  This paper concerns the recently proposed quasi-balanced truncation model
reduction method for linear quantum stochastic systems. It has previously been
shown that the quasi-balanceable class of systems (i.e. systems that can be
truncated via the quasi-balanced method) includes the class of completely
passive systems. In this work, we refine the previously established
characterization of quasi-balanceable systems and show that the class of
quasi-balanceable systems is strictly larger than the class of completely
passive systems. In particular, we derive a novel characterization of
completely passive linear quantum stochastic systems solely in terms of the
controllability Gramian of such systems. Exploiting this result, we prove that
all linear quantum stochastic systems with a pure Gaussian steady-state (active
systems included) are all quasi-balanceable, and establish a new complete
parameterization for this important class of systems. Examples are provided to
illustrate our results.
