  The topological classifications of quadratic bosonic systems according to the
symmetries of the Hamiltonian, the dynamic matrix from the equation of motion,
and the effective Hamiltonian from the Lindblad equation are summarized. The
Hermitian Hamiltonian leads to a three-fold way table while the non-Hermitian
dynamic matrix and effective Hamiltonian lead to a ten-fold way table. The
difference comes from to the bosonic commutation relation and particle-hole
symmetry. Moreover, the system-reservoir coupling may cause a system with or
without coupling to a reservoir to fall into different classifications. We
present a 1D bosonic superconductor model inspired by the fermionic Kitaev
chain to contrast the topological behavior from the Hamiltonian, dynamic
matrix, and effective Hamiltonian. In contrast, a 2D Chern insulator is
insensitive to the different classifications.
