Interference depends on the relationship between alternatives, not simply on the brightness of each path. A physical marker can carry information that distinguishes the paths and changes the interference available in the unconditioned output. An explicit two-state calculation shows what that statement means.
Add a path marker
Begin with two equally populated path states, |0⟩ and |1⟩. Associate the first with a normalized marker state |d₀⟩ and the second with |d₁⟩. A joint pure state can be written
|Ψ⟩ = (|0⟩|d₀⟩ + eiφ|1⟩|d₁⟩)/√2.
The marker could be another physical degree of freedom. It need not be read by a conscious observer. For this example, choose phases so that ⟨d₀|d₁⟩ = μ is real and lies between zero and one. This restricts the example to a simple family; a complex overlap would also shift the fringe phase.
Ignore the marker and calculate
Measure the path in the balanced output basis |+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ − |1⟩)/√2, without conditioning on a marker outcome. Projecting the joint state onto the first output leaves the marker vector (|d₀⟩ + eiφ|d₁⟩)/2. Taking its squared norm gives
P₊ = (1 + μ cos φ)/2, P₋ = (1 − μ cos φ)/2.
At μ = 1, the marker states coincide and the full phase dependence remains. At μ = 0, they are orthogonal and both unconditioned probabilities are one half for every phase. At μ = 0.6 and φ = 0°, the probabilities are 0.8 and 0.2; at 180° they exchange.
For this balanced setup, fringe visibility V = (Pmax − Pmin)/(Pmax + Pmin) equals μ. The word “visibility” here has a mathematical definition across a phase scan. A single probability measured at one phase does not determine it.
How distinguishable are the markers?
Choose |d₀⟩ = |0⟩ and |d₁⟩ = μ|0⟩ + √(1 − μ²)|1⟩ in a separate two-dimensional marker space. The states are normalized and have the required overlap. For equally likely pure markers, optimal binary discrimination has distinguishability D = √(1 − μ²), where the best success probability is (1 + D)/2.
At μ = 0.6, D = 0.8 and the optimal success probability is 0.9. In this ideal family, V² + D² = 1. Englert's published inequality places the visibility–distinguishability relation in a more general two-way setting. Equality in our chosen pure, balanced construction should not be imposed on every noisy experiment.
A lost fringe does not identify one cause
The same reduced visibility can arise through different physical histories. Suppose no path marker is introduced, but uncontrolled phase variation averages the cross term. A suitable phase distribution can reproduce the same measured fringe visibility. Reduced output contrast alone then cannot decide which mechanism occurred.
This is a question of experimental discrimination. Measurements of the marker, independent phase stabilization or changed preparation conditions may distinguish accounts that fit the same original scan. A broader record can retain information absent from the path subsystem alone.
Conditioning on selected marker outcomes may reveal interference in subensembles even when the unconditioned path distribution is flat. The selected records must be compared using the ordinary classical communication required to identify those subensembles. Sorting data after reception does not send an earlier controllable message through the interferometer.
Continue in the existing laboratory
The site's interference laboratory already lets you vary phase and coherence, inspect probabilities and export calculated values. Use μ as the laboratory's visibility input to explore the reduced path state in this example. The new contribution here is the explicit joint-state construction and the distinction between a reduced description and its possible physical causes.
For SMM/DTT, the published phase-selection paper concerns the mathematical compatibility of complex structure with a given representation. An account of coherence loss must additionally specify coupling, records and which subsystem is observed. Calling a correlation inner-time memory does not by itself supply those dynamics or distinguish the model from the standard calculation.
Sources and further reading
- Englert, B.-G. (1996), Fringe Visibility and Which-Way Information: An Inequality, Physical Review Letters 77, 2154–2157. DOI: 10.1103/PhysRevLett.77.2154.
- Entanglement across separation; waves and individual detections; real and complex descriptions.
- Duality of Time: book contents.