Automata, Languages and Programming

Volume 5126 of the series Lecture Notes in Computer Science pp 298-310

Interacting Quantum Observables

  • Bob CoeckeAffiliated withOxford University Computing Laboratory
  • , Ross DuncanAffiliated withOxford University Computing Laboratory

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We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum observables form bialgebra-like structures. We also provide an abstract account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus.