Title: Topological phases of many-body non-Hermitian systems

Abstract: We show that many-body fermionic non-Hermitian systems require two distinct sets of topological invariants to describe the topology of energy bands and quantum states respectively, with the latter yet to be explored. We identify 10 symmetry classes -- determined by particle-hole, linearized time-reversal, and linearized chiral symmetries. Each class has topological invariant associated with each dimension, dictating the topology of quantum states. These findings pave the way for deeper understanding of the topological phases of many-body non-Hermitian systems.