  The spectral and dynamical properties of dissipative quantum systems, as
modeled by a damped oscillator in the Fock space, are investigated from a
topological point of view. Unlike a physical lattice system that is naturally
under the open boundary condition, the bounded-from-below nature of the Fock
space offers a unique setting for understanding and verifying non-Hermitian
skin modes under semi-infinity boundary conditions that are elusive in actual
physical lattices. A topological characterization based on the complex spectra
of the Liouvillian superoperator is proposed and the associated complete set of
topologically protected skin modes can be identified, thus reflecting the
complete bulk-boundary correspondence of point-gap topology generally absent in
realistic materials. Moreover, we discover anomalous skin modes with
exponential amplification even though the quantum system is purely dissipative.
Our results indicate that current studies of non-Hermitian topological matter
can greatly benefit research on quantum open systems and vice versa.
