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An important class of physical systems that are of interest in practice areinput-output open quantum systems that can be described by quantum stochasticdifferential equations and defined on an infinite-dimensional underlyingHilbert space. Most commonly, these systems involve coupling to a quantumharmonic oscillator as a system component. This paper is concerned with errorbounds in the finite-dimensional approximations of input-output open quantumsystems defined on an infinite-dimensional Hilbert space. We develop aframework for developing error bounds between the time evolution of the stateof a class of infinite-dimensional quantum systems and its approximation on afinite-dimensional subspace of the original, when both are initialized in thelatter subspace. This framework is then applied to two approaches for obtainingfinite-dimensional approximations: subspace truncation and adiabaticelimination. Applications of the bounds to some physical examples drawn fromthe literature are provided to illustrate our results.

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