  We present a systematic study of statistical mechanics for non-Hermitian
quantum systems. Our work reveals that the stability of a non-Hermitian system
necessitates the existence of a single path-dependent conserved quantity,
which, in conjunction with the system's Hamiltonian, dictates the equilibrium
state. By elucidating the relationship between the Hamiltonian and the
supported conserved quantity, we propose criteria for discerning equilibrium
states with finite relaxation times. Although our findings indicate that only
non-Hermitian systems with real energy spectrum precisely possess such
conserved quantities, we also demonstrate that an effective conserved quantity
can manifest in certain systems with complex energy spectra. The effective
conserved quantity, alongside the effective transitions within their associated
subspace, collectively determines the system's equilibrium state. Our results
provide valuable insights into non-Hermitian systems across more realistic
contexts and hold potential for applications in a diverse range of physical
systems.
