Dagger and Dilation in the Category of Von Neumann algebras

Bas Westerbaan

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Dagger and Dilation in the Category of Von Neumann algebras

Bas Westerbaan

Operator Algebras · Submitted 5 Mar 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics.

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This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics. First up are dilations, covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the second topic: effectus theory. Both chapters, or rather parts, can be read separately and feature a comprehensive introduction of their own.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics.	contribution 0.69	0.51	0.03	asserted 1.00
s1	First up are dilations, covered in chapter 2.	contribution 0.90	0.30	0.02	asserted 0.84
s2	In chapter 3 "diamond, andthen, dagger" we turn to the second topic: effectus theory.	contribution 0.87	0.45	0.02	asserted 0.90
s3	Both chapters, or rather parts, can be read separately and feature a comprehensive introduction of their own.	method 0.38	0.36	0.03	asserted 1.00

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The paper

Original abstract

This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics. First up are dilations, covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the second topic: effectus theory. Both chapters, or rather parts, can be read separately and feature a comprehensive introduction of their own.

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10 arXiv papers cite this in 40 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 31 sentences without a confident label

Uses its method or tool 1 paper

This will allow us to define a dagger structure on the category of pure maps in Section [ref] via an abstract argument found in Bas Westerbaan's PhD thesis [ref] .

Quantum Theory from Principles, Quantum Software from Diagrams · Diamond and dagger · 0.85

Compares with it 1 paper

This bares similarities to Westerbaan's notion of purity in effectuses [ref] .

Categorical Operational Physics · Deriving the CP axiom · 0.98

Cites it as support 2 papers

It can be shown that an effectus has comprehensions iff it has kernels [ref] .

A computer scientist's reconstruction of quantum theory · Filters and comprehensions · 0.98
Our definition of pure maps then consists of maps that send pure states ( states of maximal information) to pure states[As further evidence to the naturalness of this definition of purity, Theorem 171VII of Ref. [ref] shows that a normal completely positive map between von Neumann algebras is pure if and only if its Paschke dilation, i.e. its purification (cf. Section [ref]), is surjective.].

Quantum Theory from Principles, Quantum Software from Diagrams · Pure effect theories · 1.00

Cites it as background 3 papers

We could impose any of a number of additional assumptions that would require there to be truly quantum systems, such as the existence of dilations, which are explored in the context of effectus theory in [ref] .

A computer scientist's reconstruction of quantum theory · Reconstruction for monoidal effectuses · 0.84
Westerbaan in the context of effectus theory [ref] .

Categorical Operational Physics · Alternative notions of purity · 0.97
For more results concerning effectus theory we refer the interested reader to Refs. [ref] .

Quantum Theory from Principles, Quantum Software from Diagrams · Properties of pure maps · 0.99

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