Interacting Quantum Observables: Categorical Algebra and Diagrammatics TeX Export

@article{citeulike:5017560, 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.}, archivePrefix = {arXiv}, author = {Coecke, Bob and Duncan, Ross}, citeulike-article-id = {5017560}, citeulike-linkout-0 = {http://arxiv.org/abs/0906.4725}, citeulike-linkout-1 = {http://arxiv.org/pdf/0906.4725}, day = {25}, eprint = {0906.4725}, keywords = {algebra, category-theory, mathematics, physics, quantum, quantum-information}, month = {Jun}, posted-at = {2009-06-30 03:01:36}, priority = {3}, title = {Interacting Quantum Observables: Categorical Algebra and Diagrammatics}, url = {http://arxiv.org/abs/0906.4725}, year = {2009} }

TY - JOUR ID - citeulike:5017560 L3 - citeulike-article-id:5017560 N2 - 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. TI - Interacting Quantum Observables: Categorical Algebra and Diagrammatics KW - algebra KW - category-theory KW - mathematics KW - physics KW - quantum KW - quantum-information AU - Coecke, Bob AU - Duncan, Ross PY - 2009/Jun/25/ UR - http://arxiv.org/abs/0906.4725 ER -