Interacting Quantum Observables: Categorical Algebra and Diagrammatics 论文

摘要

Abstract: Within an intuitive diagrammatic calculus and corresponding high-level category-theoretic algebraic description we axiomatise complementary ob-servables for quantum systems described in finite dimensional Hilbert spaces, and study their interaction. We also axiomatise the phase shifts relative to an observable. The resulting graphical language is expressive enough to denote any quantum physical state of an arbitrary number of qubits, and any processes thereof. The rules for manipulating these result in very concise and straightfor-ward computations with elementary quantum gates, translations between dis-tinct quantum computational models, and simulations of quantum algorithms such as the quantum Fourier transform. They enable the description of the in-teraction between classical and quantum data in quantum informatic protocols. More specifically, we rely on the previously established fact that in the sym-metric monoidal category of Hilbert spaces and linear maps non-degenerate ob-servables correspond to special commutative †-Frobenius algebras. This leads to a generalisation of the notion of observable that extends to arbitrary †-symmetric monoidal categories (†-SMC). We show that any observable in a †-SMC comes with an abelian group of phases. We define complementarity of observables in arbitrary †-SMCs and prove an elegant diagrammatic characterisation thereof. We show that an important class of complementary observables give rise to a Hopf-algebraic structure, and provide equivalent characterisations thereof. Contents 1. Introduction................................