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Reference: Bob Coecke, (2007). De-linearizing linearity : projective quantum axiomatics from strong compact closure. Electronic Notes in Theoretical Computer Science, 170, 49–72.Citable link to this page:

 

De-linearizing linearity : projective quantum axiomatics from strong compact closure

Abstract: Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases.First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843].Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.

Publication status:PublishedPeer Review status:Peer reviewedVersion:Publisher's versionNotes:© 2007 Published by Elsevier B.V. Open access under CC BY-NC-ND license. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0/

Bibliographic Details

Publisher: Elsevier

Publisher Website: http://www.elsevier.com/

Host: Electronic Notes in Theoretical Computer Sciencesee more from them

Publication Website: http://www.journals.elsevier.com/electronic-notes-in-theoretical-computer-science

Issue Date: 2007-3

Copyright Date: 2007

pages:49–72Identifiers

Doi: https://doi.org/10.1016/j.entcs.2006.12.011

Issn: 1571-0661

Urn: uuid:2fdf97c7-6b2e-4996-9758-2f3cf15d73b2 Item Description

Type: Article: post-print;

Language: en

Version: Publisher's versionKeywords: strong compact closure quantum mechanics global phases projective geometry categorical trace quantum logicSubjects: Mathematics Tiny URL: ora:9723

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Author: Bob Coecke - institutionUniversity of Oxford facultyMathematical, Physical and Life Sciences Division - Department of Computer Sc

Source: https://ora.ox.ac.uk/objects/uuid:2fdf97c7-6b2e-4996-9758-2f3cf15d73b2



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