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arXiv:2610.04556

The collapse time of weak measurement on NNN entangled sites

Truong-Son P. Van, David W. Snoke

quant-phSubmitted Oct 3, 2026• Announced Oct 6, 2026• 0 citations
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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 $1-δ$, we prove $T_N/\ln N\to1/2$ in $L^1$, with time measured in units of the characteristic measurement time $τ_m$ of each detector. We also connect the asymptotic stabilization theorem of that work to a finite measurement: the first site to reach $1-δ$ is the eventual outcome with conditional probability exactly $1-δ$. 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.

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21 pages, 1 figure

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We determine the time required for local weak measurements to collapse to one occupied site in an initial superposition of NNN alternatives in the model of PRA 111, 032217 (2025). For equal initial occupation probabilities and a fixed confidence threshold 1−δ1-δ1−δ, we prove TN/ln⁡N→1/2T_N/\ln N\to1/2TN​/lnN→1/2 in L1L^1L1, with time measured in units of the characteristic measurement time τmτ_mτm​ of each detector. We also connect the asymptotic stabilization theorem of that work to a finite measurement: the first site to reach 1−δ1-δ1−δ is the eventual outcome with conditional probability exactly 1−δ1-δ1−δ. 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-NNN law.
It is well established that starting only with strong, projective quantum measurements, experiments can be designed to allow weak measurements, which lead to a random walk between the possible final measurement outcomes. However, one can ask the reverse question: starting with only weak measurements, can all the results of standard strong measurements be recovered? Prior work has shown that some results can be, such as the Born rule for the probability of measurement outcomes as a function of wave intensity. In this paper, we show that another crucial result can be reproduced by purely weak measurements, namely, the collapse of a many-body, nonlocally entangled wave function on a timescale comparable to the characteristic time of a single, local measurement. For an entangled state of a single excitation among NNN qubits, we find the collapse time scales as a double logarithm of NNN. This result affirms the self-consistency of the hypothesis that spontaneous weak measurements lie at the base of all physical measurements, independent of human observers.
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.
We numerically investigate the long-time behavior of the density-density auto-correlation function in driven lattice gases, with particle exclusion and periodic boundary conditions in one, two, and three dimensions using precise Monte Carlo simulations of larger system sizes than previous studies. In the one-dimensional asymmetric exclusion process on a ring with half the lattice sites occupied, we find that correlations induce extremely slow relaxation to the asymptotic power law decay We compare the crossover functions obtained from our simulations with various analytic results in the literature, and analyze the characteristic oscillations that occur in finite systems away from half-filling. As expected, correlations are weak in three dimensions and consequently the mean-field description is adequate. We also investigate the relaxation towards the non-equilibrium steady state in the two-time density-density auto-correlations, starting from strongly correlated initial conditions. We obtain simple aging scaling behavior in one, two, and three dimensions, with the expected power laws. We numerically investigate the behavior of driven lattice gases with nearest neighbor interactions at half-filling with periodic boundary conditions below and at the critical temperature using Monte Carlo simulations of very large lattices in two dimensions. This work is one of few that explores the relaxation to a non-equilibrium steady state. We obtain data collapse for the finite-size scaling form of density-density auto-correlation function at the critical point. We achieve data collapse using finite-size scaling of the time-dependent order parameter during the transient regime starting from strongly correlated initial conditions. We present simple aging scaling of the density-density auto-correlation function at the critical point starting from strongly correlated initial conditions using Monte Carlo simulations of two different lattice anisotropies. We thus unambiguously confirm the critical exponents determined by renormalization group methods using measurement of dynamic quantities in the transient regime. Measuring these dynamic quantities in the transient regime provides more conclusive measurements of the critical exponents than previous studies measuring static quantities in the stationary state. We provide qualitative arguments that the lattice anisotropy determines the steady-state for sub-critical quenches.
The quantum measurement problem—why observation appears to collapse superposition states—has resisted resolution despite decades of progress in decoherence theory. While environmental decoherence explains the emergence of classicality, it lacks a quantitative criterion distinguishing "measured" from "merely decohered" systems. This work proposes such a criterion: measurement occurs when environmental coupling strength, quantified by the dimensionless parameter W = (I * tau) / tau0, exceeds a universal threshold W0. Here I is path distinguishability (Englert’s D), tau is information retention time, and tau0 is system coherence time. The framework predicts continuous visibility decay V(W) = 1 / [1 + exp(kappa * (W - W0))] rather than binary collapse, with threshold W0 ≈ 1–3 across platforms after normalization. Testable predictions include: (1) I × tau trade-off (different combinations of distinguishability and retention yielding identical visibility at matched W); (2) quantum eraser timing dependence (recovery window ~ tau0); and (3) finite collapse propagation speed, formalized as the Causal Limit Postulate (v = c in vacuum). The model predicts a measurable delay (Delta t ≈ L / c) and introduces Causal Viscosity, where n_eff ~ Gamma^(1/2), to describe propagation drag in decohering media. A cavity-QED toy model with 87Rb atoms validates the framework with realistic parameters, independently confirmed by dispersive regression.
We study discrete-time quantum trajectories of a finite-dimensional system in a disordered environment, where the instrument applied at each step is determined by an invertible measure-preserving dynamical system. For general quantum instruments on a standard Borel outcome space, we construct the quenched probability law on sequences of measurement outcomes for a fixed realization of the disorder and a given initial state. Under a quenched Doeblin condition with a uniform deterministic minorization constant and an environment-dependent minorizing probability measure, we establish the existence and uniqueness of an equivariant family of random posterior laws and identify its barycenter as the unique dynamically stationary state for the associated non-selective channel cocycle. If the environment is ergodic, we prove a pathwise ergodic theorem valid for every measurable choice of initial state: for almost every realization of the disorder, time averages of posterior states converge almost surely under the corresponding quenched law to the disorder average of the unique dynamically stationary state. For an ergodic environment, we establish quenched variance asymptotics and, when the asymptotic variance is positive, a central limit theorem and standardized Berry--Esseen bounds for additive functionals generated by bounded measurable real-valued functions of the environment and posterior state. These results hold for every measurable initial state and almost every disorder realization. We then show that the stationary annealed joint process of instruments and posterior states inherits the ααα-mixing of the instrument process, up to an exponentially decaying term. Finally, we provide classes of examples involving perfect and imperfect measurements, including models with continuous or finite outcome spaces, that satisfy the standing Doeblin condition.