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Interacting quantum observables : categorical algebra and diagrammatics

Coecke, Bob and Duncan, Ross (2011) Interacting quantum observables : categorical algebra and diagrammatics. New Journal of Physics, 13 (April). ISSN 1367-2630

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This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z- and X-spin observables, which yields a scaled variant of a bialgebra.