Beyond the arcsine law: exact time-in-lead statistics of a continuously monitored qubit
Paper 19B · Pødenphant Lund, T. (2026) · Technical note · Live on Zenodo
Occupation-time laws are among the oldest exact results about stochastic paths, and interest has returned to physics with the demonstration that arcsine laws govern thermodynamic currents in mesoscopic machines. This note asks the corresponding question for a canonical quantum-stochastic process, the diffusive quantum trajectory of a measured qubit, and answers it exactly. For an ideal diffusive QND readout, the population obeys a closed scalar diffusion; in half-log-odds coordinates it becomes an autonomous tanh-drift equation whose path law is exactly a Born-weighted mixture of two Brownian motions with constant drift ±1, as measures on the full path σ-algebra. From that reduction the time-in-lead occupation law up to localisation follows, with an exact joint Laplace transform.
| DOI (concept) | 10.5281/zenodo.21471598 |
| Type | Technical note (statistical mechanics of quantum trajectories; exact results with numerical verification) |
| Target venue | J. Stat. Mech. (primary); J. Phys. A (secondary) |
| Regime | Ideal: unit detector efficiency, no Hamiltonian drive, QND coupling |
| Author | Tomas Pødenphant Lund [ORCID] |
TL;DR
Two classical facts anchor the process. The population is a bounded martingale, so it converges and cannot stall in the middle: it lands on an outcome with Born statistics, the continuous version of the martingale argument for repeated QND measurement. And the noise amplitude is largest at the halfway line and vanishes at the poles, so the process moves fastest exactly where the lead changes hands and lingers near the outcomes. That asymmetry is the mechanism behind every result here.
Fix a threshold ε, let Tε be the first time the state is localised within ε of an outcome, and let the time-in-lead fraction be the share of that horizon spent above the halfway line. In half-log-odds the localisation band is a symmetric interval, and the family parameter turns out to be its half-width together with the starting point. There is no fixed time horizon anywhere in the problem.
The reduction
Theorem 1 makes two statements. In half-log-odds coordinates the measurement diffusion becomes the autonomous equation with tanh drift, a Brownian motion pushed by a drift that always points away from the lead line and saturates quickly to unit strength. And the path law is exactly the Born-weighted mixture of a drift +1 and a drift −1 Brownian motion, both started at the same point, as an identity of measures on the whole path σ-algebra rather than only at fixed times.
The proof reweights driftless Brownian paths by a positive martingale built from a space-time harmonic function. Girsanov then gives the reweighted process the tanh drift. The pleasant step is that the hyperbolic cosine is by definition an average of two exponentials, and exponential reweighting is exactly what gives a Brownian motion constant drift, so the reweighting is algebraically identical to a convex combination of the two constant-drift changes of measure, with weights that come out as the Born probabilities. The mixture identity is a computational identity, not a limit.
Three remarks place the theorem. The content is classical, and the note says so plainly: the tanh-drift diffusion is Beneš's exactly solvable filter (1981), the population process is Shiryaev's sequential-testing posterior with precisely this Bayesian mixture structure, and the discrete-time QND counterpart is Bauer and Bernard (2011). No priority claim is made for the representation itself, only for the use of it. Because the statement is on the full path space it transfers to stopping-time functionals without further argument. And the mixture weights being the Born probabilities means the pre-collapse ensemble is an exact classical mixture of two conditioned evolutions, each with unit drift toward its own outcome.
The occupation law
Corollary 1. The pair of times spent above and below the lead line is, in distribution, the Born-weighted mixture over drift ±1 of the corresponding pair for a drifted Brownian motion run to first exit from the symmetric interval. The joint Laplace transform is computed by Feynman–Kac: the transform solves an ODE killed at one rate above zero and another below, with value one at the boundaries; on each half the solution combines two exponentials, and four coefficients are fixed by the two boundary conditions plus smooth matching at zero, reducing to an explicit small linear system. The qubit transform is then the mixture of the two drift versions. Moments follow by differentiation, but the density requires numerical inversion, and no closed form is claimed for the density itself.
The family has two recognisable endpoints. As the band narrows to nothing the drift has no time to act and the law converges to the exit-occupation fraction of a driftless Brownian motion, a scale-invariant law that is arcsine-adjacent but measurably not arcsine: Lévy's law lives on a fixed horizon and the exit law on a random one. The difference is not normalisation, since both laws are scale invariant and no rescaling carries one into the other; what changes the law is the type of horizon, a fixed clock time versus a first commitment. In the other direction, as the threshold tightens each mixture component exits on its own drift side and the law tends to Bernoulli with the Born parameter.
Between the endpoints the law is a sharpening U, because multiplicative noise peaks at the lead line and dies at the poles. The edge mass grows with the band half-width, starting below the arcsine value at the driftless end, crossing it at moderate widths, and rising to one at the Bernoulli end. This monotone growth is reported as a numerical observation over the tested range, not a theorem. Lévy's law is a reference point the family crosses, not a limit it approaches.
Exact identities
- Exit probability, and the Born rule at finite ε. Optional stopping on the population martingale gives the exit probability as (p0 − ε)/(1 − 2ε), and the drifted Brownian exit formula reproduces the same number on the mixture side as a consistency check. As ε goes to zero this is the Born rule, recovered as the limit of an exact finite-threshold law.
- The loser's total lead time. Remove the absorption and ask how long the eventually-losing outcome leads in total. From the symmetric start the mean is exactly half a measurement time, with the full Laplace transform available in closed form. The value one-half is specific to the symmetric start; the general expression for the drift-aligned component decays exponentially in the initial log-odds.
- The mixture mean. Averaging over both components, where the counter-aligned one must first cross zero, the sum collapses to a closed form: the mean time the wrong outcome leads is that outcome's Born weight times one plus the magnitude of the initial log-odds, in measurement-time units.
Numerical verification
Everything is checked against direct simulation of the measurement diffusion itself, with the drift mixture simulated independently as a second ensemble. Fixed-step Euler is known to bias exit and occupation functionals through missed barrier crossings, so a step-size refinement study bounds the effect, with bootstrap intervals throughout. The path-measure identity holds at fixed horizon across three ensembles within Monte Carlo error at both a symmetric and a skewed start. Two-sample Kolmogorov–Smirnov tests between the occupation fraction from the population simulation and from the mixture show no trend in step size, indicating the same law at every refinement level. The exact transform is matched by Monte Carlo from both ensembles on a grid of transform arguments at both starting points, with misses in both directions and no systematic trend, so any Euler bias sits below Monte Carlo resolution. The exit formula reproduces its closed form to about 10−9, and the driftless limit law was tested directly: its edge mass and second moment both exclude the arcsine values from below, with a one-sample KS distance to the arcsine law that is decisive.
Discussion and scope
What the law is for. Diffusive qubit trajectories of exactly this type are recorded routinely in circuit-QED experiments, and the time-in-lead fraction is computed from the sign process of an already-reconstructed trajectory, so only the sign of the population offset and the localisation time enter, with no free parameters once the starting odds and threshold are fixed. The reconstruction carries the calibration burden, however: it needs the measurement rate, the detector efficiency, and the whole filtering model. The statistic economises on what is extracted from a reconstructed trajectory, not on the calibration behind it.
Because the law holds in the ideal case, its role is that of the exactly solvable reference. It serves as a null hypothesis for the ideal diffusive QND unravelling, so that jump unravellings, detector inefficiency and residual dynamics turn from qualitative deformations into quantitative deviations; and as a localisation diagnostic, since the edge mass interpolates along the family, so a measured edge mass locates the effective threshold at which an experiment's trajectories have committed.
Relation to near neighbours. Exact spike statistics for the same kind of monitored qubit have been computed before; occupation statistics are complementary, resolving the coarse who-leads structure rather than the fine excursion structure. The Beneš tanh diffusion of Theorem 1 has been studied for first-passage, bridges and conditioned processes, but occupation and time-in-lead functionals are not treated there and the quantum-measurement identification is not made. Read classically, the result is an occupation law for how long a sequential test's posterior favours the hypothesis that ends up rejected; the note states that it knows of no such exit-occupation law in either the quantum-trajectory or the sequential-analysis literature, and that a predecessor would be welcome.
Open problems. A Rabi drive breaks the QND structure, and whether any exact statement survives a weak drive is open. Detector inefficiency makes the diffusion two-dimensional; the martingale structure of the population survives but the log-odds reduction does not obviously follow, and the fate of the mixture representation under inefficiency is called the most experimentally pressing question. Jump unravellings pose the parallel in a different trajectory topology, and the multi-outcome generalisation is open already at the level of definition, since it is unclear what "the lead" should mean on a simplex.
Connections to other papers in the series
- Paper 19A (One mixer suffices) — the sibling note. 19A asks why the Born exponent is two, from the algebra of the probability rule; 19B asks how the measurement race actually unfolds in time and recovers the same Born probabilities as the mixture weights of its diffusion.
- Paper 10 (Race architecture) — the substrate-general race vocabulary. In race and commit models of sequential decision the same functional measures how long the eventually losing option was winning, so an exactly solvable instance in a physical system fixes a useful reference point. The mathematics does not depend on that reading, and the note stands on its own as a statistical-mechanics result.
Read the paper
The full note is on Zenodo (concept DOI 10.5281/zenodo.21471598):