  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.
