Connecting the probability distributions of different operators and generalization of the Chernoff–Hoeffding inequality

Abstract

This work aimed to explore the fundamental aspects of the spectral properties of few-body general operators. We first consider the following question: when we know the probability distributions of a set of observables, what do we know about the probability distribution of their summation? When considering arbitrary operators, we could not obtain useful information over the third-order moment, while under the assumption of k-locality, we can rigorously prove a much stronger bound on the moment generating function for arbitrary quantum states. Second, with the use of this bound, we generalize the Chernoff inequality (or the Hoeffding inequality), which characterizes the asymptotic decay of the probability distribution for the product states by Gaussian decay. In the present form, the Chernoff inequality can be applied to a summation of independent local observables (e.g. single-site operators). We extend the range of application of the Chernoff inequality to the generic few-body observables.