  We study quantum information and computation from a novel point of view. Our
approach is based on recasting the standard axiomatic presentation of quantum
mechanics, due to von Neumann, at a more abstract level, of compact closed
categories with biproducts. We show how the essential structures found in key
quantum information protocols such as teleportation, logic-gate teleportation,
and entanglement-swapping can be captured at this abstract level. Moreover,
from the combination of the --apparently purely qualitative-- structures of
compact closure and biproducts there emerge `scalars` and a `Born rule'. This
abstract and structural point of view opens up new possibilities for describing
and reasoning about quantum systems. It also shows the degrees of axiomatic
freedom: we can show what requirements are placed on the (semi)ring of scalars
C(I,I), where C is the category and I is the tensor unit, in order to perform
various protocols such as teleportation. Our formalism captures both the
information-flow aspect of the protocols (see quant-ph/0402014), and the
branching due to quantum indeterminism. This contrasts with the standard
accounts, in which the classical information flows are `outside' the usual
quantum-mechanical formalism.
