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Big Toy Models:          Representing Physical Systems As Chu Spaces                            Samson Abramsky           ...
Introduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms o...
ThemesIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorph...
ThemesIntroduction• Themes                • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spac...
ThemesIntroduction• Themes                • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spac...
ThemesIntroduction• Themes                • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spac...
ThemesIntroduction• Themes                • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spac...
Chu SpacesIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuM...
Chu SpacesIntroduction                        We should understand Chu spaces as providing a very general (and, we• Themes...
Chu SpacesIntroduction                        We should understand Chu spaces as providing a very general (and, we• Themes...
Chu SpacesIntroduction                        We should understand Chu spaces as providing a very general (and, we• Themes...
Outline IIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMo...
Outline IIntroduction• Themes                • Chu spaces as a setting. We can find natural representations of• Chu Spaces•...
Outline IIntroduction• Themes                • Chu spaces as a setting. We can find natural representations of• Chu Spaces•...
Outline IIntroduction• Themes                • Chu spaces as a setting. We can find natural representations of• Chu Spaces•...
Outline IIntroduction• Themes                • Chu spaces as a setting. We can find natural representations of• Chu Spaces•...
Outline IIntroduction• Themes                • Chu spaces as a setting. We can find natural representations of• Chu Spaces•...
Outline IIBig Toy Models   Workshop on Informatic Penomena 2009 – 6
Outline II         • This leads to a further question of conceptual interest: is this representation            preserved ...
Outline II         • This leads to a further question of conceptual interest: is this representation            preserved ...
Outline II         • This leads to a further question of conceptual interest: is this representation            preserved ...
Outline II         • This leads to a further question of conceptual interest: is this representation            preserved ...
Outline II         • This leads to a further question of conceptual interest: is this representation            preserved ...
IntroductionChu Spaces• Chu Spaces• Definitions• Extensionality andSeparability• BiextensionalCollapseRepresenting Physical...
Chu SpacesBig Toy Models   Workshop on Informatic Penomena 2009 – 8
Chu Spaces     History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by     Po-Hsiang Chu. A general...
Chu Spaces     History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by     Po-Hsiang Chu. A general...
Chu Spaces     History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by     Po-Hsiang Chu. A general...
Chu Spaces     History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by     Po-Hsiang Chu. A general...
Chu Spaces     History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by     Po-Hsiang Chu. A general...
Chu Spaces     History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by     Po-Hsiang Chu. A general...
DefinitionsBig Toy Models   Workshop on Informatic Penomena 2009 – 9
Definitions     Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of     ‘points’ or ‘objects’, A...
Definitions     Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of     ‘points’ or ‘objects’, A...
Definitions     Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of     ‘points’ or ‘objects’, A...
Definitions     Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of     ‘points’ or ‘objects’, A...
Extensionality and SeparabilityBig Toy Models                        Workshop on Informatic Penomena 2009 – 10
Extensionality and Separability     Given a Chu space C = (X, A, e), we say that C is:Big Toy Models                      ...
Extensionality and Separability     Given a Chu space C = (X, A, e), we say that C is:         • extensional if for all a1...
Extensionality and Separability     Given a Chu space C = (X, A, e), we say that C is:         • extensional if for all a1...
Extensionality and Separability     Given a Chu space C = (X, A, e), we say that C is:         • extensional if for all a1...
Extensionality and Separability     Given a Chu space C = (X, A, e), we say that C is:         • extensional if for all a1...
Biextensional CollapseIntroduction                        Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morph...
Biextensional CollapseIntroduction                        Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morph...
Biextensional CollapseIntroduction                        Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morph...
Biextensional CollapseIntroduction                        Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morph...
IntroductionChu SpacesRepresenting PhysicalSystems• The GeneralParadigm• RepresentingQuantum Systems AsChu SpacesCharacter...
The General ParadigmBig Toy Models             Workshop on Informatic Penomena 2009 – 13
The General Paradigm     We take a system to be specified by its set of states S , and the set of questions Q     which can...
The General Paradigm     We take a system to be specified by its set of states S , and the set of questions Q     which can...
The General Paradigm     We take a system to be specified by its set of states S , and the set of questions Q     which can...
The General Paradigm     We take a system to be specified by its set of states S , and the set of questions Q     which can...
The General Paradigm     We take a system to be specified by its set of states S , and the set of questions Q     which can...
Representing Quantum Systems As Chu SpacesIntroductionChu SpacesRepresenting PhysicalSystems• The GeneralParadigm• Represe...
Representing Quantum Systems As Chu Spaces                        A quantum system with a Hilbert space H as its state spa...
Representing Quantum Systems As Chu Spaces                        A quantum system with a Hilbert space H as its state spa...
Representing Quantum Systems As Chu Spaces                        A quantum system with a Hilbert space H as its state spa...
Representing Quantum Systems As Chu Spaces                        A quantum system with a Hilbert space H as its state spa...
Representing Quantum Systems As Chu Spaces                        A quantum system with a Hilbert space H as its state spa...
IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensiona...
OverviewIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biex...
OverviewIntroduction                          We shall now see how the simple, discrete notions of Chu spaces sufficeChu Sp...
OverviewIntroduction                          We shall now see how the simple, discrete notions of Chu spaces sufficeChu Sp...
OverviewIntroduction                          We shall now see how the simple, discrete notions of Chu spaces sufficeChu Sp...
BiextensionaityIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overvie...
BiextensionaityIntroduction                          Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu ...
BiextensionaityIntroduction                          Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu ...
BiextensionaityIntroduction                          Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu ...
BiextensionaityIntroduction                          Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu ...
Projectivity = BiextensionalityIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu...
Projectivity = BiextensionalityIntroduction                          We shall now use some notions and results from projec...
Projectivity = BiextensionalityIntroduction                          We shall now use some notions and results from projec...
Projectivity = BiextensionalityIntroduction                          We shall now use some notions and results from projec...
Projectivity = BiextensionalityIntroduction                          We shall now use some notions and results from projec...
Characterizing Chu MorphismsBig Toy Models                     Workshop on Informatic Penomena 2009 – 19
Characterizing Chu Morphisms     To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism               ...
Characterizing Chu Morphisms     To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism               ...
Characterizing Chu Morphisms     To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism               ...
Injectivity AssumptionBig Toy Models               Workshop on Informatic Penomena 2009 – 20
Injectivity Assumption     Proposition 7     If f∗ is injective, then the following diagram commutes:                     ...
Injectivity Assumption     Proposition 7     If f∗ is injective, then the following diagram commutes:                     ...
Orthogonality is Preserved                          Another basic property of the evaluation.IntroductionChu Spaces       ...
Orthogonality is Preserved                          Another basic property of the evaluation.IntroductionChu Spaces       ...
Orthogonality is Preserved                          Another basic property of the evaluation.IntroductionChu Spaces       ...
Constructing the Left AdjointIntroduction                          We define a map f → : L(H) → L(K):Chu SpacesRepresenting...
Constructing the Left AdjointIntroduction                          We define a map f → : L(H) → L(K):Chu SpacesRepresenting...
Constructing the Left AdjointIntroduction                          We define a map f → : L(H) → L(K):Chu SpacesRepresenting...
Using Projective DualityBig Toy Models                 Workshop on Informatic Penomena 2009 – 23
Using Projective Duality     By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a     left...
Using Projective Duality     By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a     left...
Using Projective Duality     By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a     left...
Using Projective Duality     By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a     left...
Using Projective Duality     By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a     left...
Using Projective Duality     By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a     left...
Wigner’s TheoremIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overvi...
Wigner’s TheoremIntroduction                          Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 byChu S...
Wigner’s TheoremIntroduction                          Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 byChu S...
RemarksIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biext...
RemarksIntroductionChu Spaces                          • Note that in our case, taking f∗ = f , Pg is just the action of t...
RemarksIntroductionChu Spaces                          • Note that in our case, taking f∗ = f , Pg is just the action of t...
RemarksIntroductionChu Spaces                          • Note that in our case, taking f∗ = f , Pg is just the action of t...
A Surprise: Surjectivity Comes for Free!Big Toy Models                                 Workshop on Informatic Penomena 200...
A Surprise: Surjectivity Comes for Free!     Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗    ...
A Surprise: Surjectivity Comes for Free!     Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗    ...
A Surprise: Surjectivity Comes for Free!     Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗    ...
A Surprise: Surjectivity Comes for Free!     Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗    ...
Putting The Pieces TogetherIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spa...
Putting The Pieces TogetherIntroduction                          We say that a map U : H → K is semiunitary if it is eithe...
Putting The Pieces TogetherIntroduction                          We say that a map U : H → K is semiunitary if it is eithe...
IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheore...
The Big PictureBig Toy Models        Workshop on Informatic Penomena 2009 – 29
The Big Picture     We define a category SymmH as follows:Big Toy Models                               Workshop on Informat...
The Big Picture     We define a category SymmH as follows:         • The objects are Hilbert spaces of dimension  2.Big Toy...
The Big Picture     We define a category SymmH as follows:         • The objects are Hilbert spaces of dimension  2.       ...
The Big Picture     We define a category SymmH as follows:         • The objects are Hilbert spaces of dimension  2.       ...
The Big Picture     We define a category SymmH as follows:         • The objects are Hilbert spaces of dimension  2.       ...
The Big Picture     We define a category SymmH as follows:         • The objects are Hilbert spaces of dimension  2.       ...
RemarksBig Toy Models   Workshop on Informatic Penomena 2009 – 30
Remarks         • By the results of the previous subsection, Chu morphisms essentially force us            to consider the...
Remarks         • By the results of the previous subsection, Chu morphisms essentially force us            to consider the...
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
Big Toy Models: Representing Physical Systems as Chu Spaces
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http://dauns.math.tulane.edu/~mwm/WIP2009/slides/samson.pdf

WORKSHOP ON INFORMATIC PHENOMENA (2009):
http://dauns.math.tulane.edu/~mwm/WIP2009/Titles_and_Abstracts.html

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  • 'Samson Abramsky, Information Flow in Physics, Geometry, Logic and Computation

    In my lectures, I will describe a broad program that has been initiated to model information flow in these and related areas. This includes a high-level reformulation of quantum information and quantum computing using category theory that have been shown to capture all of the fundamental components of the theory. The approach supports reasoning about classical and quantum communication in the same model. The approach also has provided what are arguably the first completely formal descriptions and proofs of correctness of several key quantum informatic protocols, e.g. (logic-gate) teleportation, superdense coding, and one-way computational schemes. It also provides a description of the quantum state, as well as the flow of information from the quantum state to the classical world (measurements), and from the classical world to the quantum state (control), all of which are important for reasoning about security in a quantum setting.'
    http://129.81.170.14/~mwm/clifford/Site/Abstracts.html

    Speaker: Samson Abramsky (University of Oxford)
    Title: Information flow in physics, geometry, logic and computation V
    Event: Clifford Lectures 2008 (March 2008, Tulane University)
    http://www.youtube.com/watch?v=pmwbcW4E35c
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Transcript of "Big Toy Models: Representing Physical Systems as Chu Spaces"

  1. 1. Big Toy Models: Representing Physical Systems As Chu Spaces Samson Abramsky Oxford University Computing LaboratoryBig Toy Models Workshop on Informatic Penomena 2009 – 1
  2. 2. Introduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces IntroductionThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One
  3. 3. ThemesIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  4. 4. ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  5. 5. ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  6. 6. ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view ofReducing The Value quantum states: A toy theory’.SetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  7. 7. ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view ofReducing The Value quantum states: A toy theory’.SetDiscussionChu Spaces and • Big toy models.CoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  8. 8. Chu SpacesIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  9. 9. Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information flow.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  10. 10. Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information flow.Representing PhysicalSystems This logic of Chu spaces was in no way biassed in its conception towardsCharacterizing Chu the description of quantum mechanics or any other kind of physicalMorphisms onQuantum Chu Spaces system.The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  11. 11. Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information flow.Representing PhysicalSystems This logic of Chu spaces was in no way biassed in its conception towardsCharacterizing Chu the description of quantum mechanics or any other kind of physicalMorphisms onQuantum Chu Spaces system.The RepresentationTheorem Just for this reason, it is interesting to see how much ofReducing The ValueSet quantum-mechanical structure and concepts can be absorbed andDiscussion essentially determined by this more general systems logic.Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  12. 12. Outline IIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  13. 13. Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  14. 14. Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  15. 15. Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms on • States as rays of Hilbert spaces fall out as the biextensionalQuantum Chu Spaces collapse of the Chu spaces.The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  16. 16. Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms on • States as rays of Hilbert spaces fall out as the biextensionalQuantum Chu Spaces collapse of the Chu spaces.The RepresentationTheorem • Chu morphisms are automatically the unitaries andReducing The ValueSet antiunitaries — the physical symmetries of quantum systems.DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  17. 17. Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms on • States as rays of Hilbert spaces fall out as the biextensionalQuantum Chu Spaces collapse of the Chu spaces.The RepresentationTheorem • Chu morphisms are automatically the unitaries andReducing The ValueSet antiunitaries — the physical symmetries of quantum systems.Discussion • This leads to a full and faithful representation of theChu Spaces andCoalgebras groupoid of Hilbert spaces and their physical symmetries inPrimer on coalgebra Chu spaces over the unit interval.Basic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  18. 18. Outline IIBig Toy Models Workshop on Informatic Penomena 2009 – 6
  19. 19. Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to finitely many values?Big Toy Models Workshop on Informatic Penomena 2009 – 6
  20. 20. Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to finitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails.Big Toy Models Workshop on Informatic Penomena 2009 – 6
  21. 21. Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to finitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations?Big Toy Models Workshop on Informatic Penomena 2009 – 6
  22. 22. Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to finitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations? • We also look at coalgebras as a possible alternative setting to Chu spaces. Some interesting and novel points arise in comparing and relating these two well-studied systems models.Big Toy Models Workshop on Informatic Penomena 2009 – 6
  23. 23. Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to finitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations? • We also look at coalgebras as a possible alternative setting to Chu spaces. Some interesting and novel points arise in comparing and relating these two well-studied systems models. There is a paper available as an Oxford University Computing Laboratory Research Report: RR–09–08 at http://www.comlab.ox.ac.uk/techreports/cs/2009.htmlBig Toy Models Workshop on Informatic Penomena 2009 – 6
  24. 24. IntroductionChu Spaces• Chu Spaces• Definitions• Extensionality andSeparability• BiextensionalCollapseRepresenting PhysicalSystemsCharacterizing Chu Chu SpacesMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big
  25. 25. Chu SpacesBig Toy Models Workshop on Informatic Penomena 2009 – 8
  26. 26. Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis.Big Toy Models Workshop on Informatic Penomena 2009 – 8
  27. 27. Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects:Big Toy Models Workshop on Informatic Penomena 2009 – 8
  28. 28. Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely).Big Toy Models Workshop on Informatic Penomena 2009 – 8
  29. 29. Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely). • They have a rich representation theory; many concrete categories of interest can be fully embedded into Chu spaces (Lafont and Streicher, Pratt).Big Toy Models Workshop on Informatic Penomena 2009 – 8
  30. 30. Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely). • They have a rich representation theory; many concrete categories of interest can be fully embedded into Chu spaces (Lafont and Streicher, Pratt). • There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an interesting characterization of information transfer across Chu morphisms (van Benthem).Big Toy Models Workshop on Informatic Penomena 2009 – 8
  31. 31. Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely). • They have a rich representation theory; many concrete categories of interest can be fully embedded into Chu spaces (Lafont and Streicher, Pratt). • There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an interesting characterization of information transfer across Chu morphisms (van Benthem). Applications of Chu spaces have been proposed in a number of areas, including concurrency, hardware verification, game theory and fuzzy systems.Big Toy Models Workshop on Informatic Penomena 2009 – 8
  32. 32. DefinitionsBig Toy Models Workshop on Informatic Penomena 2009 – 9
  33. 33. Definitions Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of ‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X × A → K is an evaluation function.Big Toy Models Workshop on Informatic Penomena 2009 – 9
  34. 34. Definitions Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of ‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X × A → K is an evaluation function. A morphism of Chu spaces f : (X, A, e) → (X ′ , A′ , e′ ) is a pair of functions f = (f∗ : X → X ′ , f ∗ : A′ → A) such that, for all x ∈ X and a′ ∈ A′ : e(x, f ∗ (a′ )) = e′ (f∗ (x), a′ ).Big Toy Models Workshop on Informatic Penomena 2009 – 9
  35. 35. Definitions Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of ‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X × A → K is an evaluation function. A morphism of Chu spaces f : (X, A, e) → (X ′ , A′ , e′ ) is a pair of functions f = (f∗ : X → X ′ , f ∗ : A′ → A) such that, for all x ∈ X and a′ ∈ A′ : e(x, f ∗ (a′ )) = e′ (f∗ (x), a′ ). Chu morphisms compose componentwise: if f : (X1 , A1 , e1 ) → (X2 , A2 , e2 ) and g : (X2 , A2 , e2 ) → (X3 , A3 , e3 ), then (g ◦ f )∗ = g∗ ◦ f∗ , (g ◦ f )∗ = f ∗ ◦ g ∗ .Big Toy Models Workshop on Informatic Penomena 2009 – 9
  36. 36. Definitions Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of ‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X × A → K is an evaluation function. A morphism of Chu spaces f : (X, A, e) → (X ′ , A′ , e′ ) is a pair of functions f = (f∗ : X → X ′ , f ∗ : A′ → A) such that, for all x ∈ X and a′ ∈ A′ : e(x, f ∗ (a′ )) = e′ (f∗ (x), a′ ). Chu morphisms compose componentwise: if f : (X1 , A1 , e1 ) → (X2 , A2 , e2 ) and g : (X2 , A2 , e2 ) → (X3 , A3 , e3 ), then (g ◦ f )∗ = g∗ ◦ f∗ , (g ◦ f )∗ = f ∗ ◦ g ∗ . Chu spaces over K and their morphisms form a category ChuK .Big Toy Models Workshop on Informatic Penomena 2009 – 9
  37. 37. Extensionality and SeparabilityBig Toy Models Workshop on Informatic Penomena 2009 – 10
  38. 38. Extensionality and Separability Given a Chu space C = (X, A, e), we say that C is:Big Toy Models Workshop on Informatic Penomena 2009 – 10
  39. 39. Extensionality and Separability Given a Chu space C = (X, A, e), we say that C is: • extensional if for all a1 , a2 ∈ A: [∀x ∈ X. e(x, a1 ) = e(x, a2 )] ⇒ a1 = a2Big Toy Models Workshop on Informatic Penomena 2009 – 10
  40. 40. Extensionality and Separability Given a Chu space C = (X, A, e), we say that C is: • extensional if for all a1 , a2 ∈ A: [∀x ∈ X. e(x, a1 ) = e(x, a2 )] ⇒ a1 = a2 • separable if for all x1 , x2 ∈ X : [∀a ∈ A. e(x1 , a) = e(x2 , a)] ⇒ x1 = x2Big Toy Models Workshop on Informatic Penomena 2009 – 10
  41. 41. Extensionality and Separability Given a Chu space C = (X, A, e), we say that C is: • extensional if for all a1 , a2 ∈ A: [∀x ∈ X. e(x, a1 ) = e(x, a2 )] ⇒ a1 = a2 • separable if for all x1 , x2 ∈ X : [∀a ∈ A. e(x1 , a) = e(x2 , a)] ⇒ x1 = x2 • biextensional if it is extensional and separable. We define an equivalence relation on X by: x1 ∼ x2 ⇐⇒ ∀a ∈ A. e(x1 , a) = e(x2 , a).Big Toy Models Workshop on Informatic Penomena 2009 – 10
  42. 42. Extensionality and Separability Given a Chu space C = (X, A, e), we say that C is: • extensional if for all a1 , a2 ∈ A: [∀x ∈ X. e(x, a1 ) = e(x, a2 )] ⇒ a1 = a2 • separable if for all x1 , x2 ∈ X : [∀a ∈ A. e(x1 , a) = e(x2 , a)] ⇒ x1 = x2 • biextensional if it is extensional and separable. We define an equivalence relation on X by: x1 ∼ x2 ⇐⇒ ∀a ∈ A. e(x1 , a) = e(x2 , a). C is separable exactly when this relation is the identity. There is a Chu morphism (q, idA ) : (X, A, e) → (X/∼, A, e′ ) where e′ ([x], a) = e(x, a) and q : X → X/∼ is the quotient map.Big Toy Models Workshop on Informatic Penomena 2009 – 10
  43. 43. Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big Workshop on Informatic Penomena 2009 – 11
  44. 44. Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystems Proof For any a′ ∈ A′ :Characterizing ChuMorphisms onQuantum Chu Spaces e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big Workshop on Informatic Penomena 2009 – 11
  45. 45. Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystems Proof For any a′ ∈ A′ :Characterizing ChuMorphisms onQuantum Chu Spaces e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).The RepresentationTheoremReducing The ValueSet We shall write eChuK , sChuK and bChuK for the full subcategoriesDiscussion of ChuK determined by the extensional, separated and biextensionalChu Spaces andCoalgebras Chu spaces.Primer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big Workshop on Informatic Penomena 2009 – 11
  46. 46. Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystems Proof For any a′ ∈ A′ :Characterizing ChuMorphisms onQuantum Chu Spaces e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).The RepresentationTheoremReducing The ValueSet We shall write eChuK , sChuK and bChuK for the full subcategoriesDiscussion of ChuK determined by the extensional, separated and biextensionalChu Spaces andCoalgebras Chu spaces.Primer on coalgebra We shall mainly work with extensional and biextensional Chu spaces.Basic Concepts Obviously bChuK is a full sub-category of eChuK .Representing PhysicalSystems AsCoalgebras Proposition 2 The inclusion bChuK ⊂ - eChuK has a left adjointComparison: A First Q, the biextensional collapse..Try Toy Models Big Workshop on Informatic Penomena 2009 – 11
  47. 47. IntroductionChu SpacesRepresenting PhysicalSystems• The GeneralParadigm• RepresentingQuantum Systems AsChu SpacesCharacterizing ChuMorphisms onQuantum Chu Spaces Representing Physical SystemsThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One
  48. 48. The General ParadigmBig Toy Models Workshop on Informatic Penomena 2009 – 13
  49. 49. The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system.Big Toy Models Workshop on Informatic Penomena 2009 – 13
  50. 50. The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s.Big Toy Models Workshop on Informatic Penomena 2009 – 13
  51. 51. The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s. This is a Chu space!Big Toy Models Workshop on Informatic Penomena 2009 – 13
  52. 52. The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s. This is a Chu space! N.B. This is essentially the point of view taken by Mackey in his classic ‘Mathematical foundations of Quantum Mechanics’. Note that we prefer ‘question’ to ‘property’, since QM we cannot think in terms of static properties which are determinately possessed by a given state; questions imply a dynamic act of asking.Big Toy Models Workshop on Informatic Penomena 2009 – 13
  53. 53. The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s. This is a Chu space! N.B. This is essentially the point of view taken by Mackey in his classic ‘Mathematical foundations of Quantum Mechanics’. Note that we prefer ‘question’ to ‘property’, since QM we cannot think in terms of static properties which are determinately possessed by a given state; questions imply a dynamic act of asking. It is standard in the foundational literature on QM to focus on yes/no questions. However, the usual approaches to quantum logic avoid the direct introduction of probabilities. More on this later!Big Toy Models Workshop on Informatic Penomena 2009 – 13
  54. 54. Representing Quantum Systems As Chu SpacesIntroductionChu SpacesRepresenting PhysicalSystems• The GeneralParadigm• RepresentingQuantum Systems AsChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  55. 55. Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems AsChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  56. 56. Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  57. 57. Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  58. 58. Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The Value • The evaluation function eH is the ‘statistical algorithm’ giving theSet basic predictive content of Quantum Mechanics:DiscussionChu Spaces andCoalgebras ψ | PS ψ PS ψ | PS ψ PS ψ 2 eH (ψ, S) = = = 2 .Primer on coalgebra ψ|ψ ψ|ψ ψBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  59. 59. Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The Value • The evaluation function eH is the ‘statistical algorithm’ giving theSet basic predictive content of Quantum Mechanics:DiscussionChu Spaces andCoalgebras ψ | PS ψ PS ψ | PS ψ PS ψ 2 eH (ψ, S) = = = 2 .Primer on coalgebra ψ|ψ ψ|ψ ψBasic ConceptsRepresenting PhysicalSystems As We have thus directly transcribed the basic ingredients of the Dirac/vonCoalgebras Neumann-style formulation of Quantum Mechanics into the definition ofComparison: A FirstTry this Chu space. Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  60. 60. IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity = Characterizing Chu MorphismsBiextensionality• Characterizing ChuMorphisms on Quantum Chu Spaces• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value
  61. 61. OverviewIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  62. 62. OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufficeChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically significant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  63. 63. OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufficeChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically significant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing Chu The arguments here make use of Wigner’s theorem and the dualities ofMorphisms• Injectivity projective geometry, in the modern form developed by Faure andAssumption ¨ Frolicher, Modern Projective Geometry (2000).• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  64. 64. OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufficeChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically significant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing Chu The arguments here make use of Wigner’s theorem and the dualities ofMorphisms• Injectivity projective geometry, in the modern form developed by Faure andAssumption ¨ Frolicher, Modern Projective Geometry (2000).• Orthogonality isPreserved• Constructing the LeftAdjoint The surprising point is that unitarity/anitunitarity is essentially forced by• Using Projective the mere requirement of being a Chu morphism. This even extends toDuality• Wigner’s Theorem surjectivity, which here is derived rather than assumed.• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  65. 65. BiextensionaityIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  66. 66. BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  67. 67. BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting Physical A basic property of the evaluation.SystemsCharacterizing ChuMorphisms on Lemma 3 For ψ ∈ H◦ and S ∈ L(H):Quantum Chu Spaces• Overview• Biextensionaity ψ ∈ S ⇐⇒ eH (ψ, S) = 1.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  68. 68. BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting Physical A basic property of the evaluation.SystemsCharacterizing ChuMorphisms on Lemma 3 For ψ ∈ H◦ and S ∈ L(H):Quantum Chu Spaces• Overview• Biextensionaity ψ ∈ S ⇐⇒ eH (ψ, S) = 1.• Projectivity =Biextensionality• Characterizing Chu From this, we can prove:Morphisms• InjectivityAssumption• Orthogonality is Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but notPreserved• Constructing the Left separable. The equivalence classes of the relation ∼ on states areAdjoint exactly the rays of H. That is:• Using ProjectiveDuality• Wigner’s Theorem• Remarks φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  69. 69. BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting Physical A basic property of the evaluation.SystemsCharacterizing ChuMorphisms on Lemma 3 For ψ ∈ H◦ and S ∈ L(H):Quantum Chu Spaces• Overview• Biextensionaity ψ ∈ S ⇐⇒ eH (ψ, S) = 1.• Projectivity =Biextensionality• Characterizing Chu From this, we can prove:Morphisms• InjectivityAssumption• Orthogonality is Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but notPreserved• Constructing the Left separable. The equivalence classes of the relation ∼ on states areAdjoint exactly the rays of H. That is:• Using ProjectiveDuality• Wigner’s Theorem• Remarks φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.• A Surprise:Surjectivity Comes forFree! Thus we have recovered the standard notion of pure states as the rays of• Putting The PiecesTogether the Hilbert space from the general notion of state equivalence in ChuThe Representation spaces.Theorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  70. 70. Projectivity = BiextensionalityIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  71. 71. Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  72. 72. Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting Physical ¯ Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which itSystemsCharacterizing Chu generates. The rays are the atoms in the lattice L(H).Morphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  73. 73. Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting Physical ¯ Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which itSystemsCharacterizing Chu generates. The rays are the atoms in the lattice L(H).Morphisms onQuantum Chu Spaces We write P(H) for the set of rays of H. By virtue of Proposition 4, we can• Overview• Biextensionaity write the biextensional collapse of (H◦ , L(H), eH ) given by Proposition 2• Projectivity =Biextensionality as• Characterizing ChuMorphisms (P(H), L(H), eH) ¯• InjectivityAssumption• Orthogonality is ¯ ¯ where eH (ψ, S) = eH (ψ, S).Preserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  74. 74. Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting Physical ¯ Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which itSystemsCharacterizing Chu generates. The rays are the atoms in the lattice L(H).Morphisms onQuantum Chu Spaces We write P(H) for the set of rays of H. By virtue of Proposition 4, we can• Overview• Biextensionaity write the biextensional collapse of (H◦ , L(H), eH ) given by Proposition 2• Projectivity =Biextensionality as• Characterizing ChuMorphisms (P(H), L(H), eH) ¯• InjectivityAssumption• Orthogonality is ¯ ¯ where eH (ψ, S) = eH (ψ, S).Preserved• Constructing the LeftAdjoint We restate Lemma 3 for the biextensional case.• Using ProjectiveDuality• Wigner’s Theorem Lemma 5 For ψ ∈ H◦ and S ∈ L(H):• Remarks• A Surprise:Surjectivity Comes for ¯ ¯ ¯ eH (ψ, S) = 1 ⇐⇒ ψ ⊆ S.Free!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  75. 75. Characterizing Chu MorphismsBig Toy Models Workshop on Informatic Penomena 2009 – 19
  76. 76. Characterizing Chu Morphisms To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism (f∗ , f ∗ ) : (P(H), L(H), eH) → (P(K), L(K), eK ). ¯ ¯Big Toy Models Workshop on Informatic Penomena 2009 – 19
  77. 77. Characterizing Chu Morphisms To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism (f∗ , f ∗ ) : (P(H), L(H), eH) → (P(K), L(K), eK ). ¯ ¯ Proposition 6 For ψ ∈ H◦ and S ∈ L(K): ¯ ¯ ψ ⊆ f ∗ (S) ⇐⇒ f∗ (ψ) ⊆ S. Proof By Lemma 5: ¯ ¯ ¯ ¯ ¯ ψ ⊆ f ∗ (S) ⇔ eH (ψ, f ∗ (S)) = 1 ⇔ eK (f∗ (ψ), S) = 1 ⇔ f∗ (ψ) ⊆ S. ¯Big Toy Models Workshop on Informatic Penomena 2009 – 19
  78. 78. Characterizing Chu Morphisms To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism (f∗ , f ∗ ) : (P(H), L(H), eH) → (P(K), L(K), eK ). ¯ ¯ Proposition 6 For ψ ∈ H◦ and S ∈ L(K): ¯ ¯ ψ ⊆ f ∗ (S) ⇐⇒ f∗ (ψ) ⊆ S. Proof By Lemma 5: ¯ ¯ ¯ ¯ ¯ ψ ⊆ f ∗ (S) ⇔ eH (ψ, f ∗ (S)) = 1 ⇔ eK (f∗ (ψ), S) = 1 ⇔ f∗ (ψ) ⊆ S. ¯ Note that P(H) ⊆ L(H).Big Toy Models Workshop on Informatic Penomena 2009 – 19
  79. 79. Injectivity AssumptionBig Toy Models Workshop on Informatic Penomena 2009 – 20
  80. 80. Injectivity Assumption Proposition 7 If f∗ is injective, then the following diagram commutes: f∗ P(H) - P(K) ∩ ∩ (1) ? ? L(H) ∗ L(K) f That is, for all ψ ∈ H◦ : ¯ ¯ ψ = f ∗ (f∗ (ψ)).Big Toy Models Workshop on Informatic Penomena 2009 – 20
  81. 81. Injectivity Assumption Proposition 7 If f∗ is injective, then the following diagram commutes: f∗ P(H) - P(K) ∩ ∩ (1) ? ? L(H) ∗ L(K) f That is, for all ψ ∈ H◦ : ¯ ¯ ψ = f ∗ (f∗ (ψ)). Proof ¯ ¯ Proposition 6 implies that ψ ⊆ f ∗ (f∗ (ψ)). For the converse, suppose that ¯ ¯ ¯ ¯ φ ⊆ f ∗ (f∗ (ψ)). Applying Proposition 6 again, this implies that f∗ (φ) ⊆ f∗ (ψ). ¯ ¯ ¯ ¯ Since f∗ (φ) and f∗ (ψ) are atoms, this implies that f∗ (φ) = f∗ (ψ), which since f∗ ¯ ¯ ¯ ¯ is injective implies that φ = ψ . Thus the only atom below f ∗ (f∗ (ψ)) is ψ . Since ¯ ¯ L(H) is atomistic, this implies that f ∗ (f∗ (ψ)) ⊆ ψ .Big Toy Models Workshop on Informatic Penomena 2009 – 20
  82. 82. Orthogonality is Preserved Another basic property of the evaluation.IntroductionChu Spaces Lemma 8 For any φ, ψ ∈ H◦ :Representing PhysicalSystems ¯ ¯ ¯ eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.Characterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 21
  83. 83. Orthogonality is Preserved Another basic property of the evaluation.IntroductionChu Spaces Lemma 8 For any φ, ψ ∈ H◦ :Representing PhysicalSystems ¯ ¯ ¯ eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.Characterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity Proposition 9 If f∗ is injective, it preserves and reflects• Projectivity =Biextensionality orthogonality. That is, for all φ, ψ ∈ H◦ :• Characterizing ChuMorphisms• Injectivity ¯ ¯ φ ⊥ ψ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ).Assumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 21
  84. 84. Orthogonality is Preserved Another basic property of the evaluation.IntroductionChu Spaces Lemma 8 For any φ, ψ ∈ H◦ :Representing PhysicalSystems ¯ ¯ ¯ eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.Characterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity Proposition 9 If f∗ is injective, it preserves and reflects• Projectivity =Biextensionality orthogonality. That is, for all φ, ψ ∈ H◦ :• Characterizing ChuMorphisms• Injectivity ¯ ¯ φ ⊥ ψ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ).Assumption• Orthogonality isPreserved• Constructing the Left ProofAdjoint• Using Projective ¯ ¯ ¯Duality• Wigner’s Theorem φ ⊥ ψ ⇐⇒ eH (φ, ψ) = 0 Lemma 8• Remarks• A Surprise:Surjectivity Comes for ¯ ¯ ¯ ⇐⇒ eH (φ, f ∗ (f∗ (ψ))) = 0 Proposition 7Free!• Putting The Pieces ¯ ¯ ⇐⇒ eK (f∗ (φ), f∗ (ψ)) = 0 ¯TogetherThe RepresentationTheorem ¯ ¯ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ) Lemma 8. Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 21
  85. 85. Constructing the Left AdjointIntroduction We define a map f → : L(H) → L(K):Chu SpacesRepresenting PhysicalSystems f → (S) = ¯ {f∗ (ψ) | ψ ∈ S◦ }Characterizing ChuMorphisms onQuantum Chu Spaces where S◦ = S {0}.• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 22
  86. 86. Constructing the Left AdjointIntroduction We define a map f → : L(H) → L(K):Chu SpacesRepresenting PhysicalSystems f → (S) = ¯ {f∗ (ψ) | ψ ∈ S◦ }Characterizing ChuMorphisms onQuantum Chu Spaces where S◦ = S {0}.• Overview• Biextensionaity• Projectivity = Lemma 10 The map f → is left adjoint to f ∗ :Biextensionality• Characterizing ChuMorphisms• Injectivity f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ).Assumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 22
  87. 87. Constructing the Left AdjointIntroduction We define a map f → : L(H) → L(K):Chu SpacesRepresenting PhysicalSystems f → (S) = ¯ {f∗ (ψ) | ψ ∈ S◦ }Characterizing ChuMorphisms onQuantum Chu Spaces where S◦ = S {0}.• Overview• Biextensionaity• Projectivity = Lemma 10 The map f → is left adjoint to f ∗ :Biextensionality• Characterizing ChuMorphisms• Injectivity f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ).Assumption• Orthogonality isPreserved We can now extend the diagram (1):• Constructing the LeftAdjoint• Using Projective f∗Duality• Wigner’s Theorem P(H) - P(K) ∩ ∩• Remarks• A Surprise:Surjectivity Comes for (2)Free!• Putting The PiecesTogether ? f→ - ?The Representation L(H) ⊥ L(K)Theorem Big Toy ModelsReducing The Value f∗ Workshop on Informatic Penomena 2009 – 22
  88. 88. Using Projective DualityBig Toy Models Workshop on Informatic Penomena 2009 – 23
  89. 89. Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries.Big Toy Models Workshop on Informatic Penomena 2009 – 23
  90. 90. Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries.Big Toy Models Workshop on Informatic Penomena 2009 – 23
  91. 91. Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002).Big Toy Models Workshop on Informatic Penomena 2009 – 23
  92. 92. Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002). Let V1 be a vector space over the field F and V2 a vector space over the field G. A semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a field homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and v ∈ V1 : f (λv) = α(λ)f (v).Big Toy Models Workshop on Informatic Penomena 2009 – 23
  93. 93. Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002). Let V1 be a vector space over the field F and V2 a vector space over the field G. A semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a field homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and v ∈ V1 : f (λv) = α(λ)f (v). Note that semilinear maps compose: if (f, α) : V1 → V2 and (g, β) : V2 → V3 , then (g ◦ f, β ◦ α) : V1 → V2 is a semilinear map.Big Toy Models Workshop on Informatic Penomena 2009 – 23
  94. 94. Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002). Let V1 be a vector space over the field F and V2 a vector space over the field G. A semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a field homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and v ∈ V1 : f (λv) = α(λ)f (v). Note that semilinear maps compose: if (f, α) : V1 → V2 and (g, β) : V2 → V3 , then (g ◦ f, β ◦ α) : V1 → V2 is a semilinear map. N.B. There are lots of (horrible) automorphisms, and non-surjective endomorphisms, of the complex field!Big Toy Models Workshop on Informatic Penomena 2009 – 23
  95. 95. Wigner’s TheoremIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 24
  96. 96. Wigner’s TheoremIntroduction Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 byChu SpacesRepresenting PhysicalSystems ¯ P(g)(ψ) = g(ψ).Characterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 24
  97. 97. Wigner’s TheoremIntroduction Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 byChu SpacesRepresenting PhysicalSystems ¯ P(g)(ψ) = g(ψ).Characterizing ChuMorphisms onQuantum Chu Spaces We can now state Wigner’s Theorem in the form we shall use it.• Overview• Biextensionaity Theorem 12 Let f : P(H) → P(K) be a total map of projective• Projectivity =Biextensionality geometries, where dim H 2. If f preserves orthogonality, meaning• Characterizing ChuMorphisms that• InjectivityAssumption ¯ ¯ ¯ ¯ φ ⊥ ψ ⇒ f (φ) ⊥ f (ψ)• Orthogonality isPreserved• Constructing the Left then there is a semilinear map g : H → K such that P(g) = f , andAdjoint• Using ProjectiveDuality g(φ) | g(ψ) = σ( φ | ψ ),• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes for where σ is the homomorphism associated with g . Moreover, thisFree!• Putting The Pieces homomorphism is either the identity or complex conjugation, so g is eitherTogether linear or antilinear. The map g is unique up to a phase, i.e. a scalar ofThe RepresentationTheorem modulus 1. Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 24
  98. 98. RemarksIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  99. 99. RemarksIntroductionChu Spaces • Note that in our case, taking f∗ = f , Pg is just the action of theRepresenting Physical biextensional collapse functor on Chu morphisms.SystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  100. 100. RemarksIntroductionChu Spaces • Note that in our case, taking f∗ = f , Pg is just the action of theRepresenting Physical biextensional collapse functor on Chu morphisms.SystemsCharacterizing ChuMorphisms on • Note that a total map of projective geometries must necessarilyQuantum Chu Spaces come from an injective map g on the underlying vector spaces,• Overview• Biextensionaity since P(g) maps rays to rays, and hence g must have trivial kernel.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  101. 101. RemarksIntroductionChu Spaces • Note that in our case, taking f∗ = f , Pg is just the action of theRepresenting Physical biextensional collapse functor on Chu morphisms.SystemsCharacterizing ChuMorphisms on • Note that a total map of projective geometries must necessarilyQuantum Chu Spaces come from an injective map g on the underlying vector spaces,• Overview• Biextensionaity since P(g) maps rays to rays, and hence g must have trivial kernel.• Projectivity =Biextensionality• Characterizing Chu • For this reason, partial maps of projective geometries areMorphisms• Injectivity ¨ considered in the Faure-Frolicher approach. However, we areAssumption• Orthogonality is simply following the ‘logic’ of Chu space morphisms here.Preserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  102. 102. A Surprise: Surjectivity Comes for Free!Big Toy Models Workshop on Informatic Penomena 2009 – 26
  103. 103. A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective.Big Toy Models Workshop on Informatic Penomena 2009 – 26
  104. 104. A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective. Proof We write Im g for the set-theoretic direct image of g , which is a linear subspace of K, since σ is an automorphism. In particular, g carries rays to rays, since λg(φ) = g(σ −1 (λ)φ).Big Toy Models Workshop on Informatic Penomena 2009 – 26
  105. 105. A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective. Proof We write Im g for the set-theoretic direct image of g , which is a linear subspace of K, since σ is an automorphism. In particular, g carries rays to rays, since λg(φ) = g(σ −1 (λ)φ). We claim that for any vector ψ ∈ K◦ which is not in the image of g , ψ ⊥ Im g . ¯ ¯ Given such a ψ , for any φ ∈ H◦ it is not the case that f∗ (φ) ⊆ ψ ; for otherwise, for some λ, g(φ) = λψ , and hence g(σ −1 (λ−1 )φ) = ψ . Then by Proposition 6, ¯ f ∗ (ψ) = {0}. It follows that for all φ ∈ H◦ , ¯ ¯ ¯ ¯ eK (f∗ (φ), ψ) = eH (φ, {0}) = 0, ¯ and hence by Lemma 8 that ψ ⊥ Im g .Big Toy Models Workshop on Informatic Penomena 2009 – 26
  106. 106. A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective. Proof We write Im g for the set-theoretic direct image of g , which is a linear subspace of K, since σ is an automorphism. In particular, g carries rays to rays, since λg(φ) = g(σ −1 (λ)φ). We claim that for any vector ψ ∈ K◦ which is not in the image of g , ψ ⊥ Im g . ¯ ¯ Given such a ψ , for any φ ∈ H◦ it is not the case that f∗ (φ) ⊆ ψ ; for otherwise, for some λ, g(φ) = λψ , and hence g(σ −1 (λ−1 )φ) = ψ . Then by Proposition 6, ¯ f ∗ (ψ) = {0}. It follows that for all φ ∈ H◦ , ¯ ¯ ¯ ¯ eK (f∗ (φ), ψ) = eH (φ, {0}) = 0, ¯ and hence by Lemma 8 that ψ ⊥ Im g . Now suppose for a contradiction that such a ψ exists. Consider the vector ψ + χ where χ is a non-zero vector in Im g , which must exist since g is injective and H has positive dimension. This vector is not in Im g , nor is it orthogonal to Im g , since e.g. ψ + χ | χ = χ | χ = 0. This yields the required contradiction.Big Toy Models Workshop on Informatic Penomena 2009 – 26
  107. 107. Putting The Pieces TogetherIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 27
  108. 108. Putting The Pieces TogetherIntroduction We say that a map U : H → K is semiunitary if it is either unitary orChu Spaces antiunitary; that is, if it is a bijective map satisfyingRepresenting PhysicalSystemsCharacterizing Chu U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, U φ | U ψ = σ( φ | ψ )Morphisms onQuantum Chu Spaces• Overview where σ is the identity if U is unitary, and complex conjugation if U is• Biextensionaity• Projectivity = antiunitary. Note that semiunitaries preserve norm, so if U and V areBiextensionality• Characterizing Chu semiunitaries and U = λV , then |λ| = 1.Morphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 27
  109. 109. Putting The Pieces TogetherIntroduction We say that a map U : H → K is semiunitary if it is either unitary orChu Spaces antiunitary; that is, if it is a bijective map satisfyingRepresenting PhysicalSystemsCharacterizing Chu U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, U φ | U ψ = σ( φ | ψ )Morphisms onQuantum Chu Spaces• Overview where σ is the identity if U is unitary, and complex conjugation if U is• Biextensionaity• Projectivity = antiunitary. Note that semiunitaries preserve norm, so if U and V areBiextensionality• Characterizing Chu semiunitaries and U = λV , then |λ| = 1.Morphisms• InjectivityAssumption• Orthogonality is Theorem 14 Let H, K be Hilbert spaces of dimension greater than 2.Preserved• Constructing the Left Consider a Chu morphismAdjoint (f∗ , f ∗ ) : (P(H), L(H), eH) → (P(K), L(K), eK ).• Using ProjectiveDuality ¯ ¯• Wigner’s Theorem• Remarks• A Surprise: where f∗ is injective. Then there is a semiunitary U : H → K such thatSurjectivity Comes forFree! f∗ = P(U ). U is unique up to a phase.• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 27
  110. 110. IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheorem• The Big Picture• Remarks The Representation Theorem• Functors• Not Quite Right Yet• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy ModelsBasic Concepts
  111. 111. The Big PictureBig Toy Models Workshop on Informatic Penomena 2009 – 29
  112. 112. The Big Picture We define a category SymmH as follows:Big Toy Models Workshop on Informatic Penomena 2009 – 29
  113. 113. The Big Picture We define a category SymmH as follows: • The objects are Hilbert spaces of dimension 2.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  114. 114. The Big Picture We define a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  115. 115. The Big Picture We define a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps. • Semiunitaries compose as explained more generally for semilinear maps in the previous subsection. Since complex conjugation is an involution, semiunitaries compose according to the rule of signs: two antiunitaries or two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  116. 116. The Big Picture We define a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps. • Semiunitaries compose as explained more generally for semilinear maps in the previous subsection. Since complex conjugation is an involution, semiunitaries compose according to the rule of signs: two antiunitaries or two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary. This category is a groupoid, i.e. every arrow is an isomorphism.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  117. 117. The Big Picture We define a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps. • Semiunitaries compose as explained more generally for semilinear maps in the previous subsection. Since complex conjugation is an involution, semiunitaries compose according to the rule of signs: two antiunitaries or two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary. This category is a groupoid, i.e. every arrow is an isomorphism. The seminunitaries are the physically significant symmetries of Hilbert space from the point of view of Quantum Mechanics. The usual dynamics according to the Schrodinger equation is given by a continuous one-parameter group {U (t)} of ¨ these symmetries; the requirement of continuity forces the U (t) to be unitaries. However, some important physical symmetries are represented by antiunitaries, e.g. time reversal and charge conjugation.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  118. 118. RemarksBig Toy Models Workshop on Informatic Penomena 2009 – 30
  119. 119. Remarks • By the results of the previous subsection, Chu morphisms essentially force us to consider the symmetries on Hilbert space. As pointed out there, linear maps which can be represented as Chu morphisms in the biextensional category must be injective; and if L : H → K is an injective linear or antilinear map, then P(L) is injective.Big Toy Models Workshop on Informatic Penomena 2009 – 30
  120. 120. Remarks • By the results of the previous subsection, Chu morphisms essentially force us to consider the symmetries on Hilbert space. As pointed out there, linear maps which can be represented as Chu morphisms in the biextensional category must be injective; and if L : H → K is an injective linear or antilinear map, then P(L) is injective. • Our results then show that if L can be represented as a Chu morphism, it must in fact be semiunitary.Big Toy Models Workshop on Informatic Penomena 2009 – 30
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