Spectral Theorem approach to the Characteristic Function of Quantum Observables

TL;DR

This paper applies the spectral theorem to compute the quantum characteristic function for observables expressed as sums of Lie algebra generators, offering an alternative to traditional methods using Stone's formula.

Contribution

It introduces a spectral theorem-based approach to calculating the quantum Fourier transform of certain observables, bypassing the need for operator exponential splitting formulas.

Findings

Demonstrates the use of Stone's formula for spectral resolution

Provides a new method for computing quantum characteristic functions

Offers an alternative to traditional disentanglement techniques

Abstract

Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\langle \Phi, e^{itH}\Phi\rangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $\Phi$ is a unit vector in a Hilbert space $\mathcal{H}$. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential.