A categorical semantics of quantum protocols
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, logicgate teleportation, and entanglementswapping 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 informationflow aspect of the protocols (see quantph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantummechanical formalism.
 Publication:

arXiv eprints
 Pub Date:
 February 2004
 arXiv:
 arXiv:quantph/0402130
 Bibcode:
 2004quant.ph..2130A
 Keywords:

 Quantum Physics;
 Computer Science  Logic in Computer Science;
 Mathematical Physics;
 Mathematics  Category Theory;
 Mathematics  Mathematical Physics
 EPrint:
 Significant additions and modifications as compared to the previous version (abstract innerproducts, 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