Interacting Quantum Observables: Categorical Algebra and Diagrammatics

Bob Coecke; Ross Duncan

doi:10.1088/1367-2630/13/4/043016

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arXiv · 0906.4725

Interacting Quantum Observables: Categorical Algebra and Diagrammatics

Bob Coecke·Ross Duncan

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Abstract

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 axiomatise 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 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.

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Bob Coecke, Ross Duncan. 2011-04-21. Interacting Quantum Observables: Categorical Algebra and Diagrammatics. https://doi.org/10.1088/1367-2630%2F13%2F4%2F043016

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