Convergence of repeated QND measurements and the wave function collapse

H. A. Haus

Michel Bauer, Denis Bernard

Motivated by recent experiments on quantum trapped fields, we give a rigorous proof that repeated indirect quantum nondemolition (QND) measurements converge to the collapse of the wave function as predicted by the postulates of quantum mechanics for direct measurements. We also relate the rate of convergence toward the collapsed wave function to the relative entropy of each indirect measurement, a result which makes contact with information theory.

Gordey B. Lesovik

Christopher M Struck

The Bare Binary Biconditional selected 1/2 as the unique exchange-invariant law on a bare binary alternative and connected independent repetitions of that law to half-frequency and Lebesgue measure. This paper develops the separate dynamic question that originally motivated the biconditional and the present research series: in what precise sense can an accumulating observation record approach a definite binary outcome? A self-contained classical result is given first. Let Θ ∈ {0,1} have the symmetric prior, let observations be conditionally independent with strictly positive and distinguishable likelihoods, and let 𝑞𝑛 = 𝑃(Θ = 1 ∣ 𝑌1,…,𝑌𝑛). Then (𝑞𝑛) is a bounded martingale beginning at 1/2, and identifiability, not martingale convergence alone, gives 𝑞𝑛 → 1{Θ=1} almost surely. Its two endpoints therefore have equal probability. A binary-probe specialization places the observation record itself in {0,1}ℕ; a Gaussian model is retained as a continuous-observation comparison. Repeated independent runs, rather than the dependent observations within one run, produce the fair binary endpoint record connected to [1]. Bauer and Bernard’s repeated quantum non-demolition measurement theorem supplies a bounded quantum counterpart: pointer weights are martingales; under their nondegeneracy condition they converge to pointer-state endpoints; the endpoint probabilities are the initial pointer weights; and relative entropy governs the asymptotic decay of losing weights. Their theorem does not choose 1/2. This paper imports that starting value only under the explicit bare symmetric premise of [1]. Reproducible classical simulations illustrate finite-depth posterior behavior and information-rate scaling without simulating quantum dynamics or treating finite posteriors as completed limits. The remaining claim that observation gives physical significance to this convergence structure is stated separately as the author’s quantum convergence conjecture. Note clarifying this paper’s conclusion: “Particle-like” refers here to a definite observation outcome. In the repeated-QND setting analyzed, the quantum wave-function measurement model mathematically predicts convergence to a definite, particle-like observation outcome. The phrase “not evidence that the object has ceased to be a wave” means only that a particle-like observation is consistent with how the quantum wave function predicts the system should behave upon measurement. This clarification does not deny wave-function collapse or address wave or particle behavior outside the observation process discussed.

Ronald F. Peierls

Orly Alter, Y. Yamamoto

It is shown that a series of repeated weak quantum nondemolition measurements performed on a single quantum system gives no information about the wave function of the system. The physical explanation, based on the quantum Brownian motion and the continuous collapse of the wave function which originate in the projection postulate, is discussed in two specific examples.