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

TL;DR

This paper explores the spectral properties of few-body operators, establishing bounds on their probability distributions, and generalizes the Chernoff-Hoeffding inequality to apply to a broader class of quantum observables.

Contribution

It introduces a rigorous bound on the moment generating function for few-body operators and extends the Chernoff inequality to these operators, broadening its applicability.

Findings

Established a stronger bound on the moment generating function for few-body operators.

Generalized the Chernoff inequality to apply to arbitrary few-body observables.

Extended the asymptotic decay characterization from product states to more general quantum states.

Abstract

This work is devoted to explore 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 can we way on the probability distribution of the summation of them? In considering arbitrary operators, we could not obtain a useful information over third order moment, while under the assumption of the few-body operators, we can rigorously prove a much stronger bound on the moment generating function for arbitrary quantum states. Second, by 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 the 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.