# Diagnosing quantum chaos in many-body systems using entanglement as a resource

## TL;DR

This paper introduces a novel protocol for diagnosing quantum chaos in many-body systems using entanglement and thermofield double states, avoiding the need for backward time evolution and enabling experimental measurement of chaos indicators.

## Contribution

It proposes a new method to measure out-of-time-ordered correlators via ordinary measurements on thermofield double states, facilitating experimental detection of quantum chaos.

## Key findings

- Lyapunov exponent can be extracted from two-point correlations in TFD states.
- The method yields the regularized OTOC, a key indicator of quantum chaos.
- Numerical simulations on the Sachdev-Ye-Kitaev model support the protocol's effectiveness.

## Abstract

Classical chaotic systems exhibit exponentially diverging trajectories due to small differences in their initial state. The analogous diagnostic in quantum many-body systems is an exponential growth of out-of-time-ordered correlation functions (OTOCs). These quantities can be computed for various models, but their experimental study requires the ability to evolve quantum states backward in time, similar to the canonical Loschmidt echo measurement. In some simple systems, backward time evolution can be achieved by reversing the sign of the Hamiltonian; however in most interacting many-body systems, this is not a viable option. Here we propose a new family of protocols for OTOC measurement that do not require backward time evolution. Instead, they rely on ordinary time-ordered measurements performed in the thermofield double (TFD) state, an entangled state formed between two identical copies of the system. We show that, remarkably, in this situation the Lyapunov chaos exponent $\lambda_L$ can be extracted from the measurement of an ordinary two-point correlation function. As an unexpected bonus, we find that our proposed method yields the so-called "regularized" OTOC -- a quantity that is believed to most directly indicate quantum chaos. According to recent theoretical work, the TFD state can be prepared as the ground state of two weakly coupled identical systems and is therefore amenable to experimental study. We illustrate the utility of these protocols on the example of the maximally chaotic Sachdev-Ye-Kitaev model and support our findings by extensive numerical simulations.