  There exist several graphical languages for quantum information processing,
like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages
forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an
interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces.
In the recent years, one of the main achievements of the categorical approach
to quantum mechanics has been to provide several equational theories for most
of these graphical languages, making them complete for various fragments of
pure quantum mechanics. We address the question of the extension of these
languages beyond pure quantum mechanics, in order to reason on mixed states and
general quantum operations, i.e. completely positive maps. Intuitively, such an
extension relies on the axiomatisation of a discard map which allows one to get
rid of a quantum system, operation which is not allowed in pure quantum
mechanics. We introduce a new construction, the discard construction, which
transforms any dagger-symmetric monoidal category into a symmetric monoidal
category equipped with a discard map. Roughly speaking this construction
consists in making any isometry causal. Using this construction we provide an
extension for several graphical languages that we prove to be complete for
general quantum operations. However this construction fails for some fringe
cases like the Clifford+T quantum mechanics, as the category does not have
enough isometries.
