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

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 enables fluctuation relations to provide insights into holography, condensed matter, and quantum information and vice versa.

Figure 1

Quantum processes described by the complex amplitudes in the Jarzynski-like equality for the out-oftime-ordered correlator (OTOC): Theorem 1 shows that the OTOC depends on a complex distribution $P(W,W^{\prime })$. This $P(W,W^{\prime })$ parallels the probability distribution over possible values of thermodynamic work in Jarzynski's equality. $P(W,W^{\prime })$ results from summing products $A_{\rho }^{*}(.)A_{\rho }(.)$. Each $A_{\rho }(.)$ denotes a probability amplitude [Eq. (2)], so each product resembles a probability. However, the amplitudes' arguments differ, due to the OTOC's out-of-time ordering: The amplitudes correspond to different quantum processes. Panel (a) illustrates the process associated with $A_{\rho }^{*}(.)$; and panel (b) illustrates the process associated with $A_{\rho }(.)$. Time runs from left to right. Each process begins with the preparation of the state $\rho ={\sum }_j{p}_j|j\rangle \langle j|$ and a measurement of the state's eigenbasis. Three evolutions ($U, U^{†}, U$) 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. Panels (a) and (b) are used to define $P(W,W^{\prime })$, rather than illustrating protocols for measuring $P(W,W^{\prime })$. $P(W,W^{\prime })$ can be inferred from weak measurements and from interference.