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Contextuality and the fundamental theorems of quantum mechanics
[article]
2020
arXiv
pre-print
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This shows that a very substantial part of the structure of quantum theory is encoded by contextuality. ...
We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. ...
(Generalised Gleason's theorem in contextual form.) Let N be a von Neumann algebra with no direct summand of type I 2 . ...
arXiv:1910.09591v2
fatcat:lxsdlthzzfdf5kjsjwuqxs7m5i
Spectral Presheaves, Kochen-Specker Contextuality, and Quantale-Valued Relations
2018
Electronic Proceedings in Theoretical Computer Science
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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 ...
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. ...
This theory can be modelled in the category of sets and relations Rel using Frobenius algebras to represent observables [6] . ...
doi:10.4204/eptcs.266.24
fatcat:pyxpk74umjcsvgt3sdfpmfbqe4
From dependent type theory to higher algebraic structures
[article]
2021
arXiv
pre-print
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We characterise dependently typed algebraic theories as finitary monads on certain presheaf categories, generalising a well-known result due to Lawvere, Bénabou and Linton for ordinary multisorted algebraic ...
The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. ...
Contextual categories were introduced in [Car78] , where Cartmell also gives an equivalence [Car78, §2.4] between the category of contextual categories and the category of generalised algebraic theories ...
arXiv:2110.02804v1
fatcat:bdwbbll3ivct3prpbtworvblo4
Generalised Mermin-type non-locality arguments
[article]
2019
arXiv
pre-print
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Our proofs are carried out in the graphical language of string diagrams for dagger compact categories, and their validity extends beyond quantum theory to any theory featuring the relevant algebraic structures ...
We broadly generalise Mermin-type arguments on GHZ states, and we provide exact group-theoretic conditions for non-locality to be achieved. ...
Acknowledgements The authors would like to thank Samson Abramsky, Bob Coecke, Aleks Kissinger and Rui Soares Barbosa for comments, suggestions and useful discussions, as well as Sukrita Chatterji and Nicolò ...
arXiv:1702.01772v4
fatcat:jned434spbcx5egy6bfyf7eyre
Generalised Mermin-type non-locality arguments
2019
Logical Methods in Computer Science
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Our proofs are carried out in the graphical language of string diagrams for dagger compact categories, and their validity extends beyond quantum theory to any theory featuring the relevant algebraic structures ...
We broadly generalise Mermin-type arguments on GHZ states, and we provide exact group-theoretic conditions for non-locality to be achieved. ...
Acknowledgements The authors would like to thank Samson Abramsky, Bob Coecke, Aleks Kissinger and Rui Soares Barbosa for comments, suggestions and useful discussions, as well as Sukrita Chatterji and Nicolò ...
doi:10.23638/lmcs-15(2:3)2019
fatcat:t32h7jmqk5bnlouxa5kkw5cnlq
Page 5055 of Mathematical Reviews Vol. , Issue 88j
[page]
1988
Mathematical Reviews
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{For the entire collection see MR 88i:68002.}
03H Nonstandard models
88j:03048
88j:03046 03G30 18A15 18C10
Cartmell, John (4-EDIN-C)
Generalised algebraic theories and contextual categories.
Ann. ...
Among the topics discussed in the paper are: (1) an equivalence between the category of gener- alized algebraic theories and the category of contextual categories and (2) the coincidence of the descriptive ...
Pre-nets, Read Arcs and Unfolding: A Functorial Presentation
[chapter]
2003
Lecture Notes in Computer Science
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Then the algebraic and unfolding view are reconciled: we exploit the algebraic semantics to define a functor from the category of prenets to the category of domains that is shown to be naturally isomorphic ...
Any pre-net is mapped to an acyclic branching net, representing its behaviour, then to a prime event structure and finally to a finitary prime algebraic domain. ...
monoidal categories [Mac71] ) and it can be expressed as a theory morphism, accounting for the algebraic and logical views. ...
doi:10.1007/978-3-540-40020-2_8
fatcat:rg7lnbd5v5acjdaygfm5cmmy2q
Page 2450 of Mathematical Reviews Vol. , Issue 2003d
[page]
2003
Mathematical Reviews
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See also 03104, 05150, 54017, 54022, 58001.
18 CATEGORY THEORY; HOMOLOGICAL ALGEBRA
18C Categories and theories
2003d:18009 18C20 68Q55
Mulry, P.S. (1-COLG-C; Hamilton, NY)
Lifting results for categories ...
The authors present a general construction of a crossed prod- uct Hopf algebra A, x H from two Hopf algebras and a normal twisted cocycle. This construction generalises A x H due to S. Ma- jid [J. ...
Contextuality and Noncommutative Geometry in Quantum Mechanics
2019
Communications in Mathematical Physics
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Here, we consider the geometrically dual objects of noncommutative algebras of observables as being generalisations of classical state spaces to the quantum setting and argue that these generalised geometric ...
To each unital C^*-algebra we associate a geometric object acting as a generalised Gel'fand spectrum. ...
the Operator Algebras Seminar at the Fields Institute for their guidance, encouragement, and mathematical insights. ...
doi:10.1007/s00220-018-3222-9
fatcat:7eazfr3jzjcofjlx4gcis4s5h4
Foundations of Software Science and Computational Structures: Selected papers from FOSSACS 2005
2008
Theoretical Computer Science
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Typical topics include: algebraic models; automata and language theory; behavioural equivalences; categorical models; computation processes over discrete and continuous data; computation structures; logics ...
semantics of programming languages; software specification and refinement; type systems and type theory. ...
Murawski and Igor Walukiewicz This paper merges techniques from automata theory and results from game semantics to formulate algorithms for contextual equivalence and approximation of programs in the third-order ...
doi:10.1016/j.tcs.2007.09.019
fatcat:sknvkrnpozfl7d365rws52wwke
On the Structure of Abstract H*-Algebras
2018
Electronic Proceedings in Theoretical Computer Science
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Previously we have shown that the topos approach to quantum theory of Doering and Isham can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky ...
Here we show that H*-algebras naturally correspond with the notions of states and observables in the generalised topos approach to quantum theory. ...
approach to quantum theory and the generalised topos approach to quantum theory have compatible interpretations of states and observables. ...
doi:10.4204/eptcs.266.13
fatcat:bcieghtwffavpgleszhxv5z4zq
Coreflective Concurrent Semantics for Single-Pushout Graph Grammars
[chapter]
2003
Lecture Notes in Computer Science
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During the last years, a rich theory of concurrency for the algebraic approach to graph transformation has been developed, including the generalisation of various classical Petri net concurrency models ...
algebraic domains. ...
Secondly, the functor from the category of occurrence grammars to the category of ies's does not admit a left adjoint and thus the whole semantic transformation is not expressed as a coreflection. ...
doi:10.1007/978-3-540-40020-2_9
fatcat:jjqueul3mjfl7olk4bfd6iw7r4
The semantic marriage of monads and effects
[article]
2014
arXiv
pre-print
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Dually, this approach yields indexed comonads which gives a unified semantics and effect system to contextual notions of effect (called coeffects), which we have previously described. ...
Wadler and Thiemann unified type-and-effect systems with monadic semantics via a syntactic correspondence and soundness results with respect to an operational semantics. ...
Relating indexed monads and monads Indexed monads collapse to regular monads when I is a single-object monoidal category. Thus, indexed monads generalise monads. ...
arXiv:1401.5391v1
fatcat:hdntf3yjijf33l6jg6j3hocjsy
Categorical Quantum Dynamics
[article]
2017
arXiv
pre-print
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All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future. ...
duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence ...
generalise classical observables and configuration spaces in Algebraic Quantum Theory [HLS09] . ...
arXiv:1709.09772v1
fatcat:idax6g6zkfdwdksdh3zwea3mpa
Models of Computation: A Tribute to Ugo Montanari's Vision
[chapter]
2008
Lecture Notes in Computer Science
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Notably, the use of category theory is a pervasive common trait. Algebraic techniques. ...
To mention just three (in alphabetic order): algebraic techniques, concurrency theory, and unification mechanisms. ...
Notably, the use of category theory is a pervasive common trait. Algebraic techniques. ...
doi:10.1007/978-3-540-68679-8_31
fatcat:2ten745a2fe37ahbgbc4wukcsi
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