Q041 · Random at each end, correlated together

Bell Correlations & CHSH

Can two stations whose results each look random still beat the limit of any local theory? Yes. With entangled pairs the CHSH value can pass the local limit of two and reach two times the square root of two, about 2.83, even though each station alone sees random results. The evidence is fair only if the settings are chosen at random and every trial, losses included, is counted.

Turn two polarization analyzers, collect all four setting combinations, and compare entangled photons with a local model while keeping missed detections. Then check an ideal event-ready test against its source, switching and detection timestamps; the check never drops trials from the Bell data.

Model signed CHSH sum $\text{—}$

Each station: positive result $\text{—}$

Recorded CHSH · all trials; local magnitude at most two $\text{—}$

First, make a prediction

Will the current model exceed the local bound?

How to read the scene

Each station can measure along two axes, and every measurement has two outcomes. The randomness at one station and the correlation between the two stations are different observations.

Choose a prediction, then compare the cases and acquire events.

No events acquired yet. Replaying the inputs does not create detector records.

Change one condition at a time

Compare the cases

Displayed setting pairAlice axis, first setting

a_0

0\,\mathrm{deg}

Alice axis, second setting

a_1

45\,\mathrm{deg}

Bob axis, first setting

b_0

22.5\,\mathrm{deg}

Bob axis, second setting

b_1

-22.5\,\mathrm{deg}

Drag either analyzer axis, or use the sliders from the keyboard.

Source, detectors and sampling

Correlated-state fraction

v

1

Alice detection efficiency

\eta_A

1

Bob detection efficiency

\eta_B

1

Trials per acquisition / nominal setting

N

800

Acquisition seed

s

1969

Missed detections stay in the raw record. In the local comparison model, the two stations share one uniformly random angle, and each responds only to its own axis; mixing adds independent random bits.

This is an ideal two-qubit simulation. It also assumes a source-ready signal that does not disturb the photons, and optical delays with no loss. Statistics from coincidences alone rely on fair sampling; the all-trial estimate keeps every miss and scores it as a positive outcome. Timing is checked separately and never decides which trials count. The error bounds hold for a fixed number of trials, not when you decide when to stop by watching the data. The simulation cannot certify a real loophole-free Bell test.

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{—}

Complete the scan, or acquire all four settings, to see an inference.

Inspect counts and individual trials

Four setting combinations from the full record
Settings	All / retained pairs	Joint counts: positive-positive, positive-negative, negative-positive, negative-negative	Correlation

Can the settings and outcomes be spacelike separated?

Timing is a separate test

Keep every trial and check the light cones

Ideal protocol: a nondestructive source-ready flag confirms a photon pair without measuring either tested photon. This is an assumed device, not ordinary pair coincidence heralding or the Hensen apparatus. Two ideal lossless optical delay lines wait for the ready broadcast and local switching. Source at the midpoint; all times are nanoseconds from emission, positions are metres.

The source accepts a ready flag only in its fixed first 100 ns. After that deadline its accept bit travels at light speed to each station. Accepted trials are fixed before fresh choices. Missing or late flags remain rejected attempt records. Every accepted trial, including misses and timing failures, enters the same all-trial Bell sum; there is no timing-qualified subset.

Station separation (m)300Extra optical delay (ns)350Choice after ready signal (ns)40Axis switching time (ns)20Bounded physical jitter (ns)2Each clock uncertainty (ns)2Each station position bound (m)0.2Source-ready probability0.7Source-ready latency (ns)50

Nominal photon path per arm is half the station separation plus light speed times extra delay. Delay coils are ideal and lossless, with vacuum group speed. Bounded jitter perturbs the delay; negative waiting times are clipped to zero. A shutter blocks photons arriving before the requested axis settles. Clock and position errors are worst-case calibration bounds, separate from physical jitter.

No attempts yet. The Bell angles, source model, efficiencies and seed come from the controls above.

Smallest margin between remote choice and the whole outcome window

\text{—}

Worst local switching deadline margin

\text{—}

All accepted trials, fixed miss assignment

\text{—}

Nominal per-arm optical path

\text{—}

Acquire records to check the time order and the spacelike margins.

Diagram: orange photon paths, gray accept broadcast, blue choice events, green switch completion, red complete outcome windows. Dotted lines mark light rays from the choices. Bent optical paths show propagation time, not a measured trajectory. The table holds the same record for readers who cannot use the diagram.

Latest accepted attempt: observed event tags (ns); source is at zero metres
Event	Source	Alice	Bob

Latest 10 attempts; export preserves every attempt and timestamp
Attempt	Accepted	Ready (ns)	Settings	Outcomes / misses	Timing check

What this timing check shows

m=\frac{\max(0,|x_1-x_2|-2\delta x)}{c}-|t_1-t_2|-2\delta t

A positive margin means every event coordinate allowed by the error bounds stays spacelike separated. Separately, local switching must finish before the photon arrives. The remote-choice check covers both ends of each fixed 10 ns outcome window, including trials with no click. The model treats choice events as instantaneous. Independence of the inputs is still an explicit assumption about the pseudorandom source: the choices come after the source broadcast, so they are not spacelike separated from the emission.

All-trial assignment removes the need to discard misses, but a violation still depends on efficiency, settings independence and the declared trial protocol. Coincidence-only evidence above needs fair sampling. Timing flags are diagnostics, never a selection rule. The displayed Bell sum is descriptive; repeated pause/resume and optional stopping do not provide an anytime-valid significance test. No device-memory model, physical random-generator calibration, relativistic clock network or real loophole-free certification is provided.

Hensen et al. (2015) · Shalm et al. (2015) · Giustina et al. (2015)

Physics tutorial

Test a joint record, not a single outcome

BackgroundEach photon of a Bell pair gives a completely random result on its own, yet the two results can be strongly correlated. Each station picks one of two polarization axes, and the four setting combinations together give the CHSH value.

Why it mattersA concrete local model shows what the bound means in practice. Its two stations share a random angle, but neither station's answer depends on the far setting. Such a model can agree with quantum mechanics at single settings and still differ in the sum of all four terms.

Start with the essentials

Focus question: What changes if missed detections are kept instead of thrown away?
One-sentence intuition: Choose each pair of settings independently at random, keep every heralded trial, and compute the Bell sum two ways: from coincidences only, and from all trials with each miss scored as a fixed outcome.

Core mathematical model

Prepared state and axes

\rho=v|\Phi^+\rangle\langle\Phi^+|+(1-v)I_4/4,\quad |\Phi^+\rangle=(|HH\rangle+|VV\rangle)/\sqrt2

Polarizer axes are real angles in degrees, and a half turn brings an axis back to the same setting. On the Bloch sphere the angle doubles.

Joint probabilities and marginals

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

Each outcome is a plus or a minus. Turning the far analyzer changes the joint statistics but never the ideal probabilities at the near station. The model offers no way to signal faster than light.

Four-term Bell sum

S=E_{00}+E_{01}+E_{10}-E_{11},\quad |S|_{\mathrm{local}}\le2,\quad |S|_{\mathrm{quantum}}\le2\sqrt2

The default polarizer settings reach the largest value quantum mechanics allows. The fourth term in the plot gets a minus sign before the sum.

Response rule of the local model

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 uniformly random and has nothing to do with the settings. On a set fraction of trials, each station gives an independent random output instead. Any fixed table of responses gives a Bell sum of plus or minus two.

Counts and a conservative range

\widehat E=\frac{n_{++}+n_{--}-n_{+-}-n_{-+}}{n},\quad r_S=\sum_{x,y}\sqrt{\frac{2\ln160}{n_{xy}}}

A union bound combines four Hoeffding bounds for a fixed number of trials. The range errs on the safe side, but it is not valid if you keep checking the data and choose when to stop. If any setting has no data, no estimate is given.

Common difficulties

Correlations cannot carry a message

Typical misconceptionWhen Bob turns his axis, the statistics of Alice’s random results should change.

Better mental modelOnly the correlations between the stations change. In the ideal model each station still sees each outcome half the time, and you need results from both stations to see the change.

One local model does not stand for all of them

Typical misconceptionBeating the local curve on the plot is enough to prove a Bell violation.

Better mental modelWhen settings are chosen independently, the local bound holds for every local response table. The plotted model is only one example; the real test compares the measured sum and its uncertainty with that bound.

A simulator does not close experimental loopholes

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

Better mental modelThe timing protocol checks a fixed, ideal geometry using the saved timestamps. Independent settings, a source herald that does not disturb the photons and the clock error bounds are still assumptions. Device memory, calibration of a physical random-number generator and validation of real detectors are left out.

Run the experiment

01
Look at each station alone
Acquire pairs one at a time at a single setting and look at each station’s results.
What to observe: At each station, plus and minus come out about equally often. How often the two stations agree depends on the angle between their axes.
02
Collect all four settings
Predict the result, then acquire data at all four settings. Look at the signed terms and the raw CSV.
What to observe: The settings are chosen at random, independently of the source. The sum comes only from the measured outcomes, not from the model reference.
03
Use a concrete local model
Choose Local response model and repeat the four-setting acquisition.
What to observe: The expected value stays inside the local bound. A finite sample can still land past the bound by chance, and the conservative interval shows how large that uncertainty is.
04
See what fair sampling assumes
Try imperfect detection and compare the retained-pair estimate with the all-trial estimate.
What to observe: The first throws away missed outcomes and assumes the rest are a fair sample. The second keeps every heralded trial and scores each miss as a positive outcome, so its violation shrinks as loss grows.
05
Check the event-ready timing
Open the timing protocol and acquire attempts. Then compare stations placed too close, late switching and too much optical delay. Pause, save the partial run and resume it.
What to observe: A source-ready signal that does not disturb the photons and lossless delay lines are ideal assumptions. Which trials count is fixed before any setting is chosen. The timestamps are checked against switching deadlines and the full outcome window at the far station, and no trial is removed from the Bell denominator.

Bell Correlations & CHSH

Four correlations, one Bell sum

Keep every trial and check the light cones

Test a joint record, not a single outcome

Core mathematical model

Prepared state and axes

Joint probabilities and marginals

Four-term Bell sum

Response rule of the local model

Counts and a conservative range

Common difficulties

Correlations cannot carry a message

One local model does not stand for all of them

A simulator does not close experimental loopholes

Run the experiment

Look at each station alone

Collect all four settings

Use a concrete local model

See what fair sampling assumes

Check the event-ready timing

Related experiments and articles

Bell tests and photon information