Double-slit visibility and path information

Let the two scalar detector fields include the same polarization and geometry, and define their mutual coherence by a time or ensemble average. The detector averages a squared field over many optical cycles. Expanding that square leaves two individual intensities and a complex cross correlation. Scanning an extra phase changes the real part of the cross term while leaving the individual intensities fixed. Its magnitude determines the possible contrast if the envelopes vary negligibly during the scan.

Define $\gamma_{12}=\langle U_1^*U_2\rangle/\sqrt{\langle|U_1|^2\rangle\langle|U_2|^2\rangle}$ for the scalar phasor envelopes $U_1,U_2$. Cauchy–Schwarz bounds its magnitude by one. Temporal fluctuations, extended-source spatial incoherence, or uncontrolled phase drift can reduce that magnitude. Orthogonal vector polarizations make the vector cross term vanish even for otherwise coherent waves. One may factor temporal coherence and polarization overlap only when the statistical and polarization assumptions actually allow that separation.

Even perfectly coherent fields cannot cancel completely if their magnitudes differ. For an amplitude ratio $a$, the visibility is $2a/(1+a^2)$ when the polarizations match. Taking one slit almost dark drives visibility toward zero, despite perfect relative-phase stability. Conversely, equal mean intensities do not imply full contrast: a fluctuating phase can wash out the average pattern while each short-time realization remains strongly modulated.

Finite slit widths multiply the two-slit modulation by a diffraction envelope, and distant fringes may sample unequal envelope values. Visibility should therefore be interpreted locally or through a controlled phase scan. Stray light and detector dark offsets raise the minimum without being a fundamental coherence effect. Before interpreting a contrast measurement, measure the individual slit signals and background separately, using the same collection geometry and integration time as the interference measurement.

For an ideal balanced coherent two-path quantum input with normalized pure marker states $|d_1\rangle$ and $|d_2\rangle$, ignoring the marker multiplies the interference cross term by their overlap. The unconditioned visibility is the magnitude of that overlap. Here distinguishability means the optimal advantage in guessing which marker state was prepared, with equal prior probabilities. It is not a vague statement about how much someone happens to know.

Under those pure-state, balanced, otherwise ideal assumptions, visibility and optimal distinguishability obey an equality. Mixed markers, additional dephasing, or imperfect collection generally permit an inequality instead; unequal priors require an appropriately defined discrimination problem. Measuring a marker in another basis and selecting subensembles may reveal conditional fringes, but adding those subensembles restores the original unconditioned result. Human attention is not the mechanism: correlations and averaging are.