---
title: "Three-Photon GHZ Correlation Bench Interactive Lab | Sandbox Physics"
description: "Three random outcomes. One shared parity. 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."
canonical_url: "https://sandboxphysics.com/labs/three-photon-ghz-correlations/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/three-photon-ghz-correlations/"
last_modified: "2026-10-19"
---

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 simulation: open this page in a web browser to change the parameters and run the model.

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.

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.

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

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.

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

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.

$i,\quad \max(t_A,t_B,t_C)-\min(t_A,t_B,t_C)\;(\mathrm{ns})$

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.

| Trigger | Basis | A | B | C | Accepted |
| --- | --- | --- | --- | --- | --- |

### 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

$$
\begin{aligned}|G_\phi\rangle&=\frac{|HHH\rangle+e^{i\phi}|VVV\rangle}{\sqrt2}\\\rho&=V|G_\phi\rangle\langle G_\phi|+\frac{1-V}{8}I\end{aligned}
$$

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

Equatorial analyzers

$$
\begin{aligned}A(\vartheta)&=\cos\vartheta\,X+\sin\vartheta\,Y\\P(a,b,c)&=\frac{1+abc\,E}{8}\\E&=V\cos(\vartheta_A+\vartheta_B+\vartheta_C-\phi)\end{aligned}
$$

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

$$
\begin{aligned}M&=E_{XXX}-E_{XYY}\\&\quad-E_{YXY}-E_{YYX}\\|M_{\rm local}|&\le2,\quad M_{G_0}=4\end{aligned}
$$

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

$$
\begin{aligned}\widehat E_{\rm cond}&=\frac{\sum_{r\in C}a_rb_rc_r}{|C|}\\\widehat E_{\rm all}&=\frac{\sum_{r=1}^{N_{\rm trig}}a_rb_rc_r}{N_{\rm trig}}\end{aligned}
$$

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)](https://web.physics.ucsb.edu/~quopt/ghz.pdf) [Mermin · Extreme quantum entanglement (1990)](https://doi.org/10.1103/PhysRevLett.65.1838)

[Two-Ion Quantum Logic Spectroscopy](https://sandboxphysics.com/labs/two-ion-quantum-logic-spectroscopy/)

Physics tutorial

## Three-Photon GHZ Correlation Bench

**Background** 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.

**Why it matters** A 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

$$
\begin{aligned}|G_\phi\rangle&=\frac{|HHH\rangle+e^{i\phi}|VVV\rangle}{\sqrt2}\\\rho&=V|G_\phi\rangle\langle G_\phi|+\frac{1-V}{8}I\end{aligned}
$$

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

#### Equatorial analyzers

$$
\begin{aligned}A(\vartheta)&=\cos\vartheta\,X+\sin\vartheta\,Y\\P(a,b,c)&=\frac{1+abc\,E}{8}\\E&=V\cos(\vartheta_A+\vartheta_B+\vartheta_C-\phi)\end{aligned}
$$

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

$$
\begin{aligned}M&=E_{XXX}-E_{XYY}\\&\quad-E_{YXY}-E_{YYX}\\|M_{\rm local}|&\le2,\quad M_{G_0}=4\end{aligned}
$$

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

$$
\begin{aligned}\widehat E_{\rm cond}&=\frac{\sum_{r\in C}a_rb_rc_r}{|C|}\\\widehat E_{\rm all}&=\frac{\sum_{r=1}^{N_{\rm trig}}a_rb_rc_r}{N_{\rm trig}}\end{aligned}
$$

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 misconception** Recovering a source parameter independently confirms the law.

**Better mental model** These 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.

Physical conclusion

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.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)