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Just as conventional functional programs may be understood as proofs in anintuitionistic logic, so quantum processes can also be viewed as proofs in asuitable logic. We describe such a logic, the logic of compact closedcategories and biproducts, presented both as a sequent calculus and as a systemof proof-nets. This logic captures much of the necessary structure needed torepresent quantum processes under classical control, while remaining agnosticto the fine details. We demonstrate how to represent quantum processes asproof-nets, and show that the dynamic behaviour of a quantum process iscaptured by the cut-elimination procedure for the logic. We show that the cutelimination procedure is strongly normalising: that is, that every legal way ofsimplifying a proof-net leads to the same, unique, normal form. Finally, takingsome initial set of operations as non-logical axioms, we show that that theresulting category of proof-nets is a representation of the free compact closedcategory with biproducts generated by those operations.

Generalised Proof-Nets for Compact Categories with Biproducts