TY - JOUR

T1 - Connecting the probability distributions of different operators and generalization of the Chernoff-Hoeffding inequality

AU - Kuwahara, Tomotaka

N1 - Funding Information:
This work was supported by World Premier International Research Center Initiative (WPI), Mext, Japan. TK also acknowledges support from JSPS grant no. 2611111.
Publisher Copyright:
© 2016 IOP Publishing Ltd and SISSA Medialab srl.

PY - 2016/11/16

Y1 - 2016/11/16

N2 - 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.

AB - 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.

KW - Exact results

KW - Large deviation

KW - Random/ordered microstructures

KW - Rigorous results in statistical mechanics

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U2 - 10.1088/1742-5468/2016/11/113103

DO - 10.1088/1742-5468/2016/11/113103

M3 - Article

AN - SCOPUS:85000979519

SN - 1742-5468

VL - 2016

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

IS - 11

M1 - 113103

ER -