The collapse time of weak measurement on $N$ entangled sites

Abstract

We determine the time required for local weak measurements to collapse to one occupied site in an initial superposition of $N$ alternatives in the model of PRA 111, 032217 (2025). For equal initial occupation probabilities and a fixed confidence threshold $1-\delta$, we prove $T_N/\ln N\to 1/2$ in $L^1$, with time measured in units of the characteristic measurement time ${\tau }_m$ of each detector. We also connect the asymptotic stabilization theorem of that work to a finite measurement: the first site to reach $1-\delta$ is the eventual outcome with conditional probability exactly $1-\delta$. The result corrects the previously proposed double-logarithmic asymptotic scaling. The finite-range numerical fit does not determine a constant asymptotic correction or a precise finite-$N$ law.

21 pages, 1 figure