  This paper is concerned with the generation of Gaussian invariant states in
cascades of open quantum harmonic oscillators governed by linear quantum
stochastic differential equations. We carry out infinitesimal perturbation
analysis of the covariance matrix for the invariant Gaussian state of such a
system and the related purity functional subject to inaccuracies in the energy
and coupling matrices of the subsystems. This leads to the problem of balancing
the state-space realizations of the component oscillators through symplectic
similarity transformations in order to minimize the mean square sensitivity of
the purity functional to small random perturbations of the parameters. This
results in a quadratic optimization problem with an effective solution in the
case of cascaded one-mode oscillators, which is demonstrated by a numerical
example. We also discuss a connection of the sensitivity index with classical
statistical distances and outline infinitesimal perturbation analysis for
translation invariant cascades of identical oscillators. The findings of the
paper are applicable to robust state generation in quantum stochastic networks.
