  This paper is concerned with the original risk-sensitive performance
criterion for quantum stochastic systems and its recent quadratic-exponential
counterpart. These functionals are of different structure because of the
noncommutativity of quantum variables and have their own useful features such
as tractability of evolution equations and robustness properties. We discuss a
Lie algebraic connection between these two classes of cost functionals for open
quantum harmonic oscillators using an apparatus of complex Hamiltonian kernels
and symplectic factorizations. These results are aimed to extend useful
properties from one of the classes of risk-sensitive costs to the other and
develop state-space equations for computation and optimization of these
criteria in quantum robust control and filtering problems.
