  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.
