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, 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. We give detailed formal descriptions and proofs of correctness of the example protocols.