  Non-Hermitian systems and the Lindblad form master equation have always been
regarded as reliable tools in dissipative modeling. Intriguingly, existing
literature often obtains an equivalent non-Hermitian Hamiltonian by neglecting
the quantum jumping terms in the master equation. However, there lacks
investigation into the effects of discarded terms as well as the unified
connection between these two approaches. In this study, we study the
Su-Schrieffer-Heeger model with collective loss and gain from a topological
perspective. When the system undergoes no quantum jump events, the
corresponding shape matrix exhibits the same topological properties in contrast
to the traditional non-Hermitian theory. Conversely, the occurrence of quantum
jumps can result in a shift in the positions of the phase transition. Our study
provides a qualitative analysis of the impact of quantum jumping terms and
reveals their unique role in quantum systems.
