Error Bounds for Finite-Dimensional Approximations of Input-Output Open Quantum Systems by Subspace Truncation and Adiabatic Elimination

TL;DR

This paper develops a framework to quantify the error in finite-dimensional approximations of infinite-dimensional input-output open quantum systems, focusing on subspace truncation and adiabatic elimination methods.

Contribution

It introduces a novel framework for deriving error bounds between the true quantum system evolution and its finite-dimensional approximations.

Findings

Error bounds are established for subspace truncation.

Error bounds are established for adiabatic elimination.

Applications demonstrate the bounds' effectiveness on physical examples.

Abstract

An important class of physical systems that are of interest in practice are input-output open quantum systems that can be described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with error bounds in the finite-dimensional approximations of input-output open quantum systems defined on an infinite-dimensional Hilbert space. We develop a framework for developing error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. This framework is then applied to two approaches for obtaining finite-dimensional approximations: subspace truncation and adiabatic elimination. Applications of the bounds to some physical examples drawn from the literature are provided to illustrate our results.