Planckian bound on quantum dynamical entropy

TL;DR

The paper defines a quantum dynamical entropy for systems under continuous monitoring by a CNT-like construction and shows that a nonzero entropy rate can arise in generic many-body quantum systems away from large-N limits. By focusing on mesoscopic observables and Gaussian fluctuations, it derives an exact rate formula in the thermodynamic and long-time limit: $s_{\text{CNT}} = \frac{1}{2} \int \frac{d \omega}{2\pi} \left[ \ln \left(1 + \tilde{G}(\omega)\right) - \tilde{G}(\omega) + \frac{\beta \omega}{\sinh (\beta \omega)} G_K(\omega) \right]$ with $\tilde{G}(\omega) = G_K(\omega) + |G_R(\omega)|^2$, and the purification rate $J/t = \frac{1}{4\pi} \int d\omega \; \frac{\beta \omega}{\sinh (\beta \omega)} \; \frac{G_K(\omega)}{1 + G_K(\omega) + |G_R(\omega)|^2}$. It proves a Planckian bound on the entropy rate, $\beta s_{\text{CNT}} \le c$, with $c \approx 0.375$, and shows that microscopic monitoring yields $J/t \to 0$ while mesoscopic monitoring yields nonzero rates; purification obeys a related bound as well. These results connect quantum dynamical entropy to low-temperature quantum chaos and linear-response theory through the fluctuation-dissipation framework and support the use of CNT entropy as a diagnostic for chaos in many-body quantum systems.

Abstract

We introduce a simple definition of dynamical entropy for quantum systems under continuous monitoring, inspired by Connes, Narnhofer and Thirring. It quantifies the amount of information gained about the initial condition. A nonzero entropy rate can be obtained by monitoring the thermal fluctuation of an extensive observable in a generic many-body system (away from classical or large N limit). We explicitly compute the entropy rate in the thermodynamic and long-time limit, in terms of the two-point correlation functions. We conjecture a universal Planckian bound for the entropy rate. Related results on the purification rate are also obtained.

Figures (3)

• Figure 1: The purification setup. Two identical systems $A\bar{A}$ are initialized in an thermal field double state $\sqrt{\rho}, \rho = e^{-\beta H} / \mathrm{Tr}[e^{-\beta H}]$. $A$ is evolved by $U = e^{-i t H}$ and repeatedly measured, yielding outcomes $m_{\delta t}, \dots, m_t$. The latter disentangles (purifies) $\bar{A}$, revealing information about the initial condition.
• Figure 2: Numerical calculation of the quantity $J_s$\ref{['eq:Js-def']} and the purification rate with micro- and mesoscopic monitoring, no the on mixed-field Ising chain $H = \sum_{j=1}^L X_j + h Z_j + J Z_j Z_{j+1}$ ($L + 1 \equiv 1$), $h = 1.245, J = 0.945$, with $L = 6, 8, 10$ [with color code in panel (a)], at $\beta = 0$ by default. (a)$J_s$ with microscopic monitoring, $\delta t = 1$, with Kraus operators $K_{m = \pm} = \sqrt{(\mathbf{1} \pm O)/2 }$ with $O = 0.75 Z_0 Z_2$. (b)$J_s$ with mesoscopic monitoring \ref{['eq:Km']}, with $\delta t = 0.2, 1$, $Q = \sqrt{\gamma / L} \sum_j Z_j$, $\gamma = 0.1, 0.2$. The horizontal lines show the analytical prediction. The time is rescaled by $1/\sqrt{L}$. (c) Purification $J$ with the same microscopic monitoring as in panel (a). The three curves $L = 6, 8, 10$ are indistinguishable. (d) Purification ($J$) with mesoscopic monitoring $Q = \sqrt{\gamma / L} \sum_j (Y_j - \left< Y_j \right>)$$\gamma = 0.1$, $\delta t = 0.25$, $\beta = 0, 0.5, 1$. The straight lines represent the analytical prediction \ref{['eq:jmeso']}. See supp for numerical methods.
• Figure 3: Keldysh correlation function $G_K$ (as function of $\beta \omega$) that maximizes $\beta s_{\text{CNT}}$\ref{['eq:rateformula']}. $G_K(-\omega) = G_K(\omega)$, obtained from numerically maximizing \ref{['eq:rateformula']} subject to the fluctuation-dissipation theorem. Inset: comparison with $4 / |\beta \omega |^{- 1}$ (dashed line).