  We describe a graphical calculus for completely positive maps and in doing so
review the theory of open quantum systems and other fundamental primitives of
quantum information theory using the language of tensor networks. In particular
we demonstrate the construction of tensor networks to pictographically
represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and
system-environment representations for the evolution of quantum states, review
how these representations interrelate, and illustrate how graphical
manipulations of the tensor networks may be used to concisely transform between
them. To further demonstrate the utility of the presented graphical calculus we
include several examples where we provide arguably simpler graphical proofs of
several useful quantities in quantum information theory including the
composition and contraction of multipartite channels, a condition for whether
an arbitrary bipartite state may be used for ancilla assisted process
tomography, and the derivation of expressions for the average gate fidelity and
entanglement fidelity of a channel in terms of each of the different
representations of the channel.
