Decoherent histories with(out) objectivity in a (broken) apparatus

TL;DR

The paper investigates how classicality emerges in quantum dynamics by contrasting environment-induced decoherence (and quantum Darwinism) with decoherent histories in a solvable inflationary tree circuit model. It introduces a MERA-like, scale-invariant architecture with a tunable threshold $\theta_c = \pi/4$ that separates an apparatus phase from an encoding phase, and analyzes both via real-space RG and a decoherent-histories framework using coarse-grained observables. The authors show that approximate decoherent histories arise in both phases, but only the apparatus phase exhibits non-ergodic histories correlated with the measured system, yielding a pointer-states ensemble and explicit objectivity for finite fractions of the apparatus; in the encoding phase, information about the system becomes inaccessible and histories remain ergodic. They develop an exact, numerically tractable method based on the Heisenberg picture and Hubbard–Stratonovich transformation to compute history probabilities and pointer states, and they demonstrate Leggett-Garg inequality violations at late times, highlighting persistent non-classicality alongside emergent classical structure. Overall, the work clarifies the relationship and distinctions between decoherent histories and environment-induced objectivity, and provides a concrete, scalable setting to study classicality in quantum many-body dynamics, with potential implications for cosmology and quantum information.

Abstract

We characterize monitored quantum dynamics in a solvable model exhibiting a phase transition between a measurement apparatus and a scrambler. We show that approximate decoherent histories emerge in both phases with respect to a coarse-grained extensive observable. However, the apparatus phase, where quantum Darwinism emerges, is distinguished by the non-ergodicity of the histories and their correlation with the measured qubit, which selects an ensemble of preferred pointer states. Our results demonstrate a clear distinction between two notion of classicality, decoherent histories and environment-induced decoherence.

Figures (4)

• Figure 1: A dynamically expanding tree model where every line is a qubit and every node is an isometry $v = v(\theta)$\ref{['eq:vdef']}. The root qubit $A$ is entangled with the qubit $S$\ref{['eq:Psi']}. The model has a transition at $\theta_c$\ref{['eq:threshold']}; it is an apparatus measuring $S$ if $\theta < \theta_c$, and a scrambler if $\theta > \theta_c$. In both phases, coarse-grained monitoring of the model \ref{['eq:coarse']},\ref{['eq:Km']} yields decoherent histories \ref{['eq:decay']} (Fig. \ref{['fig:Delta']}). In the encoding phase, the histories are Ornstein-Uhlenbeck like \ref{['eq:Emtmt']} and uncorrelated with $S$. In the apparatus phase, the histories are non-ergodic \ref{['eq:freezinglaw']} and selects an ensemble of pointer states of $S$ (see also Fig. \ref{['fig:pointer']}).
• Figure 2: Decoherent histories probe $\Vert \Delta^{\mathbf{t}} \Vert_{1}$ with $\mathbf{t} = \{L+ \tau\}$, as function of $\tau$ (measurement starting time), for (a) $\theta = 0.15\pi < \theta_c$ and (b) $\theta = 0.3 \pi > \theta_c$. Approximate decoherent histories emerge for coarse-grained measurements ($\Gamma = 0.1$) and $\tau \gg 1$, but not for $\tau \sim 1$ or for fine-grained measurements (fg, $\Gamma \eta_t = 0.1$). Insets (i, ii) show the pdf $p^{\mathbf{t}}$ (no third-party) and marginal distributions in non-decoherent scenarios ($T = 11$). Straight lines show decay rate of $\Vert \Delta^{\mathbf{t}} \Vert_{1}$\ref{['eq:decay']}, and \ref{['eq:decayrate']} in supp. (a) For $\theta < \theta_c$, $\Vert \Delta^{\mathbf{t}} \Vert_{1} \sim \Gamma^{-2} 2^{2\tau (x-1)}$ with no $L$ scaling. (b) For $\theta > \theta_c$, : $\Vert \Delta^{\mathbf{t}} \Vert_{1} \sim \Gamma^{-2} 2^{-T} \max(1, 2 \cos \theta)^{2L}$. Inset (iii): $L$-dependence is shown for two $\theta > \theta_c$ ($\tau = 6$).
• Figure 3: Distribution of pointer states and marginal outcome distribution $p_m$ for $\theta = 0.05\pi < \theta_c$ (a,d), $\theta = 0.15\pi < \theta_c$ (b,e) and $\pi = 0.3 \pi > \theta_c$ (c,f). The color code relates the outcome to the condition state of $S$ inside the Bloch sphere. The points are distributed uniformly with respect to the probability. ($t = 20$ and $\Gamma = 0.05$.)
• Figure 4: The violation of the Leggett-Garg inequality in the apparatus phase (a, $\theta = 0.15 \pi$), and in the encoding phase (b, $\theta = 0.35\pi$), for $t_1, \dots, t_4 = t, \dots, t+3$, and for the operators defined by \ref{['eq:Qt-def']}, \ref{['eq:abc-def']}. The violation, indicated by the data points going above the red line, can happen at arbitrarily late time, controlled by the parameter $t_m$.