  As is well-known in the context of topological insulators and
superconductors, short-range-correlated fermionic pure Gaussian states with
fundamental symmetries are systematically classified by the periodic table. We
revisit this topic from a quantum-information-inspired operational perspective
without referring to any Hamiltonians, and apply the formalism to bosonic
Gaussian states as well as (both fermionic and bosonic) locality-preserving
unitary Gaussian operations. We find that while bosonic Gaussian states are all
trivial, there exist nontrivial bosonic Gaussian operations that cannot be
continuously deformed into the identity under the locality and symmetry
constraint. Moreover, we unveil unexpectedly complicated relations between
fermionic Gaussian states and operations, pointing especially out that some of
the former can be disentangled by the latter under the same symmetry
constraint, while some cannot. In turn, we find that some topological
operations are genuinely dynamical, in the sense that they cannot create any
topological states from a trivial one, yet they are not connected to the
identity. The notions of disentanglability and genuinely dynamical topology can
be unified in a general picture of unitary-to-state homomorphism, and apply
equally to interacting topological phases and quantum cellular automata.
