  The operator algebra of fermionic modes is isomorphic to that of qubits, the
difference between them is twofold: the embedding of subalgebras corresponding
to mode subsets and multiqubit subsystems on the one hand, and the parity
superselection in the fermionic case on the other. We discuss these two
fundamental differences extensively, and illustrate these through the
Jordan--Wigner representation in a coherent, self-contained, pedagogical way,
from the point of view of quantum information theory. Our perspective leads us
to develop useful new tools for the treatment of fermionic systems, such as the
fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic
partial trace, fermionic products of maps and fermionic embeddings of maps. We
formulate these by direct, easily applicable formulas, without mode
permutations, for arbitrary partitionings of the modes. It is also shown that
fermionic reduced states can be calculated by the fermionic partial trace,
containing the proper phase factors. We also consider variants of the notions
of fermionic mode correlation and entanglement, which can be endowed with the
usual, local operation based motivation, if the parity superselection rule is
imposed. We also elucidate some other fundamental points, related to joint map
extensions, which make the parity superselection inevitable in the description
of fermionic systems.
