Replicating a published physics result
Reconstruct Bell’s original three-setting inequality and a finite singlet-state violation while preserving its assumptions.
On this page
The question for today
What you’ll understand
- Reproduce the assumptions and algebra of a specified published inequality.
- Calculate a finite quantum violation using declared spin conventions.
- Distinguish analytic replication from reproducing an experiment or its data analysis.
Replication begins by identifying exactly which published result is being reconstructed. Bell’s 1964 paper contains a mathematical incompatibility between specified local hidden-variable assumptions and singlet-state correlations. Reproducing that argument is an analytic replication; it is not a repeat of a physical Bell experiment or a reanalysis of measured counts. Its value lies in rebuilding the assumptions, signs and finite comparison independently, so another reader can locate what would change if an assumption were relaxed. The original perfect-anticorrelation setting is particularly useful because its algebra is short and its limitations are visible.
Translate notation without changing the experiment
Use two separated spin-one-half systems in the singlet state. Measurement along a unit direction returns a dimensionless outcome plus or minus one, corresponding to a Pauli spin component; physical angular momentum would include a factor of hbar over two. The quantum correlation is minus the dot product of the two measurement directions. These are spin directions, not linear-polarizer angles, for which a different angle dependence would require a different translation.
For the hidden-variable comparison, assign outcomes A(a,lambda) and B(b,lambda), each plus or minus one. Local response means that A does not depend on the remote setting b and B does not depend on a. Use a single normalized, nonnegative hidden-variable distribution independent of the chosen settings. Perfect anticorrelation for equal settings implies B(a,lambda)=−A(a,lambda) almost everywhere. These assumptions define the class of models to which the following bound applies.
Recover the bound rather than quoting it
After using perfect anticorrelation, subtract the correlations involving settings b and c on the second side. Their difference contains a unit-magnitude factor A(a,lambda) multiplying A(c,lambda) minus A(b,lambda). For two numbers that are each plus or minus one, the absolute difference equals one minus their product. Applying the triangle inequality inside the nonnegative average gives the three-setting bound, with the remaining product expressed as minus the b,c correlation.
This derivation identifies where each assumption enters. If the hidden-variable distribution depends on the settings, the common average is unavailable. If remote settings change local responses, the assigned functions have changed. If equal-setting anticorrelation is imperfect, the substitution used here is no longer exact. A noisy experiment needs a suitable inequality and statistical analysis; multiplying all singlet correlations by a visibility and retaining this exact perfect-anticorrelation bound would silently change its premises.
Record what the reconstruction has and has not reproduced
A transparent replication identifies the source equation, fixes conventions, writes each transformation and evaluates a finite case that exposes the result. Independently check that both candidate descriptions use the same settings and outcome normalization. A sign reversal on both recorded outcomes leaves a correlation unchanged; reversing just one changes it. Such details are part of reproducibility, because a superficially different inequality may be an equivalent relabeling or an actual mistake.
An experimental replication adds source preparation, setting choices, timing, detection efficiencies, raw outcomes and an inference procedure with its own assumptions. The analytic contradiction alone supplies no experimental significance level. Nor does it show that all hidden-variable descriptions are impossible: it excludes the stipulated local response and setting-independent class when asked to reproduce the quantum predictions. The practice ensemble demonstrates the bound directly without reproducing the singlet, making this scope concrete rather than rhetorical.
Work it through
Reproduce a finite violation of the original bound
Take coplanar spin directions a, b and c at angles 0°, 60° and 120°. Use the ideal singlet correlation E=−cos of the angle difference. Calculate both sides of Bell’s stated inequality and identify the incompatible prediction.
The a,b and b,c separations are both 60°, so E(a,b)=E(b,c)=−1/2. The a,c separation is 120°, whose cosine is −1/2, giving E(a,c)=+1/2.
The left side is |−1/2−(+1/2)|=1. The right side is 1+(−1/2)=1/2. The quantum predictions therefore violate the bound by 1/2.
Every hidden-variable model satisfying all stated premises obeys the bound. Consequently such a model cannot reproduce these three ideal singlet correlations together. This arithmetic reconstructs the published incompatibility; it does not report observed experimental counts or a p-value.
The quantum comparison is 1≤1/2, which is false. The contradiction concerns the specified class of local assignments with setting independence and perfect equal-setting anticorrelation.
The living notebook
Explore the model
Interactive lab · vectors
Check the geometric part of the singlet correlation
Let a be the fixed positive x direction. Reset and set both controls to Angle 60° and Magnitude 1; predict x=0.5, hence geometric singlet correlation −x=−0.5. For a fresh comparison, Reset and set both controls to Angle 120° and Magnitude 1; predict x=−0.5 and inferred correlation +0.5.
- x component
- 1.393
- y component
- 0.975
- Magnitude
- 1.7
Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forThe vector lab checks a dot-product factor only. It prepares no entangled pair, generates no measurement outcomes, tests no locality assumption and supplies no experimental Bell statistic.
Pause & reason
Check your understanding
Your turn
Try it yourself
Construct a discrete local model with two equally probable hidden-variable types. Their A assignments at (a,b,c) are (+1,+1,+1) and (+1,+1,−1). For each type set B=−A at every setting. Compute E(a,b), E(a,c), E(b,c) and both sides of Bell’s inequality. Does this ensemble reproduce the worked singlet correlations?
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
For every pair multiply A at the first setting by B at the second, then average over the two equally weighted types.
Compare with the solution
For a,b the product is −1 in both types, so E(a,b)=−1. For a,c the products are −1 and +1, so E(a,c)=0. The b,c products are likewise −1 and +1, giving E(b,c)=0. The bound becomes |−1−0|≤1+0, or 1≤1: this local ensemble saturates it. It does not reproduce the singlet values (−1/2,+1/2,−1/2). A model may realize some correlations and perfect equal-setting anticorrelation while still failing the full set of quantum predictions.
Keep the idea
Take it with you
- Replication requires preserving the published claim’s assumptions and conventions.
- Bell’s original three-setting bound uses perfect equal-setting anticorrelation.
- An analytic incompatibility, a measured violation and a statistical significance are different results.
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