De-linearizing linearity : projective quantum axiomatics from strong compact closure
<p>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&apos;04), IEEE Computer Science Press. Extended version including proofs a...
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| Format: | Journal article |
| Language: | English |
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Elsevier
2007
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| _version_ | 1826265679589801984 |
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| author | Coecke, B |
| author_facet | Coecke, B |
| author_sort | Coecke, B |
| collection | OXFORD |
| description | <p>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&apos;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.</p> <p>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&apos;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 <em>Hilbert-Schmidt norm</em>, <em>Hilbert-Schmidt inner-product</em>, and in particular, the <em>preparation-state agreement axiom</em> 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].</p> <p>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&apos;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 <em>linear</em> and satisfies the <em>diagonal axiom</em>, 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.</p> |
| first_indexed | 2024-03-06T20:27:28Z |
| format | Journal article |
| id | oxford-uuid:2fdf97c7-6b2e-4996-9758-2f3cf15d73b2 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T20:27:28Z |
| publishDate | 2007 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:2fdf97c7-6b2e-4996-9758-2f3cf15d73b22022-03-26T12:58:07ZDe-linearizing linearity : projective quantum axiomatics from strong compact closureJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2fdf97c7-6b2e-4996-9758-2f3cf15d73b2MathematicsEnglishOxford University Research Archive - ValetElsevier2007Coecke, B<p>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&apos;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.</p> <p>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&apos;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 <em>Hilbert-Schmidt norm</em>, <em>Hilbert-Schmidt inner-product</em>, and in particular, the <em>preparation-state agreement axiom</em> 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].</p> <p>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&apos;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 <em>linear</em> and satisfies the <em>diagonal axiom</em>, 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.</p> |
| spellingShingle | Mathematics Coecke, B De-linearizing linearity : projective quantum axiomatics from strong compact closure |
| title | De-linearizing linearity : projective quantum axiomatics from strong compact closure |
| title_full | De-linearizing linearity : projective quantum axiomatics from strong compact closure |
| title_fullStr | De-linearizing linearity : projective quantum axiomatics from strong compact closure |
| title_full_unstemmed | De-linearizing linearity : projective quantum axiomatics from strong compact closure |
| title_short | De-linearizing linearity : projective quantum axiomatics from strong compact closure |
| title_sort | de linearizing linearity projective quantum axiomatics from strong compact closure |
| topic | Mathematics |
| work_keys_str_mv | AT coeckeb delinearizinglinearityprojectivequantumaxiomaticsfromstrongcompactclosure |