Interacting Quantum Observables: Categorical Algebra and Diagrammatics

TL;DR

This paper introduces the ZX-calculus, a graphical language for quantum systems, and formalizes the concept of complementarity of quantum observables within a categorical framework, enhancing understanding and manipulation of quantum phenomena.

Contribution

It provides a universal graphical calculus for multi-qubit systems and axiomatizes quantum observable complementarity using dagger symmetric monoidal categories, unifying graphical and algebraic approaches.

Findings

Graphical calculus simplifies quantum derivations

Axiomatization of complementarity in categorical terms

Identification of a strong form of complementarity for Z and X observables

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.