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.