Bound on the exponential growth rate of out-of-time-ordered correlators.

It has been conjectured by Maldacena, Shenker, and Stanford [J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 10610.1007/JHEP08(2016)106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) F(t) has a universal upper bound 2πk_{B}T/ℏ. Here we introduce a one-parameter family of out-of-time-ordered correlators F_{γ}(t) (0≤γ≤1), which has as good properties as F(t) as a regularization of the out-of-time-ordered part of the squared commutator 〈[A[over ̂](t),B[over ̂](0)]^{2}〉 that diagnoses quantum many-body chaos, and coincides with F(t) at γ=1/2. We rigorously prove that if F_{γ}(t) shows a transient exponential growth for all γ in 0≤γ≤1, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate λ does not depend on the regularization parameter γ and satisfies the inequality λ≤2πk_{B}T/ℏ.

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.

Work fluctuation and total entropy production in nonequilibrium processes.

Work fluctuation and total entropy production play crucial roles in small thermodynamic systems subject to large thermal fluctuations. We investigate a trade-off relation between them in a nonequilibrium situation in which a system starts from an arbitrary nonequilibrium state. We apply a variational method to study this problem and find a stationary solution against variations over protocols that describe the time dependence of the Hamiltonian of the system. Using the stationary solution, we find the minimum of the total entropy production for a given amount of work fluctuation. An explicit protocol that achieves this is constructed from an adiabatic process followed by a quasistatic process. The obtained results suggest how one can control the nonequilibrium dynamics of the system while suppressing its work fluctuation and total entropy production.