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Out-of-time-order fluctuation-dissipation theorem

Naoto Tsuji, Tomohiro Shitara, Masahito Ueda

TL;DR

This work addresses generalizing the fluctuation-dissipation theorem to out-of-time-ordered correlators in quantum thermal systems by introducing bipartite OTOCs and quantifying the bipartite–physical difference with the Wigner–Yanase skew information. The authors prove a universal out-of-time-order FDT for bipartite OTOCs, relate it to a nonlinear response under a time-reversed protocol, and extend it to $n$-partite OTOCs and generalized covariance using a canonical OTOC $\Phi_{(AB)^n}$. The key contribution is a second-moment relation $C_{\{A,B\}^2}(\omega)+C_{[A,B]^2}(\omega)=2\coth\left(\frac{\beta\hbar\omega}{4}\right) C_{\{A,B\}[A,B]}(\omega)$, plus its higher-order and generalized-covariance extensions, which unify special cases (e.g., $n=2$) and connect chaotic dynamics to nonlinear response in thermal quantum systems. This framework enables new routes to probe quantum chaos via nonlinear response measurements and deepens the link between quantum fluctuations, chaos, and dissipation.

Abstract

We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order $n$-partite OTOCs as well as in the form of generalized covariance.

Out-of-time-order fluctuation-dissipation theorem

TL;DR

This work addresses generalizing the fluctuation-dissipation theorem to out-of-time-ordered correlators in quantum thermal systems by introducing bipartite OTOCs and quantifying the bipartite–physical difference with the Wigner–Yanase skew information. The authors prove a universal out-of-time-order FDT for bipartite OTOCs, relate it to a nonlinear response under a time-reversed protocol, and extend it to -partite OTOCs and generalized covariance using a canonical OTOC . The key contribution is a second-moment relation , plus its higher-order and generalized-covariance extensions, which unify special cases (e.g., ) and connect chaotic dynamics to nonlinear response in thermal quantum systems. This framework enables new routes to probe quantum chaos via nonlinear response measurements and deepens the link between quantum fluctuations, chaos, and dissipation.

Abstract

We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order -partite OTOCs as well as in the form of generalized covariance.

Paper Structure

This paper contains 7 sections, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Physical meaning of the out-of-time-order fluctuation-dissipation theorem
  4. Proof of the out-of-time-order fluctuation-dissipation theorem
  5. Generalization of the out-of-time-order fluctuation-dissipation theorem
  6. Conclusion
  7. Proof of Eqs. (\ref{['FDT2 Phi']}), (\ref{['reciprocal relation n']}), (\ref{['FDT n Phi']}), (\ref{['reciprocal GC']}), (\ref{['GC FDT n']}), and (\ref{['GC FDT n Phi']})

Figures (1)

  • Figure 1: Measurement protocol for the nonlinear response function $L_{(AB)^2}^{(3)}(t_0,0)$. The system is perturbed by the pulsed fields at $t=0$ and $t=t_0$. The system evolves with the Hamiltonian $+\hat{H}$ from $t=0$ to $t=t_0$ and then with $-\hat{H}$ from $t=t_0$ to $t=2t_0$. The measurement of $\hat{B}$ is performed at $t=2t_0$.