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Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Abramsky and Coecke. == Mathematical setup == Mathematically, the basic setup is captured by a dagger symmetric monoidal category: composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes. The role of the dagger is to assign to each state a corresponding test. These can then be adorned with more structure to study various aspects, including: * A dagger compact category allows one to distinguish between "input" and "output" of a process. In the diagrammatic calculus, it allows wires to be bent, allowing for a less restricted transfer of information. In particular, it allows entangled states and measurements, and gives elegant descriptions of protocols such as quantum teleportation.〔Samson Abramsky and Bob Coecke, ''( A categorical semantics of quantum protocols )'', Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).〕 * Considering only the morphisms that are completely positive maps, one can also handle mixed states, allowing the study of quantum channels categorically.〔P. Selinger, ''( Dagger compact closed categories and completely positive maps )'', Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1 (2005).〕 * Wires are always two-ended (and can never be split into a Y), reflecting the no-cloning and no-deleting theorems of quantum mechanics. * Special commutative dagger Frobenius algebras model the fact that certain processes yield classical information, that can be cloned or deleted, thus capturing classical communication.〔B. Coecke and D. Pavlovic, ''(Quantum measurements without sums )''. In: Mathematics of Quantum Computing and Technology, pages 567–604, Taylor and Francis (2007).〕 * In early works, dagger biproducts were used to study both classical communication and the superposition principle. Later, these two features have been separated.〔B. Coecke and S. Perdrix, ''(Environment and classical channels in categorical quantum mechanics )''In: Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL), Lecture Notes in Computer Science 6247, Springer-Verlag.〕 * Complementary Frobenius algebras embody the principle of complementarity, which is used to great effect in quantum computation.〔B. Coecke and R. Duncan, ''(Interacting quantum observables )'' In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), pp. 298–310, Lecture Notes in Computer Science 5126, Springer.〕 A substantial portion of the mathematical backbone to this approach is drawn from Australian category theory, most notably from work by Kelly and Laplaza,〔G.M. Kelly and M.L. Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra 19, 193–213 (1980).〕 Joyal and Street,〔A. Joyal and R. Street, The Geometry of tensor calculus I, Advances in Mathematics 88, 55–112 (1991).〕 Carboni and Walters,〔A. Carboni and R. F. C. Walters, Cartesian bicategories I, Journal of Pure and Applied Algebra 49, 11–32 (1987).〕 and Lack.〔S. Lack, Composing PROPs, Theory and Applications of Categories 13, 147–163 (2004).〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Categorical quantum mechanics」の詳細全文を読む スポンサード リンク
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