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Parity correlations of three photons

Three-Photon GHZ Correlation Bench

How can three photons whose individual results are random still beat every local explanation? Through the product of the three results. Any one photon, or any pair, gives plus or minus at random, but in an ideal GHZ state the product of all three is certain at the right analyzer settings. Combining the products from four settings reaches 4, while any local recipe for the outcomes reaches at most 2. With missed clicks, the comparison holds only if the kept triples form a fair sample.

Record outcomes at three stations, with arrival times, missed clicks and a set coincidence window. Build histograms of the eight outcomes and four parity correlations from the saved triggers, then compare the Mermin value from accepted triples with the one from all triggers.

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.

Reproducible synthetic observations

Every trigger remains in the record.

Ready to collect simulated data.

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 the records

Claim only what the measurement supports

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{—}

The main statistic needs a complete run with at least fifty accepted events in every basis. The error is a binomial shot-noise approximation, not a p-value. These four equatorial settings cannot separate visibility from phase, and they cannot 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 check.

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

The trigger is ideal and the three-qubit GHZ state is assumed already prepared, with white noise, independent detector loss, Gaussian timing and incoherent accidental triples added. The source is an abstract module, not a rebuilt SPDC optical source. The analyzers measure only in the equatorial plane. There is no full tomography, no certification of fidelity or genuine three-party entanglement, no spacelike timing, no loophole closure and no reconstruction of historical data. Statistics from accepted triples alone hold only if acceptance is assumed to be fair.

The assumed three-photon state

∣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}

The Lab assumes this three-photon state is already prepared; it does not simulate how optics would generate it. Accidental triples add a separate component in which all outcomes are equally likely.

Outcome probabilities for 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 one or minus one. Before any loss, the results at one station, or at any two stations, are completely random. Only the parity of all three outcomes carries the coherence.

The Mermin combination of four 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 comes from checking all 64 deterministic assignments, in which each of the three stations fixes one outcome for each of its two settings. The ideal value assumes aligned analyzers and zero phase.

Two estimates of a parity correlation

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}

A trigger is accepted only if all three stations click and the arrival times span no more than the time gate. The all-trigger statistic ignores the gate and counts each missed click as zero. Comparing the accepted-only value with the bound assumes that acceptance is fair. 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

BackgroundRecord outcomes at three stations, with arrival times, missed clicks and a set coincidence window. Build histograms of the eight outcomes and four parity correlations from the saved triggers, then compare the Mermin value from accepted triples with the one from all triggers.

Why it mattersA measurement protocol is the chain of steps that links a prepared quantum system to the detector counts you keep.

Start with the essentials

Focus question
How can three photons whose individual results are random still beat every local explanation?
One-sentence intuition
Keep what is assumed about the source apart from what is estimated from the recorded data.

Core mathematical model

The assumed three-photon state

∣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}

The Lab assumes this three-photon state is already prepared; it does not simulate how optics would generate it. Accidental triples add a separate component in which all outcomes are equally likely.

Outcome probabilities for 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 one or minus one. Before any loss, the results at one station, or at any two stations, are completely random. Only the parity of all three outcomes carries the coherence.

The Mermin combination of four 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 comes from checking all 64 deterministic assignments, in which each of the three stations fixes one outcome for each of its two settings. The ideal value assumes aligned analyzers and zero phase.

Two estimates of a parity correlation

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}

A trigger is accepted only if all three stations click and the arrival times span no more than the time gate. The all-trigger statistic ignores the gate and counts each missed click as zero. Comparing the accepted-only value with the bound assumes that acceptance is fair. Neither statistic certifies a real loophole-free experiment.

Common difficulties

A simulation already contains its law

Typical misconceptionRecovering a source parameter from the data is an independent check of the law.

Better mental modelThese synthetic records test how the measurement and the inference work. They are not historical observations, they give no confidence interval for the parameter, and they are not external scientific certification.

Run the experiment

  1. 01

    Record all four settings

    Acquire triggers with randomly chosen bases, then look at every outcome bin.

    What to observe: All eight bins are shown, including empty ones. The CSV keeps missed clicks and the times of rejected coincidences.
  2. 02

    Change the phase and the coherence

    Save a comparison, choose Phase reversal, then choose Incoherent mixture.

    What to observe: Reversing the phase flips the signs of the correlations. The incoherent mixture removes the three-way parity, and each station's own results stay random throughout.
  3. 03

    See what selection does

    Lower the detection efficiency, or narrow the time gate.

    What to observe: The accepted-only and all-trigger statistics now disagree. The exported data still contain every missed click and every rejected trigger.

Related experiments and articles

Quantum evidence · Reading path

Bell tests and photon information

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleThree-photon GHZ correlations defeat a shared answer sheet
  2. LabBell Correlations & CHSHCompare a separate two-photon Bell test
  3. LabQuantum State & Process TomographySee how a parity test differs from state reconstruction
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