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

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.

23 pages, 6 figures

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• Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interaction
• Eigenstate thermalization from the clustering property of correlation
• Quantum concentration inequalities
• Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations
• Matrix product state approximations to quantum states of low energy variance
• Efficient Quantum Algorithm for Filtering Product States
• Trotterization is substantially efficient for low-energy states
• Properties of thermal quantum states: locality of temperature, decay of correlations, and more
• Hardware-efficient quantum phase estimation via local control
• Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach