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      A categorical semantics of quantum protocols

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          Abstract

          We study quantum information and computation from a novel point of view. Our approach is based on recasting the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of compact closed categories with biproducts. We show how the essential structures found in key quantum information protocols such as teleportation, logic-gate teleportation, and entanglement-swapping can be captured at this abstract level. Moreover, from the combination of the --apparently purely qualitative-- structures of compact closure and biproducts there emerge `scalars` and a `Born rule'. This abstract and structural point of view opens up new possibilities for describing and reasoning about quantum systems. It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. Our formalism captures both the information-flow aspect of the protocols (see quant-ph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantum-mechanical formalism.

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          Catégories tannakiennes

          P. Deligne (2007)
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            Nuclear and trace ideals in tensored ∗-categories

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              Interaction Categories and the Foundations of Typed Concurrent Programming

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                Author and article information

                Journal
                18 February 2004
                2007-03-05
                Article
                quant-ph/0402130
                77b13df6-f190-49b5-b7ca-d9dd1b64366d
                History
                Custom metadata
                Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004)
                Significant additions and modifications as compared to the previous version (abstract inner-products, simplified presentation of abstract quantum mechanics). 21 pages, some pictures, some diagrams. 2007 replacement: source had become incompatible with Paul taylor's latest update of his Diagrams package so update was needed
                quant-ph cs.LO math-ph math.CT math.MP

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