Beyond the arcsine law: exact time-in-lead statistics of a continuously monitored qubit

Abstract

For a continuously monitored qubit under an ideal diffusive quantum-nondemolition measurement, the fraction of the pre-collapse trajectory during which a given outcome leads is derived exactly—and it is not Lévy’s arcsine law, but an explicitly solvable one-parameter deformation of it. In half-log-odds coordinates the measurement diffusion becomes the exactly solvable tanh-drift (Beneš) filter, whose path law is exactly the initial-population-weighted mixture of two unit-drift Brownian motions—the sequential-testing posterior of Shiryaev. The time-in-lead fraction up to a localization threshold then equals a mixture of drifted-Brownian exit-occupation laws, with a joint Laplace transform in closed form; the family runs from an arcsine-adjacent (but numerically non-arcsine) law to a Bernoulli law in the collapse limit. A second exact identity gives the mean time the eventually-losing outcome leads: exactly half a measurement time from a symmetric start. The reduction is classical filtering theory; what is new is the observable and its exact law for the monitored qubit. All results are verified by simulation, with a step-size study and bootstrap confidence intervals. Target venue: Journal of Statistical Mechanics: Theory and Experiment. v2 update (August 2026). Adds the classical discrete-walk references for the arcsine tradition (Sparre Andersen 1953, 1954, Mathematica Scandinavica) to the introduction's lineage. No results, proofs, or claims change. v3 (August 2026) — the promised inversion is now performed. The closed-form joint transform is inverted numerically with a stated error budget rather than left as a promise; the analytic steps the inversion relies on are justified rather than asserted; and an appendix lemma bound is corrected and proved. Earlier versions remain in the version history.