Out-of-time-order fluctuation-dissipation theorem

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.

Figure 1

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