  The topological classifications of quadratic bosonic systems according to the
symmetries of the dynamic matrices from the equations of motion of closed
systems and the effective Hamiltonians from the Lindblad equations of open
systems are analyzed. While the non-Hermitian dynamic matrix and effective
Hamiltonian both lead to a ten-fold way table, the system-reservoir coupling
may cause a system with or without coupling to a reservoir to fall into
different classes. A 2D Chern insulator is shown to be insensitive to the
different classifications. In contrast, we present a 1D bosonic
Su-Schrieffer-Heeger model with chiral symmetry and a 2D bosonic topological
insulator with time-reversal symmetry to show the corresponding open systems
may fall into different classes.
