  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.
