Q-based, objective-field model for wave-function collapse: Analyzing measurement on a macroscopic superposition state

TL;DR

The paper tackles the quantum measurement problem by adopting a Q-based, objective-field framework in which a system’s phase-space amplitudes $x$ and $p$ evolve under forward-backward stochastic dynamics of the Husimi $Q$-function. By analyzing both single-mode and two-mode (system–meter) Cat states, it shows that amplification of a quadrature leads to macroscopic distinguishability and that Born’s rule emerges from the postselected distribution of amplified amplitudes, while the inferred post-measurement state behaves as a projection onto the measured eigenstate or coherent state. The authors argue for macroscopic realism in the amplified outcomes and present a two-stage collapse: first amplification creates distinct branches, then inference conditioned on the meter outcome yields the final state with information loss about the meter’s complementary variable. They further demonstrate that postselected “hidden” distributions cannot be identified with quantum $Q$-functions, aligning the MR view with a nonclassical, measurement-induced collapse. Overall, the work provides a concrete, testable mechanism for quantum-to-classical transition within a deterministic, retrocausal $Q$-based description that remains compatible with Bell-type constraints while challenging conventional notions of quantum state completeness.

Abstract

The measurement problem remains unaddressed in modern physics, with an array of proposed solutions but as of yet no agreed resolution. In this paper, we examine measurement using the Q-based, objective-field model for quantum mechanics. Schrodinger considered a microscopic system prepared in a superposition of states which is then coupled to a macroscopic meter. We analyze the entangled meter and system, and measurements on it, by solving forward-backward stochastic differential equations for real amplitudes $x(t)$ and $p(t)$ that correspond to the phase-space variables of the Q function of the system at a time $t$. We model the system and meter as single-mode fields, and measurement of $\hat{x}$ by amplification of the amplitude $x(t)$. Our conclusion is that the outcome for the measurement is determined at (or by) the time $t_{m}$, when the coupling to the meter is complete, the meter states being macroscopically distinguishable. There is consistency with macroscopic realism. By evaluating the distribution of the amplitudes $x$ and $p$ postselected on a given outcome of the meter, we show how the $Q$-based model represents a more complete description of quantum mechanics: The variances associated with amplitudes $x$ and $p$ are too narrow to comply with the uncertainty principle, ruling out that the distribution represents a quantum state. We conclude that the collapse of the wavefunction occurs as a two-stage process: First there is an amplification that creates branches of amplitudes $x(t)$ of the meter, associated with distinct eigenvalues. The outcome of measurement is determined by $x(t)$ once amplified, explaining Born's rule. Second, the distribution that determines the final collapse is the state inferred for the system conditioned on the outcome of the meter: information is lost about the meter, in particular, about the complementary variable $p$.

Figures (25)

• Figure 1: Solutions of the forward-backward equations (\ref{['eq:forwardSDE-2-1-1-1']}) (top) and (\ref{['eq:backwardSDE-2-1-1-1']}) (lower) , modeling the measurement of $\hat{x}$ on a system prepared in the state (Eq. (\ref{['eq:sq-state-x']})), where $x_{1}=3$. The figures are for the measurement on the highly squeezed state modeling the eigenstate $|x_{1}\rangle$ of $\hat{x}$, where $r=3$. The plots are generated with $10^{6}$ trajectories.
• Figure 2: As for \ref{['fig:fb-1']}, where $x_{1}=3$. Here, the two figures solve for the measurement on a coherent state $|\alpha_{0}\rangle$, where $\alpha_{0}=1.5$ and $r=0$. The plots are generated with $10^{6}$ trajectories.
• Figure 3: Forward-backward stochastic solutions modeling the measurement of $\hat{x}$ on a system prepared in a superposition given by $|\psi_{S}\rangle$ (Eq. (\ref{['eq:sup-sq']})) with $c_{1}=-ic_{2}=1/\sqrt{2}$. we choose $r=2$ which models measurement on a superposition of eigenstates of $\hat{x}$ (Eq. (\ref{['eq:sup-eigen']})). The top plot shows $x_{1}=0.7$ and $r=2$, which models measurement on a microscopic superposition of eigenstates of $\hat{x}$. The lower plot shows $x_{1}=6$ and $r=2$ which models measurement on a macroscopic superposition of eigenstates. Plots show $10^{6}$ trajectories. $t_{f}=3/g$.
• Figure 4: Forward-backward stochastic solutions modeling the measurement of $\hat{x}$ on a system prepared in a state $|\psi_{cat}\rangle$ [Eq. (\ref{['eq:sup-cat']})], a superposition of coherent states as given by $|\psi_{S}\rangle$ (Eq. (\ref{['eq:sup-sq']})) with $c_{1}=-ic_{2}=1/\sqrt{2}$, $r=0$ and $\varphi=\pi/2$. The plots are for $\alpha_{0}=0.5$ and $r=0$, which models measurement on a microscopic superposition of coherent states. Plots show $10^{6}$ trajectories. $t_{f}=3/g$.
• Figure 5: As for Figure \ref{['fig:sup-2-1']}. The plots are for $\alpha_{0}=4$ and $r=0$ and $\varphi=\pi/2$, which models measurement on a cat-state [Eq. (\ref{['eq:sup-cat']})] which is a macroscopic superposition of coherent states. Plots show $1.2\times10^{6}$ trajectories. $t_{f}=3/g$.