From a prepared state
to a visible fringe.

Control the quantities that determine the detector probabilities. Then separate loss of coherence from unequal path populations.

Standard two-path quantum benchmark

Phase and retained coherence

Begin with equal path populations and a balanced readout. Vary the relative phase and coherence while checking normalization and positivity.

Change phase and coherence

Supplied model inputs

Relative phase φ 60°

Coherence magnitude V 0.80

Show the incoherent comparator (V=0)

Use arrow keys on either slider; Home and End select its limits. Phase is displayed in degrees. V is a dimensionless input, held fixed while scanning the fringe.

The default example below is complete. Interactive controls become available when the script loads.

At φ=60° and V=0.80, P₀=0.7, P₁=0.3 and P₀+P₁=1. The corresponding incoherent probabilities are 0.5 and 0.5.

P₀ · blue solidP₁ · orange dashed

Curves join samples calculated every 2°. The marked phase and numerical outputs are calculated directly from the formula.

γ=0.4+0.69282i. Density-matrix eigenvalues: 0.9, 0.1; purity Tr(ρ²)=0.82.

Calculated checkpoints for coherence magnitude 0.8
Phase	P₀	P₁	Sum
0°	0.9	0.1	1
60°	0.7	0.3	1
90°	0.5	0.5	1
180°	0.1	0.9	1
270°	0.5	0.5	1
360°	0.9	0.1	1

Exports include the selected point and the calculated fringe samples. No information is uploaded.

Try this investigationSet the coherence to zero, then scan the phase. The probability must remain one half at each ideal detector.

Coherence and imbalance are different controls

A pure state can have unequal path populations and therefore less than unit fringe visibility. Keep the state and the detector model explicit.

Unequal-path preparation

Population in path 0 0.5

Relative coherence η 0.8

Relative phase / degrees 60

At p=0.5, η=0.8 and φ=60°: P₀=0.7, P₁=0.3 and visibility=0.8.

P₀ · solid blueP₁ · dashed orange

Declared state and measurementρ₀₀=p, ρ₁₁=1−p and |ρ₀₁|=η√[p(1−p)], with 0≤η≤1. For a balanced readout, P₀,₁=[1±V cosφ]/2 and V=2η√[p(1−p)]. The path predictability is D=|2p−1|, so V²+D²≤1. D here measures preparation imbalance; it is not a measurement of an arbitrary environmental which-path detector.

Try this investigationSet η=1 and move p toward zero. The state remains pure while the balanced-readout fringe loses visibility.

What would distinguish a temporal model?

A temporal interpretation that reproduces the same state and measurement probabilities agrees with the standard benchmark. A distinguishable proposal needs a separately specified law that changes a measurable quantity.

Choose a predicted phase or coherence dependence, freeze its calibration, and evaluate another protocol. The comparison workbench can inspect first- and second-harmonic fits; a harmonic residual does not uniquely identify a physical cause.

Phase and retained coherence

Change phase and coherence

Coherence and imbalance are different controls

Unequal-path preparation

What would distinguish a temporal model?

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