How long does the losing answer stay ahead?

Paper 19B · Beyond the arcsine law · Read on Zenodo

A quantum bit does not collapse the instant you look at it. If you watch it continuously, its odds wobble: first one outcome is ahead, then the other, back and forth, until the state finally settles on an answer. It is a race, and you can watch it run. This note works out exactly how that race tends to go. One answer is a small surprise: the outcome that eventually loses spends, on average, exactly half a measurement leading. And the famous coin-flipping law that governs who-is-ahead in ordinary random walks turns out not to be quite the right law here.

The race you can actually watch

In a continuous quantum measurement, the detector delivers a noisy signal, and from it you keep updating the probability of each outcome. That running probability does not march steadily to its destination. It jitters. It crosses the halfway line some finite number of times, trading the lead between the two outcomes, and then it commits.

Two facts about this wobble anchor everything else. First, the running probability is a fair process in the technical sense, so it has to converge, and it cannot get stuck in the middle. It ends up at one outcome or the other, and the chance of each is exactly the Born probability you started with. Collapse and the Born rule are outputs of the wobble, not extra ingredients.

Second, and this is the mechanism behind everything that follows, the wobble is not the same size everywhere. The noise is largest exactly at the halfway line and dies away near the outcomes. The process moves fastest precisely where the lead changes hands, and it lingers once it is near an answer. So the race is quick and skittish in the middle, and sticky at the ends.

The question, made precise

Pick a threshold: say we call the measurement finished when the probability has come within a hair of one outcome. Now ask what fraction of the time, up to that finishing point, a given outcome was ahead. Call it the time-in-lead fraction.

The classic answer to questions like this is Lévy's arcsine law from 1940, and it is famously counterintuitive: for an ordinary random walk over a fixed stretch of time, a roughly even split of the lead is the least likely outcome. One side tends to hog the lead. The law is U-shaped, piling up probability at both extremes.

So does the measured qubit follow the arcsine law? Not quite, and the near-miss is interesting.

The answer: a family, not a single law

The note derives the exact law, and it turns out to be a one-parameter family indexed by how strict your finishing threshold is. Both ends of the family are recognisable.

At a loose threshold the law converges to something scale-invariant that sits close to the arcsine law but is measurably not it. The reason is not normalisation. It is that Lévy's law lives on a fixed clock time, while this one lives on a random horizon, the moment the measurement first commits. A fixed deadline and a first commitment are different kinds of stopping, and they give different laws. The gap is decisive rather than marginal in the numerical tests.

At a strict threshold, following the race all the way to the outcome, the law collapses to the simplest thing imaginable: the winner led essentially the whole time, and which outcome wins is settled by the Born probability.

In between, the law is a U that sharpens as you tighten the threshold, because the noise peaks at the lead line and dies at the poles, so paths cross the middle quickly and dwell at the ends. The weight piled at the extremes grows as the threshold tightens, starting below the arcsine law's value, crossing it at moderate settings, and rising toward certainty at the strict end. Lévy's law is a reference point the family crosses, not a limit it approaches.

The loser's half-measurement

The cleanest single number in the paper concerns the outcome that loses. Drop the finishing threshold entirely and just ask: in total, how long was the eventually-losing outcome ahead?

Starting from even odds, the answer is exact and clean. On average, the loser leads for precisely half a measurement time, no matter how long the whole localisation takes. There is a companion formula for uneven starting odds, where the average time the wrong outcome leads works out as that outcome's Born weight times one-plus-its-deficit in log-odds.

It is worth being precise about the scope: the one-half is specific to the even-odds start, and the paper flags that rather than letting the tidy number travel further than it should.

How it is done

The trick is a change of coordinates. Written in terms of log-odds rather than probability, the wobble becomes a much simpler process: an ordinary random walk pushed by a drift that always points away from the halfway line and quickly saturates.

Then comes the pretty step. That drifting process turns out to be exactly a mixture of two much simpler ones: a random walk drifting steadily toward the first outcome, and one drifting steadily toward the second, combined with weights that are precisely the Born probabilities. Not approximately, and not only at fixed times, but as a statement about whole paths, which is what lets it be used for questions about random stopping times.

So the pre-collapse ensemble is an exact classical mixture of two conditioned evolutions, each marching to its own answer, weighted by the Born rule. Once you have that, the time-in-lead question reduces to a classical one about drifting random walks, which classical machinery solves exactly.

What is borrowed and what is new

The paper is emphatic on this point. The underlying mathematics is classical and is not its own: the drifting process is a known exactly-solvable filter from 1981, the probability process is the posterior from a classical sequential-testing problem, and the discrete-time version for repeated quantum measurements was published in 2011. The note makes no priority claim on the representation itself, only on the use of it.

What is offered as new is the time-in-lead law for the monitored qubit: the exact transform, the interpolating family it spans, and the exact identities that come with it, including the loser's half-measurement. The paper also states plainly 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.

Honest scope

The setting is ideal throughout: a perfect detector, no driving, and a measurement that does not disturb the quantity being measured. The result is therefore offered as an exactly solvable reference case, a benchmark that real detectors can be compared against, not as a description of any laboratory dataset.

Within that framing it has two uses. It is a null hypothesis for the ideal case, so that detector inefficiency and residual dynamics turn from qualitative deformations into quantitative deviations. And it is a localisation diagnostic, since a measured extreme-weight locates the effective threshold at which an experiment's trajectories have committed.

Two further caveats travel with it. The density of the law needs numerical inversion; no closed form is claimed for the density itself. And the growth of the extreme weight with the threshold is reported as a numerical observation over the range tested, not as a theorem. The open problems are named: what survives a weak drive, what happens when the detector is inefficient (called the most experimentally pressing question), and how to define the lead at all when there are more than two outcomes.

Read the paper

The full note is freely available on Zenodo (concept DOI 10.5281/zenodo.21471598):

Read on Zenodo → · Technical version · Dansk version

Related on this site:

• Paper 19A — Why is it the square? — the sibling note: the same Born probabilities approached from the algebra of the probability rule rather than the dynamics of a measurement.
• What is a race? — the picture of competing routes trading the lead, which this note's vocabulary borrows, though the mathematics stands on its own.