Interacting quantum observables: categorical algebra and diagrammatics

Bob Coecke, Ross Duncan

Research output: Contribution to journalArticlepeer-review

132 Citations (Scopus)
51 Downloads (Pure)


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.
Original languageEnglish
Article number043016
Number of pages85
JournalNew Journal of Physics
Issue numberApril
Publication statusPublished - Apr 2011


  • computational physics
  • quantum observables
  • categorical algebra
  • diagrammatics


Dive into the research topics of 'Interacting quantum observables: categorical algebra and diagrammatics'. Together they form a unique fingerprint.

Cite this