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Despite previous extensive analysis of open quantum systems described by theLindblad equation, it is unclear whether correlated topological states, such asfractional quantum Hall states, are maintained even in the presence of the jumpterm. In this paper, we introduce the pseudo-spin Chern number of theLiouvillian which is computed by twisting the boundary conditions only for oneof the subspaces of the doubled Hilbert space. The existence of such atopological invariant elucidates that the topological properties remainunchanged even in the presence of the jump term which does not close the gap ofthe effective non-Hermitian Hamiltonian (obtained by neglecting the jump term).In other words, the topological properties are encoded into an effectivenon-Hermitian Hamiltonian rather than the full Liouvillian. This isparticularly useful when the jump term can be written as a strictly block-upper(-lower) triangular matrix in the doubled Hilbert space, in which case thepresence or absence of the jump term does not affect the spectrum of theLiouvillian. With the pseudo-spin Chern number, we address the characterizationof fractional quantum Hall states with two-body loss but without gain,elucidating that the topology of the non-Hermitian fractional quantum Hallstates is preserved even in the presence of the jump term. This numericalresult also supports the use of the non-Hermitian Hamiltonian whichsignificantly reduces the numerical cost. Similar topological invariants can beextended to treat correlated topological states for other spatial dimensionsand symmetry (e.g., one-dimensional open quantum systems with inversionsymmetry), indicating the high versatility of our approach.

Fate of fractional quantum Hall states in open quantum systems: characterization of correlated topological states for the full Liouvillian