This paper presents a categorical semantics for quantum protocols, based on compact closed categories with biproducts. The authors show that essential structures of key quantum information protocols, such as teleportation, logic-gate teleportation, and entanglement swapping, can be captured at this abstract level. They demonstrate that the combination of compact closure and biproducts leads to the emergence of 'scalars' and a 'Born rule'. This abstract approach allows for a more general and structured understanding of quantum systems, revealing the degrees of axiomatic freedom in the (semi)ring of scalars. The formalism captures both the information-flow aspect of protocols and the branching due to quantum indeterminism, contrasting with the standard accounts where classical information flows are 'outside' the quantum-mechanical formalism. The authors provide detailed formal descriptions and proofs of correctness for the example protocols. The paper also discusses the generalization of these ideas to other models, such as Rel, and shows how the abstract setting can be used to derive and verify quantum protocols. The authors emphasize the importance of replacing ad hoc calculations with conceptual definitions and proofs, allowing for the identification of general underlying structures and lemmas. The paper concludes with a discussion of the implications of this approach for quantum information and computation.This paper presents a categorical semantics for quantum protocols, based on compact closed categories with biproducts. The authors show that essential structures of key quantum information protocols, such as teleportation, logic-gate teleportation, and entanglement swapping, can be captured at this abstract level. They demonstrate that the combination of compact closure and biproducts leads to the emergence of 'scalars' and a 'Born rule'. This abstract approach allows for a more general and structured understanding of quantum systems, revealing the degrees of axiomatic freedom in the (semi)ring of scalars. The formalism captures both the information-flow aspect of protocols and the branching due to quantum indeterminism, contrasting with the standard accounts where classical information flows are 'outside' the quantum-mechanical formalism. The authors provide detailed formal descriptions and proofs of correctness for the example protocols. The paper also discusses the generalization of these ideas to other models, such as Rel, and shows how the abstract setting can be used to derive and verify quantum protocols. The authors emphasize the importance of replacing ad hoc calculations with conceptual definitions and proofs, allowing for the identification of general underlying structures and lemmas. The paper concludes with a discussion of the implications of this approach for quantum information and computation.