formalism:

Probability Theory Review: They define a probability space $(\Omega, \mathcal{F}, P)$ and emphasize that expectations must be calculated based on the specific conditions (detector settings) under which data is collected.
Quantum Mechanical Model (Section 3): They derive the standard quantum mechanical expectation for entangled photon pairs using the singlet state $\Psi = \frac{1}{\sqrt{2}}(e_0 \otimes e_1 - e_1 \otimes e_0)$ and projection operators. This yields the correlation function $E[X_\alpha \otimes Y_\beta] = -\cos(2(\alpha - \beta))$ .
Critique of the "Four-Observable" Model (Section 4): They analyze the standard implicit model where four random variables ( $X_0, X_1, Y_0, Y_1$ ) are assumed to exist simultaneously for four detector settings. They show that while the algebraic inequality $|QS + QT + RS - RT| \le 2$ holds for any single realization of these variables, taking the expectation of this sum across different experimental runs (different settings) is mathematically invalid.
Proposal of the "Two-Observable" Model (Section 5): They construct an explicit probability space where the outcome space $\Omega$ $Ω$ includes the detector settings ( $A, B$ $A, B$ ) as random variables alongside the outcomes ( $X, Y$ $X, Y$ ).
- The quantum expectation is reinterpreted as a conditional expectation: $E[XY | A=\alpha, B=\beta]$ .
- They demonstrate that the Bell inequality derivation fails because it attempts to sum conditional expectations with different conditions (different $\alpha, \beta$ ) as if they were unconditional expectations of a single joint distribution.
Hidden Variable Extension (Section 6): They extend their model to include a hidden variable $\lambda$ (denoted $L$ ). They test for Bell separability (factorizability) and statistical locality.

3. Key Contributions

A. Resolution of the Bell Contradiction

The primary contribution is the demonstration that the Bell contradiction is a mathematical artifact of misapplying probability theory.

In the proposed 2-observables model, the expectation values are strictly conditional on the detector settings.
The authors show that the step from the algebraic inequality (valid for a single set of settings) to the expectation inequality (summing over different settings) is invalid because $E[X+Y|H_1] \neq E[X|H_2] + E[Y|H_3]$ unless $H_1=H_2=H_3$ .
Consequently, the model satisfies the Bell inequality when calculated correctly, and there is no violation. The "violation" only appears when one incorrectly assumes a joint probability distribution for mutually exclusive experimental settings.

B. Explicit Probability Space Construction

The paper provides a concrete probability space (Tables 2 and 3) that:

Includes detector settings as random variables.
Uses the quantum mechanical Born rule to define the conditional probabilities $P(X, Y | A, B)$ .
Is fully consistent with experimental data (Aspect et al.) and quantum predictions.

C. Analysis of Hidden Variables and Non-Separability

The authors extend their model to include a hidden variable $\lambda$ and prove two critical theorems:

Non-Separability: The extended probability measure is not Bell-separable. They prove that no factorization of the form $P(X, Y | A, B, \lambda) = P(X|A, \lambda)P(Y|B, \lambda)$ exists that can reproduce the quantum correlations.
Implication of Non-Separability: They establish the logical equivalence: Non-Separability $\implies$ (Non-Deterministic OR Non-Local).
- This aligns with Bell's original conclusion: any theory reproducing quantum mechanics must abandon either locality or determinism (or both).
- Crucially, the authors show this result is derived without assuming the Bell inequality violation; it is a property of the probability structure itself.

4. Results

Mathematical Consistency: The proposed model yields the quantum mechanical expectation $-\cos(2(\alpha-\beta))$ as a conditional expectation $E[XY | A=\alpha, B=\beta]$ .
No Violation: When the expectation is calculated correctly within the 2-observable framework, the CHSH sum does not exceed 2 in a way that contradicts probability theory; the apparent violation stems from the invalid summation of conditional expectations.
Hidden Variable Constraints: Even with a hidden variable, the model cannot be made separable. The proof (Section 6.2 and Appendix A) shows that assuming separability leads to a contradiction with the specific trigonometric correlations required by quantum mechanics (specifically involving Tsirelson's bound angles).
Determinism vs. Locality: The non-separability implies that the model must be either non-deterministic (outcomes are not fixed by $\lambda$ ) or non-local (outcomes depend on distant settings), or both.

5. Significance

Reframing the Debate: The paper shifts the focus from "Is nature non-local or non-real?" to "How do we correctly model the probability space of the experiment?" It suggests that the "paradox" is a result of modeling errors rather than a fundamental flaw in local realism or quantum mechanics.
Clarifying Realism: By explicitly including detector settings in the outcome space, the model avoids the assumption of "counterfactual definiteness" (that unmeasured values exist). It argues that realism is not a necessary assumption for the model to work; rather, the model works because it respects the conditional nature of the data.
Pedagogical Value: The paper provides a clear, step-by-step derivation showing exactly where the standard derivation of the Bell inequality breaks down mathematically (the invalid linearity of conditional expectations across different conditions).
Experimental Agreement: The model is shown to be in "full agreement" with both quantum mechanics and loophole-free experimental results, validating the approach as a viable alternative interpretation of the data.

In summary, van Hee, van Berkel, and de Graaf argue that the Bell experiment does not disprove local realism per se, but rather disproves the specific probability model (the assumption of a joint distribution for mutually exclusive settings) used to derive the inequality. Their proposed conditional probability model resolves the contradiction while maintaining consistency with quantum mechanics.