Estimating time in quantum chaotic systems and black holes

TL;DR

This paper explores how the ability to estimate time in quantum chaotic systems and black holes varies with measurement complexity, revealing universal features and implications for the black hole information paradox.

Contribution

It introduces a quantum metrology approach to analyze time estimation in chaotic systems and contrasts predictions with semiclassical gravity, offering new insights into black hole information loss.

Findings

Optimal measurements on large subsystems enable precise late-time time estimation.

Measurement restrictions lead to increased late-time uncertainty, aligning with equilibration.

Contradicts Hawking's semiclassical predictions, suggesting a resolution to the information paradox.

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.