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Sandbox Physics

Q041 · Random outcomes / joint evidence

Bell Correlations & CHSH

Send a pair to two independent polarization analyzers. Predict the joint results, turn the measurement axes and collect all four setting combinations. Compare entangled pairs with a specified local model, then compute the Bell statistic from the clicks.

Interactive modelBell Correlations & CHSH
Model signed CHSH sum—\text{—}
Each station: positive result—\text{—}
Evidence from your detector records

SIMULATED ACQUISITION

Four correlations, one Bell sum

Acquire events to begin.

Orange: retained-pair signed terms. Blue: all heralded trials, assigning every missed outcome to positive. The fourth term is subtracted.

Retained pairs: signed sum and conservative 95% range

—\text{—}

All trials: signed sum and conservative 95% range

—\text{—}

Inference requires a complete scan or all four settings.

Four setting combinations from the full record
SettingsAll / retainedJoint counts: positive-positive, positive-negative, negative-positive, negative-negativeCorrelation
Latest 12 raw trials; CSV contains every trial

Physics tutorial

Test a joint record, not a single outcome

BackgroundA Bell pair can have uniformly random local outcomes and strongly correlated joint outcomes. Each station chooses one of two polarization axes; the four setting combinations supply a CHSH statistic.

Why it mattersA specified local comparator makes the bound operational. It shares a random angle but neither response uses the remote setting. Quantum and local models can agree at individual settings while differing in their four-term sum.

Start with the essentials

Focus question
What changes when missed detections are retained rather than discarded?
One-sentence intuition
Collect independent, randomly chosen setting pairs, preserve every heralded trial, and compute both the coincidence-only and fixed-assignment all-trial statistics.

Core mathematical model

Prepared state and axes

ρ=v∣Φ+⟩⟨Φ+∣+(1−v)I4/4,∣Φ+⟩=(∣HH⟩+∣VV⟩)/2\rho=v|\Phi^+\rangle\langle\Phi^+|+(1-v)I_4/4,\quad |\Phi^+\rangle=(|HH\rangle+|VV\rangle)/\sqrt2

Linear polarization axes are physical angles in degrees, modulo a half turn. The corresponding Bloch angle is doubled.

Joint probabilities and marginals

P(r,s∣a,b)=1+rs vcos⁡[2(a−b)]4,P(r∣a)=P(s∣b)=12P(r,s|a,b)=\frac{1+rs\,v\cos[2(a-b)]}{4},\quad P(r|a)=P(s|b)=\frac12

Outcomes are signed. Changing the remote axis changes the joint record, never the ideal local marginal. This model offers no superluminal signaling operation.

Four-term Bell sum

S=E00+E01+E10−E11,∣S∣local≤2,∣S∣quantum≤22S=E_{00}+E_{01}+E_{10}-E_{11},\quad |S|_{\mathrm{local}}\le2,\quad |S|_{\mathrm{quantum}}\le2\sqrt2

The default polarization settings reach the ideal quantum extremum. The fourth plotted term is signed with a minus before summing.

Declared local response

Ax(λ)=sgn⁡cos⁡[2(ax−λ)],By(λ)=sgn⁡cos⁡[2(by−λ)]A_x(\lambda)=\operatorname{sgn}\cos[2(a_x-\lambda)],\quad B_y(\lambda)=\operatorname{sgn}\cos[2(b_y-\lambda)]

The shared angle is uniform and independent of settings. A fraction of trials uses independent random outputs. Every deterministic response table has a Bell sum of positive or negative two.

Counts and a conservative range

E^=n+++n−−−n+−−n−+n,rS=∑x,y2ln⁡160nxy\widehat E=\frac{n_{++}+n_{--}-n_{+-}-n_{-+}}{n},\quad r_S=\sum_{x,y}\sqrt{\frac{2\ln160}{n_{xy}}}

Four Hoeffding bounds are combined by a union bound for a fixed record. The range is conservative and is not an anytime-valid claim for repeated optional stopping. Missing settings leave the estimate unresolved.

Common difficulties

Correlation is not a message

Typical misconceptionTurning Bob’s axis should change Alice’s local random sequence distribution.

Better mental modelOnly the joint correlations change. Each ideal local marginal remains one half; comparisons require records from both stations.

One comparator is not every local model

Typical misconceptionBeating the plotted local curve alone proves a Bell violation.

Better mental modelThe general bound applies to all local response tables under independent settings. The displayed comparator is one example; inference tests the bound and uncertainty.

A simulator does not close experimental loopholes

Typical misconceptionAn estimate above the bound certifies a real loophole-free experiment.

Better mental modelSetting independence, heralding and outcome-independent detection are declared assumptions. Space-time separation, device memory and real detector calibration are omitted.

Run the experiment

  1. 01

    Compare local randomness

    Acquire individual pairs at one setting and inspect each station’s record.

    What to observe: Positive and negative local counts approach equality. Joint agreement depends on the angle difference.
  2. 02

    Collect all four settings

    Predict a result, then acquire all four settings. Inspect signed terms and raw CSV.

    What to observe: Random setting selection is independent of the source. The sum is computed only from acquired outcomes, not from the model reference.
  3. 03

    Use a concrete local model

    Choose Local response model and repeat the four-setting acquisition.

    What to observe: The expectation stays inside the local bound. A finite estimate can fluctuate across the bound; a conservative interval makes that uncertainty visible.
  4. 04

    Expose sampling assumptions

    Try imperfect detection and compare retained-pair and all-trial estimates.

    What to observe: The first discards missed outcomes and assumes fair sampling. The second keeps every herald and assigns a fixed positive outcome to a miss; its violation weakens under loss.