Comonotonicity and Choquet integrals of Hermitian operators and their applications

TL;DR

This paper introduces a novel framework using comonotonicity and Choquet integrals to analyze Hermitian operators in quantum systems, providing bounds and insights into their physical properties.

Contribution

It develops a new formalism combining comonotonicity and Choquet integrals for Hermitian operators, with applications to quantum state analysis and physical quantity bounds.

Findings

Boundedness of comonotonic operators established

Bounds for partition functions derived

Application to ground state analysis demonstrated

Abstract

In a quantum system with d-dimensional Hilbert space, the Q-function of a Hermitian positive semidefinite operator ?, is defined in terms of the d2 coherent states in this system. The Choquet integral CQ of the Q-function, is introduced using a ranking of the values of the Q-function, and Mobius transforms which remove the overlaps between coherent states. It is a figure of merit of the quantum properties of Hermitian operators, and it provides upper and lower bounds to various physical quantities in terms of the Q-function. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar operators. Comonotonic operators are shown to be bounded, with respect to an order based on Choquet integrals. Applications of the formalism to the study of the ground state of a physical system, are discussed. Bounds for partition functions, are also derived.