Quantum timekeeping and the dynamics of scrambling in critical systems

Abstract

In this work, we develop a quantum metrological framework for quantum chaos by showing that local subsystems of information scrambling systems naturally function as quantum stopwatches. The reduced quantum state of a subsystem encodes the passage of time through its growing distinguishability from the initial preparation. Treating time as the estimation parameter, we then derive a generalized quantum Cramer-Rao bound that directly relates the precision of time estimation to the decay of out-of-time ordered correlators (OTOCs) and subsystem quantum Fisher information (QFI). As a main result, we obtain continuity bounds for quantum Lyapunov exponent in terms of the subsystem QFI in quantumly chaotic dynamics. Furthermore, using a scaling analysis based on imaginary-time correlators, we show that the subsystem QFI exhibits universal critical amplification near quantum phase transitions. Our results are demonstrated and verified by a numerical analysis of the dynamics of a chaotic Ising chain.

Figures (2)

• Figure 1: Heatmap of $I_{F}(t,h)$ for the one-dimensional TFIM with a longitudinal field with open boundary conditions. $I_{F}(t)$ peaks approximately around $h=\pm J$ and decays over time. The parameters are $J=1.0$, $g=0.4$, $N=11$.
• Figure 2: (a) Quantum Lyapunov Exponent $\lambda_Q$ as a function of the transverse field $h$. $\lambda_Q$ peaks approximately around $h=\pm J$ demonstrating maximal chaos. (b) Bound on the Lyapunov exponent $\frac{1}{8t}\left(\int_0^t\sqrt{I_F(s,h)}\,ds\right)^2$ as a heatmap. The bound also peaks around the same region as $\lambda_Q$ corresponding to maximal chaos. The results are obtained for the one-dimensional TFIM with a longitudinal field with open boundary conditions. The parameters are $J=1.0$, $g=0.4$, $N=11$.