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Interacting Quantum Observables: Categorical Algebra and Diagrammatics | | Bob Coecke
; Ross Duncan
; | | Date: |
25 Jun 2009 | | Abstract: | Within an intuitive diagrammatic calculus and corresponding high-level
category-theoretic algebraic description we axiomatise complementary
observables 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
straightforward computations with elementary quantum gates, translations
between distinct quantum computational models, and simulations of quantum
algorithms such as the quantum Fourier transform. They enable the description
of the interaction between classical and quantum data in quantum informatic
protocols.
More specifically, we rely on the previously established fact that in the
symmetric monoidal category of Hilbert spaces and linear maps non-degenerate
observables correspond to special commutative $dag$-Frobenius algebras. This
leads to a generalisation of the notion of observable that extends to arbitrary
$dag$-symmetric monoidal categories ($dag$-SMC). We show that any observable
in a $dag$-SMC comes with an abelian group of phases. We define
complementarity of observables in arbitrary $dag$-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. | | Source: | arXiv, 0906.4725 | | Services: | Forum | Review | PDF | Favorites | |
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