A Partial Order on Classical and Quantum States

Coecke, Bob; Martin, Keye

doi:10.1007/978-3-642-12821-9_10

Cited by 27 publications

(68 citation statements)

References 4 publications

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Order By: Relevance

“…For instance, if L = [1,2,3,4,5,6], then [1,2,3], [4,5,6], [3,4,5], [2,3,4], [3,4], [5] and [ ] are all sublists of L, while [1,4,5,6], [1,3] and [2,4] are not sublists of L. The set [S] is also called the free monoid over S.…”

Section: Definitionmentioning

confidence: 99%

“…References: The results in this section are from the following sources: Section 3.1 is from some of the author's unpublished notes (1998), Section 3.2 is from [5,31], 3.3 is from [5,30] and 3.4 is from [26,32,27].…”

Section: Instances Of Partialitymentioning

confidence: 99%

“…It can then be shown [5] that for x, y ∈ ∆ n , we have x y iff there is a permutation σ ∈ S(n) such that x · σ, y · σ ∈ Λ n and…”

Section: Classical Statesmentioning

confidence: 99%

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Domain Theory and Measurement

Martin

2010

New Structures for Physics

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Section: Definitionmentioning

confidence: 99%

Section: Instances Of Partialitymentioning

confidence: 99%

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Domain Theory and Measurement

Martin

2010

New Structures for Physics

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“…The set of all pure states would thus be the set of all maximal elements. Coecke and Martin use a similar example to show that there exists a unique order on classical two-states given by the set Σ 2 and that a partial order on the more general Σ n respects a mixing law under certain restrictions [7]. The most important point here is that classical states have a unique ordering relation.…”

Section: Introductionmentioning

confidence: 96%

“…In this essay, I extend the work of Coecke and Martin [7,8] on states and measurements, within the neo-realist framework developed by Döring and Isham [9], to the concept of quantum contextuality. I begin with a review of the basic principles and definitions that will be used throughout this essay.…”

Section: Introductionmentioning

confidence: 96%

An Order-Theoretic Quantification of Contextuality

Durham

2014

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In this essay, I develop order-theoretic notions of determinism and contextuality on domains and topoi. In the process, I develop a method for quantifying contextuality and show that the order-theoretic sense of contextuality is analogous to the sense embodied in the topos-theoretic statement of the Kochen-Specker theorem. Additionally, I argue that this leads to a relation between the entropy associated with measurements on quantum systems and the second law of thermodynamics. The idea that the second law has its origin in the ordering of quantum states and processes dates to at least 1958 and possibly earlier. The suggestion that the mechanism behind this relation is contextuality, is made here for the first time.

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Unsharp Values, Domains and Topoi

Döring

Barbosa

2012

Quantum Field Theory and Gravity

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The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object Σ, analogous to the state space of a classical system, and a quantity-value object R ↔ , generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object -hence the 'values' of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as 'unsharp values'. We will motivate and explain these aspects of the topos approach and show that the structure of the quantity-value object R ↔ can be analysed using tools from domain theory, a branch of order theory that originated in theoretical computer science. Moreover, the base category of the topos associated with a quantum system turns out to be a domain if the underlying von Neumann algebra is a matrix algebra. For general algebras, the base category still is a highly structured poset. This gives a connection between the topos approach, noncommutative operator algebras and domain theory. In an outlook, we present some early ideas on how domains may become useful in the search for new models of (quantum) space and space-time.Mathematics Subject Classification (2010). Primary 81P99; Secondary 06A11, 18B25, 46L10.

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A Partial Order on Classical and Quantum States

Cited by 27 publications

References 4 publications

Domain Theory and Measurement

Domain Theory and Measurement

An Order-Theoretic Quantification of Contextuality

Unsharp Values, Domains and Topoi

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