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THREE ANALYZERS / FOUR SETTINGS / EVERY TRIGGER

Three-Photon GHZ Correlation Bench

Acquire three-station outcomes with arrival times, missed clicks and a declared coincidence gate. Build eight-outcome histograms and four parity correlations from retained triggers, then compare conditional and all-trigger Mermin statistics.

Interactive modelThree-Photon GHZ Correlation Bench

An abstract prepared-source module supplies three outgoing modes and an ideal trigger. Each station has an equatorial analyzer and two output detectors. Paths and travel times are schematic; this does not simulate a historical SPDC source or spacelike separation. Inspect a retained trigger below the scene.

SYNTHETIC OBSERVATIONS / REPRODUCIBLE RECORDS

Every trigger remains in the record.

Ready to acquire synthetic records.

Each ideal trigger randomly selects one of four joint bases. A missing detector click is zero; its time is empty. A coincidence requires three clicks and an arrival-time span within the chosen gate. Station timing offsets are fixed at zero, plus 0.4 and minus 0.3 nanoseconds. No accidental subtraction or efficiency correction is applied.

RETAINED DATA

Eight outcomes in the selected basis.

k=4[a<0]+2[b<0]+[c<0],Nk\begin{aligned}k&=4[a<0]+2[b<0]\\&\quad+[c<0],\quad N_k\end{aligned}

Eight sign triples run from all positive to all negative using binary index zero through seven. Green bars count accepted triples; zero bins remain visible. Gray outlines show the saved run at the same basis index; its recorded analyzer angles can differ. The CSV also retains every unaccepted trigger.

ESTIMATES FROM RECORDS

Keep the claim within the measurement.

Conditional Mermin statistic

—\text{—}

All-trigger statistic · missed clicks zero, no time gate

—\text{—}

Conditional shot-noise standard error

—\text{—}

Recorded three-station gate acceptance

—\text{—}

Retained station outcomes · zero means no click

—\text{—}

Accepted / triggered in inspected basis

—\text{—}

A complete run and at least fifty accepted events in every basis are required for the headline statistic. Error is a binomial shot-noise approximation, not a p-value. These four equatorial settings cannot separately determine visibility and phase or certify a density matrix.

DATA REDUCTION

Conditional and all-trigger parity.

b:  XXX,  XYY,  YXY,  YYXb:\;XXX,\;XYY,\;YXY,\;YYX

Green: accepted sample. Brown: all triggers, missing clicks zero, without time selection. Gray: saved conditional correlations with their saved angles and time gate. The local Mermin bound applies to an unconditional protocol with declared assumptions; applying it to a selected sample additionally requires fair acceptance.

Detection and time-gate audit.

i,max⁡(tA,tB,tC)−min⁡(tA,tB,tC)  (ns)i,\quad \max(t_A,t_B,t_C)-\min(t_A,t_B,t_C)\;(\mathrm{ns})

Three-click records among the first 240 triggers. Green: inside the coincidence gate; brown: outside. The horizontal line marks the declared gate. Missing-click triggers remain in CSV, with no invented arrival time.

Last eight raw records

Columns: trigger, basis index, three outcomes, accepted flag. Export includes all station angles and arrival times in nanoseconds.

TriggerBasisABCAccepted

Model and sources

Assumed ideal trigger and prepared three-qubit GHZ state with white noise, independent detector loss, Gaussian timing and incoherent accidental triples. Source preparation is abstract, not a reconstructed SPDC optical source. Equatorial analyzers only; no full tomography, fidelity or genuine-tripartite certification, spacelike timing, loophole closure or historical-data reconstruction. Conditional sampling requires an explicit fair-acceptance assumption.

Prepared source model

∣Gϕ⟩=∣HHH⟩+eiϕ∣VVV⟩2ρ=V∣Gϕ⟩⟨Gϕ∣+1−V8I\begin{aligned}|G_\phi\rangle&=\frac{|HHH\rangle+e^{i\phi}|VVV\rangle}{\sqrt2}\\\rho&=V|G_\phi\rangle\langle G_\phi|+\frac{1-V}{8}I\end{aligned}

This is an assumed three-photon preparation, not an optical generation simulation. Accidentals add a separate uniform outcome component.

Equatorial analyzers

A(ϑ)=cos⁡ϑ X+sin⁡ϑ YP(a,b,c)=1+abc E8E=Vcos⁡(ϑA+ϑB+ϑC−ϕ)\begin{aligned}A(\vartheta)&=\cos\vartheta\,X+\sin\vartheta\,Y\\P(a,b,c)&=\frac{1+abc\,E}{8}\\E&=V\cos(\vartheta_A+\vartheta_B+\vartheta_C-\phi)\end{aligned}

Each outcome is plus or minus one. All one-station and two-station marginals are uniform before loss; the triple parity carries coherence.

Four parity settings

M=EXXX−EXYY−EYXY−EYYX∣Mlocal∣≤2,MG0=4\begin{aligned}M&=E_{XXX}-E_{XYY}\\&\quad-E_{YXY}-E_{YYX}\\|M_{\rm local}|&\le2,\quad M_{G_0}=4\end{aligned}

The local bound follows by enumerating the 64 deterministic assignments of two outcomes at each of three stations. The ideal value assumes aligned analyzers and phase zero.

Keep every trigger

E^cond=∑r∈Carbrcr∣C∣E^all=∑r=1NtrigarbrcrNtrig\begin{aligned}\widehat E_{\rm cond}&=\frac{\sum_{r\in C}a_rb_rc_r}{|C|}\\\widehat E_{\rm all}&=\frac{\sum_{r=1}^{N_{\rm trig}}a_rb_rc_r}{N_{\rm trig}}\end{aligned}

The accepted set requires three clicks and a time span within the gate. The all-trigger statistic ignores the time gate; each missing local click is zero. A conditional bound comparison assumes fair acceptance. Neither statistic certifies a real loophole-free experiment.

Pan et al. · Three-photon GHZ test (2000)Mermin · Extreme quantum entanglement (1990)

Two-Ion Quantum Logic Spectroscopy

Physics tutorial

Three-Photon GHZ Correlation Bench

BackgroundAcquire three-station outcomes with arrival times, missed clicks and a declared coincidence gate. Build eight-outcome histograms and four parity correlations from retained triggers, then compare conditional and all-trigger Mermin statistics.

Why it mattersA protocol connects a prepared quantum system to retained detector observations.

Start with the essentials

Focus question
Three random outcomes. One shared parity.
One-sentence intuition
Keep source assumptions separate from estimates made from records.

Core mathematical model

Prepared source model

∣Gϕ⟩=∣HHH⟩+eiϕ∣VVV⟩2ρ=V∣Gϕ⟩⟨Gϕ∣+1−V8I\begin{aligned}|G_\phi\rangle&=\frac{|HHH\rangle+e^{i\phi}|VVV\rangle}{\sqrt2}\\\rho&=V|G_\phi\rangle\langle G_\phi|+\frac{1-V}{8}I\end{aligned}

This is an assumed three-photon preparation, not an optical generation simulation. Accidentals add a separate uniform outcome component.

Equatorial analyzers

A(ϑ)=cos⁡ϑ X+sin⁡ϑ YP(a,b,c)=1+abc E8E=Vcos⁡(ϑA+ϑB+ϑC−ϕ)\begin{aligned}A(\vartheta)&=\cos\vartheta\,X+\sin\vartheta\,Y\\P(a,b,c)&=\frac{1+abc\,E}{8}\\E&=V\cos(\vartheta_A+\vartheta_B+\vartheta_C-\phi)\end{aligned}

Each outcome is plus or minus one. All one-station and two-station marginals are uniform before loss; the triple parity carries coherence.

Four parity settings

M=EXXX−EXYY−EYXY−EYYX∣Mlocal∣≤2,MG0=4\begin{aligned}M&=E_{XXX}-E_{XYY}\\&\quad-E_{YXY}-E_{YYX}\\|M_{\rm local}|&\le2,\quad M_{G_0}=4\end{aligned}

The local bound follows by enumerating the 64 deterministic assignments of two outcomes at each of three stations. The ideal value assumes aligned analyzers and phase zero.

Keep every trigger

E^cond=∑r∈Carbrcr∣C∣E^all=∑r=1NtrigarbrcrNtrig\begin{aligned}\widehat E_{\rm cond}&=\frac{\sum_{r\in C}a_rb_rc_r}{|C|}\\\widehat E_{\rm all}&=\frac{\sum_{r=1}^{N_{\rm trig}}a_rb_rc_r}{N_{\rm trig}}\end{aligned}

The accepted set requires three clicks and a time span within the gate. The all-trigger statistic ignores the time gate; each missing local click is zero. A conditional bound comparison assumes fair acceptance. Neither statistic certifies a real loophole-free experiment.

Common difficulties

A simulation supplies its law

Typical misconceptionRecovering a source parameter independently confirms the law.

Better mental modelThese synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

Run the experiment

  1. 01

    Acquire all four settings

    Acquire triggers with randomly selected bases, then inspect every outcome bin.

    What to observe: The eight bins include zeros. The CSV retains missed clicks and rejected coincidence times.
  2. 02

    Change coherence

    Save a comparison, reverse the source phase, then use the incoherent preset.

    What to observe: The signs reverse with phase; incoherence removes triple parity while the single-station records remain random.
  3. 03

    Expose selection

    Reduce detection efficiency or narrow the time gate.

    What to observe: Conditional and all-trigger statistics differ. No missed clicks or rejected triggers are silently removed from the exported dataset.
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