Graphical processes with abelian rotation symmetry

The idea of abelian quantum rotation is applied to the well established framework of categorical quantum mechanics and we provide a novel toolbox for the simulation of finite dimensional abelian quantum rotation. Strongly complementary structures are used to give the graphical characterisation of classical aspects of abelian quantum rotation, their action on systems and the momentum observables. Weyl canonical commutation relations are identified from the axioms of strongly complementary, and the existence of dual pair of angle/momentum observables is concluded for finite dimensional abelian quantum rotation. The quantum structure of abelian quantum rotation is discussed by showing there exists a symmetry-observable duality and evolution of quantum state is described by the Eilenberg-Moore morphism. Finally, composite quantum rotational systems are constructed and it is shown that they have the synchronicity property and proved the conservation law of momentum.

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