  This paper is concerned with a class of open quantum systems whose dynamic
variables have an algebraic structure, similar to that of the Pauli matrices
pertaining to finite-level systems. The system interacts with external bosonic
fields, and its Hamiltonian and coupling operators depend linearly on the
system variables. This results in a Hudson-Parthasarathy quantum stochastic
differential equation (QSDE) whose drift and dispersion terms are affine and
linear functions of the system variables. The quasilinearity of the QSDE leads
to tractable dynamics of mean values and higher-order multi-point moments of
the system variables driven by vacuum input fields. This allows for the
closed-form computation of the quasi-characteristic function of the invariant
quantum state of the system and infinite-horizon asymptotic growth rates for a
class of cost functionals. The tractability of the moment dynamics is also used
for mean square optimal Luenberger observer design in a measurement-based
filtering problem for a quasilinear quantum plant, which leads to a Kalman-like
quantum filter.
