### Abstract

In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.

Original language | English |
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Pages (from-to) | 386-398 |

Number of pages | 13 |

Journal | Electronic Proceedings in Theoretical Computer Science, EPTCS |

Volume | 266 |

DOIs | |

Publication status | Published - 27 Feb 2018 |

Event | 14th International Conference on Quantum Physics and Logic and IQSA Quantum Structures Workshop - Nijmegen, Netherlands Duration: 3 Jul 2017 → 7 Jul 2017 |

### Fingerprint

### Keywords

- quantum theory
- Gelfand spectrum
- Kochen–Specker Theorem

### Cite this

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**Spectral presheaves, kochen-specker contextuality, and quantale-valued relations.** / Dunne, Kevin.

Research output: Contribution to journal › Conference Contribution

TY - JOUR

T1 - Spectral presheaves, kochen-specker contextuality, and quantale-valued relations

AU - Dunne, Kevin

PY - 2018/2/27

Y1 - 2018/2/27

N2 - In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.

AB - In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.

KW - quantum theory

KW - Gelfand spectrum

KW - Kochen–Specker Theorem

UR - http://www.scopus.com/inward/record.url?scp=85048440991&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1803.00709v1

U2 - 10.4204/EPTCS.266.24

DO - 10.4204/EPTCS.266.24

M3 - Conference Contribution

VL - 266

SP - 386

EP - 398

JO - Electronic Proceedings in Theoretical Computer Science

JF - Electronic Proceedings in Theoretical Computer Science

SN - 2075-2180

ER -