# Generalised Proof−Nets for Compact Categories with Biproducts

*Ross Duncan*

### Abstract

Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a system of proof-nets. This logic captures much of the necessary structure needed to represent quantum processes under classical control, while remaining agnostic to the fine details. We demonstrate how to represent quantum processes as proof-nets, and show that the dynamic behaviour of a quantum process is captured by the cut-elimination procedure for the logic. We show that the cut elimination procedure is strongly normalising: that is, that every legal way of simplifying a proof-net leads to the same, unique, normal form. Finally, taking some initial set of operations as non-logical axioms, we show that that the resulting category of proof-nets is a representation of the free compact closed category with biproducts generated by those operations.