Estimating time in quantum chaotic systems and black holes

TL;DR

This work formulates time estimation as a metrological probe of quantum chaos and black hole evaporation, quantified via the quantum Fisher information (QFI). It reveals a universal, size-dependent behavior: full systems maintain a time-invariant QFI under unitary evolution, while subsystems exhibit equilibration-like decay unless the measured region exceeds half the system, at which point the QFI becomes extensive again. The paper connects these dynamics to random-pure-state models, Brownian-GUE toy models, and an evaporating black hole scenario, arguing that Hawking’s semiclassical picture can be reconciled with unitarity only when considering computationally bounded observers. It also offers a quantum-error-correction interpretation and demonstrates a concrete time-estimation experiment via maximum-likelihood estimation. Overall, the results link metrology, chaos, and gravity, suggesting new avenues to explore the role of information-theoretic measures in quantum many-body and gravitational contexts.

Abstract

We characterize new universal features of the dynamics of chaotic quantum many-body systems, by considering a hypothetical task of "time estimation." Most macroscopic observables in a chaotic system equilibrate to nearly constant late-time values. Intuitively, it should become increasingly difficult to estimate the precise value of time by making measurements on the state. We use a quantity called the Fisher information from quantum metrology to quantify the minimum uncertainty in estimating time. Due to unitarity, the uncertainty in the time estimate does not grow with time if we have access to optimal measurements on the full system. Restricting the measurements to act on a small subsystem or to have low computational complexity leads to results expected from equilibration, where the time uncertainty becomes large at late times. With optimal measurements on a subsystem larger than half of the system, we regain the ability to estimate the time very precisely, even at late times. Hawking's calculation for the reduced density matrix of the black hole radiation in semiclassical gravity contradicts our general predictions for unitary quantum chaotic systems. Hawking's state always has a large uncertainty for attempts to estimate the time using the radiation, whereas our general results imply that the uncertainty should become small after the Page time. This gives a new version of the black hole information loss paradox in terms of the time estimation task. By restricting to simple measurements on the radiation, the time uncertainty becomes large. This indicates from a new perspective that the observations of computationally bounded agents are consistent with the semiclassical effective description of gravity.

Figures (13)

• Figure 1: We consider the evolution of the subsystem Fisher information associated with time estimation in a chaotic quantum many-body system. We divide the system into two parts $A$ and $\bar{A}$, and $n_A$, $d_A$ respectively denote the number of degrees of freedom and Hilbert space dimension of $A$. For $n_A<n_{\bar{A}}$ (left), the subsystem quantum Fisher information $F_A(t)$ decays monotonically from an extensive value to an exponentially small value at late times, consistent with the expectation from thermalization. For $n_A>n_{\bar{A}}$ (right, blue curve), $F_A(t)$ shows a surprising non-monotonic evolution and saturates to an extensive value. On restricting to simple measurements, the associated classical Fisher information $f^{\rm comp}_A(t)$ decays monotonically and becomes small even for $n_A>n/2$ (right, green dashed curve).
• Figure 2: For a pure state $\ket{\psi(t)}$ in a chaotic system, for a subsystem $A$ larger than half of the system, the QFI of the reduced density matrix $\rho_A(t)$ has a large universal late-time value. In this case, $\rho_A(t)$ is not full-rank. The late-time value of the QFI captures the speed of rotation of the support of $\rho_A(t)$ within the full Hilbert space of $A$.
• Figure 3: We plot the dynamics of $F_A(t)$ in the chaotic spin chain model \ref{['eq:chaptic ising hamiltonian']} with various $n$ and $n_A$. We evolve the system from random product initial states and average $F_A(t)$ over $200$ samples of initial states for $n\leq11$ and $84$ samples for $n=12$. Left: We focus on cases with $n_A\leq n/2$. The solid black lines show the fitting to exponential decay. The rate of exponential decay decreases with $n_A$ and is independent of $n_{\bar{A}}$. (The decay rates $\alpha_{n_A}$ as a function of $n_A$ are shown in Fig. \ref{['fig:entropy dynamics of lindbladian']}(right).) The saturation value decays exponentially with $n_{\bar{A}}$ for fixed $n_A$. Middle: We focus on the region with $n_A>n/2$, and contrast the non-monotonic time dependence with the monotonic behaviour for $n_A<n/2$ shown in left panel. The saturation value depends very weakly on $n_{\bar{A}}$ for fixed $n_A$, and we expect this weak dependence to go away in the thermodynamic limit. Middle, inset: We plot $t^*$ (the time at which $F_A(t)$ is minimized, with $n_A>n/2$) for various $n_A,n$ as a function of $n_{\bar{A}}$. Red/blue lines correspond to $n_A=7,8$. We find that $t^*\sim n_{\bar{A}}$, and does not depend on $n_A$. Right: We plot the dynamics of $F_A(t)$ with fixed $n=12$ and various subsystem sizes.
• Figure 4: We plot the late time saturation value of $F_A(t)$ (averaged over 800 samples of initial states and 10 instances of time in the interval $t\in[15,20]$) as a function of subsystem fraction $x\equiv n_A/n$, for various $n$. The solid grey line is the theoretical prediction from the random pure state model introduced in \ref{['sec:random']}, in the thermodynamic limit $n\rightarrow\infty$. Even for finite size numerical results, we see a hint of a phase transition at $x=1/2$. We expect based on the random pure state model that the result is likely to be extensive in $n_A$ in the thermodynamic limit, which may be obscured by finite size effects here. Inset: Using the same data, we plot the log of the QFI saturation value for $x<1/2$. We see a clear collapse as a function of $2n_A-n$ showing a scaling of $\sim d_{A}/d_{\bar{A}}$, as predicted in \ref{['eq:resultiv']}.
• Figure 5: We plot the dynamics of $F_A(t),F_{\bar{A}}(t)$ for $n=12,n_A=9$ (averaged over 84 samples of initial random product states), along with the two terms $F_{A,\text{ent}}$ and $F_{A,\text{rot}}$ defined in \ref{['rot_ent_def']}. We observe that the non-monotonic behavior of $F_A(t)$ comes from the competition between $F_{A,\text{ent}}(t)$ and $F_{A,\text{rot}}(t)$.