  The purpose of this paper is to study the realization theory of quantum
linear systems. It is shown that for a general quantum linear system its
controllability and observability are equivalent and they can be checked by
means of a simple matrix rank condition. Based on controllability and
observability a specific realization is proposed for general quantum linear
systems in which an uncontrollable and unobservable subspace is identified.
When restricted to the passive case, it is found that a realization is minimal
if and only if it is Hurwitz stable. Computational methods are proposed to find
the cardinality of minimal realizations of a quantum linear passive system. It
is found that the transfer function of a quantum linear passive system $G$ can
be written as a fractional form in terms of a matrix function $\Sigma$;
moreover, $G$ is lossless bounded real if and only if $\Sigma$ is lossless
positive real. A type of realization for multi-input-multi-output quantum
linear passive systems is derived, which is closely related to its
controllability and observability decomposition. Two realizations, namely the
independent-oscillator realization and the chain-mode realization, are proposed
for single-input-single-output quantum linear passive systems, and it is shown
that under the assumption of minimal realization, the independent-oscillator
realization is unique, and these two realizations are related to the lossless
positive real matrix function $\Sigma$.
