Jarzynski-like equality for the out-of-time-ordered correlator

TL;DR

This work establishes a Jarzynski-like fluctuation relation for the out-of-time-ordered correlator $F(t)$ by formulating a complex, quasiprobability–like distribution $P(W,W')$ built from a combined quantum amplitude $\tilde{A}_ρ$. It introduces two work-like random variables $W$ and $W'$ and shows that $F(t)$ can be obtained from a second mixed derivative of the moment generating function of $P(W,W')$, thereby connecting nonequilibrium thermodynamics with quantum chaos diagnostics. The paper also develops platform-nonspecific measurement protocols, including weak measurements and interference schemes, to access $F(t)$ indirectly without time reversal, and clarifies the relationship between $\tilde{A}_ρ$ and Kirkwood-Dirac quasiprobabilities. These results open new avenues for analyzing scrambling, chaos, and information flow in quantum systems and suggest practical routes to bound or measure OTOCs in diverse platforms. Potential implications span holography, condensed matter, and quantum information, where fluctuation relations can illuminate scrambling time scales and information dynamics.

Abstract

The out-of-time-ordered correlator (OTOC) diagnoses quantum chaos and the scrambling of quantum information via the spread of entanglement. The OTOC encodes forward and reverse evolutions and has deep connections with the flow of time. So do fluctuation relations such as Jarzynski's Equality, derived in nonequilibrium statistical mechanics. I unite these two powerful, seemingly disparate tools by deriving a Jarzynski-like equality for the OTOC. The equality's left-hand side equals the OTOC. The right-hand side suggests a protocol for measuring the OTOC indirectly. The protocol is platform-nonspecific and can be performed with weak measurement or with interference. Time evolution need not be reversed in any interference trial. The equality opens holography, condensed matter, and quantum information to new insights from fluctuation relations and vice versa.

Figures (1)

• Figure 1: Quantum processes described by the complex amplitudes in the Jarzynski-like equality for the out-of-time-ordered correlator (OTOC): Theorem \ref{['theorem:OTOC_FT']} shows that the OTOC depends on a complex distribution $P(W, W')$. This $P(W, W')$ parallels the probability distribution over possible values of thermodynamic work in Jarzynski's Equality. $P(W, W')$ results from summing products $A_\rho^*( . ) A_\rho( . )$. Each $A_\rho( . )$ denotes a probability amplitude [Eq. \ref{['eq:ADef']}], so each product resembles a probability. But the amplitudes' arguments differ, due to the OTOC's out-of-time ordering: The amplitudes correspond to different quantum processes. Figure \ref{['fig:Protocoll_Trial1']} illustrates the process associated with the $A_\rho^*( . )$; and Fig. \ref{['fig:Protocoll_Trial2']}, the process associated with the $A_\rho( . )$. Time runs from left to right. Each process begins with the preparation of the state $\rho = \sum_j p_j \lvert j \rangle\!\langle j \rvert$ and a measurement of the state's eigenbasis. Three evolutions ($U$, $U^\dag$, $U$) then alternate with three measurements of observables ($\tilde{ \mathcal{W} }$, $\tilde{V}$, $\tilde{ \mathcal{W} }$). If the initial state commutes with the Hamiltonian $H$ (e.g., if $\rho = e^{ - H / T } / Z$), the first $U$ can be omitted. Figures \ref{['fig:Protocoll_Trial1']} and \ref{['fig:Protocoll_Trial2']} are used to define $P(W, W')$, rather than illustrating protocols for measuring $P(W, W')$. $P(W, W')$ can be inferred from weak measurements and from interferometry.

Theorems & Definitions (2)

• Theorem 1
• proof