Path-Distinguishability Gaps at Fixed Visibility in Noisy Qubit Interferometers

Abstract

The wave--particle duality relation bounds visibility and path distinguishability but does not determine their attainable values in a fixed noisy interferometer. We characterize the complete attainable region for a two-path interferometer with a qubit marker, varying the marker input while fixing the source, preparation channel, path-dependent interaction, and subsequent marker channel. At each feasible fringe visibility, the Helstrom-optimal distinguishabilities form at most two closed intervals, which can be separated by a gap. The intervals arise from opposite axial marker polarizations with the same fringe-amplitude modulus. Fixing the complex fringe amplitude instead gives a connected interval; for balanced original-path discrimination, a gap requires nonunital marker preparation. For a balanced pure source, relative $π$ rotations, and no subsequent marker noise, we obtain a necessary and sufficient geometric criterion for a gap under some rotation axis perpendicular to the preparation displacement. Explicit channels and a time-independent Lindblad realization show how the gap depends on preparation geometry and evolution, with exact onset and optimization formulas at fixed fractional visibility. We also give sufficient bounds for excluding distinguishability intervals under calibrated finite-resolution errors.

The paper asks which visibility–distinguishability pairs are attainable in a fixed noisy two-path interferometer when only the input state of a qubit marker varies. The source, preparation channel, controlled marker unitaries, and common subsequent marker channel are fixed; distinguishability $D$ is the Helstrom optimum for inferring the original path from the marker alone. Using affine Bloch-channel representations, fixed visibility selects at most two planar disks in the input Bloch ball, corresponding to opposite axial marker polarizations. Each disk maps continuously to a closed interval of distinguishabilities, so a fixed-visibility section contains at most two intervals, potentially separated despite satisfying $V^2+D^2\leq1$. Their endpoints follow from quadratic optimization over disks. Fixing the complex fringe amplitude instead gives one connected interval. With balanced path priors, a gap requires nonunital preparation, but nonunitality alone is insufficient. For a balanced pure source, relative $\pi$ rotations, no subsequent marker noise, and axes perpendicular to the preparation displacement $\mathbf t$, some axis produces a gap exactly when $\mathbf t$ is not an eigenvector of $TT^{\mathsf T}$. An explicit CPTP channel yields $D\in[0.1718,0.2245]\cup[0.3575,0.4102]$ at $V=0.2$. A time-independent Lindblad realization gives exact gap-onset and optimization formulas. Calibrated error and input-coverage bounds provide sufficient exclusion certificates. The gaps concern optimized discrimination, not suboptimal guessing probabilities. Certification requires coverage of untested inputs and attainable values on both sides at a common visibility. Joint noise and postselection are excluded; the two-interval result relies on qubit geometry. No experimental data are reported.