  In recent work, symmetric dagger-monoidal (SDM) categories have emerged as a
convenient categorical formalization of quantum mechanics. The objects
represent physical systems, the morphisms physical operations, whereas the
tensors describe composite systems. Classical data turn out to correspond to
Frobenius algebras with some additional properties. They express the
distinguishing capabilities of classical data: in contrast with quantum data,
classical data can be copied and deleted. The algebraic approach thus shifts
the paradigm of "quantization" of a classical theory to "classicization" of a
quantum theory. Remarkably, the simple SDM framework suffices not only for this
conceptual shift, but even allows us to distinguish the deterministic classical
operations (i.e. functions) from the nondeterministic classical operations
(i.e. relations), and the probabilistic classical operations (stochastic maps).
Moreover, a combination of some basic categorical constructions (due to
Kleisli, resp. Grothendieck) with the categorical presentations of quantum
states, provides a resource sensitive account of various quantum-classical
interactions: of classical control of quantum data, of classical data arising
from quantum measurements, as well as of the classical data processing
in-between controls and measurements. A salient feature here is the graphical
calculus for categorical quantum mechanics, which allows a purely diagrammatic
representation of classical-quantum interaction.
