Original Link: 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

1. Big Toy Models:
Representing Physical Systems As Chu Spaces
Samson Abramsky
Oxford University Computing Laboratory
Big Toy Models Workshop on Informatic Penomena 2009 – 1

2. Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
Introduction
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One

3. Themes
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 3

4. Themes
Introduction
• 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 Theoretical
Chu Spaces Computer Science are ripe for use in Physics.
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 3

5. Themes
Introduction
• 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 Theoretical
Chu Spaces Computer Science are ripe for use in Physics.
Representing Physical
Systems
Characterizing Chu
• Models vs. Axioms. Examples: sheaves and toposes,
Morphisms on domain-theoretic models of the λ-calculus.
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 3

6. Themes
Introduction
• 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 Theoretical
Chu Spaces Computer Science are ripe for use in Physics.
Representing Physical
Systems
Characterizing Chu
• Models vs. Axioms. Examples: sheaves and toposes,
Morphisms on domain-theoretic models of the λ-calculus.
Quantum Chu Spaces
The Representation
Theorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of
Reducing The Value quantum states: A toy theory’.
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 3

7. Themes
Introduction
• 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 Theoretical
Chu Spaces Computer Science are ripe for use in Physics.
Representing Physical
Systems
Characterizing Chu
• Models vs. Axioms. Examples: sheaves and toposes,
Morphisms on domain-theoretic models of the λ-calculus.
Quantum Chu Spaces
The Representation
Theorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of
Reducing The Value quantum states: A toy theory’.
Set
Discussion
Chu Spaces and
• Big toy models.
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 3

8. Chu Spaces
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 4

9. Chu Spaces
Introduction
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 the
Chu Spaces
vehicle for a general logic of ‘distributed systems’ and information flow.
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 4

10. Chu Spaces
Introduction
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 the
Chu Spaces
vehicle for a general logic of ‘distributed systems’ and information flow.
Representing Physical
Systems This logic of Chu spaces was in no way biassed in its conception towards
Characterizing Chu the description of quantum mechanics or any other kind of physical
Morphisms on
Quantum Chu Spaces system.
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 4

11. Chu Spaces
Introduction
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 the
Chu Spaces
vehicle for a general logic of ‘distributed systems’ and information flow.
Representing Physical
Systems This logic of Chu spaces was in no way biassed in its conception towards
Characterizing Chu the description of quantum mechanics or any other kind of physical
Morphisms on
Quantum Chu Spaces system.
The Representation
Theorem
Just for this reason, it is interesting to see how much of
Reducing The Value
Set quantum-mechanical structure and concepts can be absorbed and
Discussion essentially determined by this more general systems logic.
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 4

12. Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 5

13. Outline I
Introduction
• Themes • Chu spaces as a setting. We can find natural representations of
• Chu Spaces
• Outline I
quantum (and other) systems as Chu spaces.
• Outline II
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 5

14. Outline I
Introduction
• Themes • Chu spaces as a setting. We can find natural representations of
• Chu Spaces
• Outline I
quantum (and other) systems as Chu spaces.
• Outline II
Chu Spaces
• The general ‘logic’ of Chu spaces and morphisms allow us to
Representing Physical ‘rationally reconstruct’ many key quantum notions:
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 5

15. Outline I
Introduction
• Themes • Chu spaces as a setting. We can find natural representations of
• Chu Spaces
• Outline I
quantum (and other) systems as Chu spaces.
• Outline II
Chu Spaces
• The general ‘logic’ of Chu spaces and morphisms allow us to
Representing Physical ‘rationally reconstruct’ many key quantum notions:
Systems
Characterizing Chu
Morphisms on
• States as rays of Hilbert spaces fall out as the biextensional
Quantum Chu Spaces
collapse of the Chu spaces.
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 5

16. Outline I
Introduction
• Themes • Chu spaces as a setting. We can find natural representations of
• Chu Spaces
• Outline I
quantum (and other) systems as Chu spaces.
• Outline II
Chu Spaces
• The general ‘logic’ of Chu spaces and morphisms allow us to
Representing Physical ‘rationally reconstruct’ many key quantum notions:
Systems
Characterizing Chu
Morphisms on
• States as rays of Hilbert spaces fall out as the biextensional
Quantum Chu Spaces
collapse of the Chu spaces.
The Representation
Theorem
• Chu morphisms are automatically the unitaries and
Reducing The Value
Set antiunitaries — the physical symmetries of quantum systems.
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 5

17. Outline I
Introduction
• Themes • Chu spaces as a setting. We can find natural representations of
• Chu Spaces
• Outline I
quantum (and other) systems as Chu spaces.
• Outline II
Chu Spaces
• The general ‘logic’ of Chu spaces and morphisms allow us to
Representing Physical ‘rationally reconstruct’ many key quantum notions:
Systems
Characterizing Chu
Morphisms on
• States as rays of Hilbert spaces fall out as the biextensional
Quantum Chu Spaces
collapse of the Chu spaces.
The Representation
Theorem
• Chu morphisms are automatically the unitaries and
Reducing The Value
Set antiunitaries — the physical symmetries of quantum systems.
Discussion
• This leads to a full and faithful representation of the
Chu Spaces and
Coalgebras groupoid of Hilbert spaces and their physical symmetries in
Primer on coalgebra Chu spaces over the unit interval.
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 5

18. Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6

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. 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. 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. 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. 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.html
Big Toy Models Workshop on Informatic Penomena 2009 – 6

24. Introduction
Chu Spaces
• Chu Spaces
• Definitions
• Extensionality and
Separability
• Biextensional
Collapse
Representing Physical
Systems
Characterizing Chu Chu Spaces
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try Toy Models
Big

25. Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8

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. 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. 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. 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. 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. 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. Definitions
Big Toy Models Workshop on Informatic Penomena 2009 – 9

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. 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. 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. 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. Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10

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. 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
Big Toy Models Workshop on Informatic Penomena 2009 – 10

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 = x2
Big Toy Models Workshop on Informatic Penomena 2009 – 10

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. 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. Biextensional Collapse
Introduction
Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then
Chu Spaces
• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,
• Definitions
• Extensionality and
Separability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).
• Biextensional
Collapse
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try Toy Models
Big Workshop on Informatic Penomena 2009 – 11

44. Biextensional Collapse
Introduction
Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then
Chu Spaces
• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,
• Definitions
• Extensionality and
Separability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).
• Biextensional
Collapse
Representing Physical
Systems
Proof For any a′ ∈ A′ :
Characterizing Chu
Morphisms on
Quantum Chu Spaces
e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try Toy Models
Big Workshop on Informatic Penomena 2009 – 11

45. Biextensional Collapse
Introduction
Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then
Chu Spaces
• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,
• Definitions
• Extensionality and
Separability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).
• Biextensional
Collapse
Representing Physical
Systems
Proof For any a′ ∈ A′ :
Characterizing Chu
Morphisms on
Quantum Chu Spaces
e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).
The Representation
Theorem
Reducing The Value
Set We shall write eChuK , sChuK and bChuK for the full subcategories
Discussion of ChuK determined by the extensional, separated and biextensional
Chu Spaces and
Coalgebras Chu spaces.
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try Toy Models
Big Workshop on Informatic Penomena 2009 – 11

46. Biextensional Collapse
Introduction
Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then
Chu Spaces
• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,
• Definitions
• Extensionality and
Separability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).
• Biextensional
Collapse
Representing Physical
Systems
Proof For any a′ ∈ A′ :
Characterizing Chu
Morphisms on
Quantum Chu Spaces
e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).
The Representation
Theorem
Reducing The Value
Set We shall write eChuK , sChuK and bChuK for the full subcategories
Discussion of ChuK determined by the extensional, separated and biextensional
Chu Spaces and
Coalgebras 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 Physical
Systems As
Coalgebras Proposition 2 The inclusion bChuK ⊂ - eChuK has a left adjoint
Comparison: A First Q, the biextensional collapse..
Try Toy Models
Big Workshop on Informatic Penomena 2009 – 11

47. Introduction
Chu Spaces
Representing Physical
Systems
• The General
Paradigm
• Representing
Quantum Systems As
Chu Spaces
Characterizing Chu
Morphisms on
Quantum Chu Spaces
Representing Physical Systems
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One

48. The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13

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. 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. 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. 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. 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. Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing Physical
Systems
• The General
Paradigm
• Representing
Quantum Systems As
Chu Spaces
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 14

55. Representing Quantum Systems As Chu Spaces
A quantum system with a Hilbert space H as its state space will be
Introduction
represented as
Chu Spaces
(H◦ , L(H), eH )
Representing Physical
Systems
• The General where
Paradigm
• Representing
Quantum Systems As
Chu Spaces
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 14

56. Representing Quantum Systems As Chu Spaces
A quantum system with a Hilbert space H as its state space will be
Introduction
represented as
Chu Spaces
(H◦ , L(H), eH )
Representing Physical
Systems
• The General where
Paradigm
• Representing
Quantum Systems As • H◦ is the set of non-zero vectors of H
Chu Spaces
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 14

57. Representing Quantum Systems As Chu Spaces
A quantum system with a Hilbert space H as its state space will be
Introduction
represented as
Chu Spaces
(H◦ , L(H), eH )
Representing Physical
Systems
• The General where
Paradigm
• Representing
Quantum Systems As • H◦ is the set of non-zero vectors of H
Chu Spaces
Characterizing Chu
Morphisms on
• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions
Quantum Chu Spaces
of QM
The Representation
Theorem
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 14

58. Representing Quantum Systems As Chu Spaces
A quantum system with a Hilbert space H as its state space will be
Introduction
represented as
Chu Spaces
(H◦ , L(H), eH )
Representing Physical
Systems
• The General where
Paradigm
• Representing
Quantum Systems As • H◦ is the set of non-zero vectors of H
Chu Spaces
Characterizing Chu
Morphisms on
• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions
Quantum Chu Spaces
of QM
The Representation
Theorem
Reducing The Value • The evaluation function eH is the ‘statistical algorithm’ giving the
Set
basic predictive content of Quantum Mechanics:
Discussion
Chu Spaces and
Coalgebras ψ | PS ψ PS ψ | PS ψ PS ψ 2
eH (ψ, S) = = = 2
.
Primer on coalgebra ψ|ψ ψ|ψ ψ
Basic Concepts
Representing Physical
Systems As
Coalgebras
Comparison: A First
Try
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 14

59. Representing Quantum Systems As Chu Spaces
A quantum system with a Hilbert space H as its state space will be
Introduction
represented as
Chu Spaces
(H◦ , L(H), eH )
Representing Physical
Systems
• The General where
Paradigm
• Representing
Quantum Systems As • H◦ is the set of non-zero vectors of H
Chu Spaces
Characterizing Chu
Morphisms on
• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions
Quantum Chu Spaces
of QM
The Representation
Theorem
Reducing The Value • The evaluation function eH is the ‘statistical algorithm’ giving the
Set
basic predictive content of Quantum Mechanics:
Discussion
Chu Spaces and
Coalgebras ψ | PS ψ PS ψ | PS ψ PS ψ 2
eH (ψ, S) = = = 2
.
Primer on coalgebra ψ|ψ ψ|ψ ψ
Basic Concepts
Representing Physical
Systems As
We have thus directly transcribed the basic ingredients of the Dirac/von
Coalgebras Neumann-style formulation of Quantum Mechanics into the definition of
Comparison: A First
Try this Chu space.
Big Toy Models
Semantics in One Workshop on Informatic Penomena 2009 – 14

60. Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity = Characterizing Chu Morphisms
Biextensionality
• Characterizing Chu
Morphisms
on Quantum Chu Spaces
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value

61. Overview
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 16

62. Overview
Introduction
We shall now see how the simple, discrete notions of Chu spaces suffice
Chu Spaces
to determine the appropriate notions of state equivalence, and to pick out
Representing Physical
Systems the physically significant symmetries on Hilbert space in a very striking
Characterizing Chu fashion. This leads to a full and faithful representation of the category of
Morphisms on
Quantum 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
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 16

63. Overview
Introduction
We shall now see how the simple, discrete notions of Chu spaces suffice
Chu Spaces
to determine the appropriate notions of state equivalence, and to pick out
Representing Physical
Systems the physically significant symmetries on Hilbert space in a very striking
Characterizing Chu fashion. This leads to a full and faithful representation of the category of
Morphisms on
Quantum 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 of
Morphisms
• Injectivity
projective geometry, in the modern form developed by Faure and
Assumption ¨
Frolicher, Modern Projective Geometry (2000).
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 16

64. Overview
Introduction
We shall now see how the simple, discrete notions of Chu spaces suffice
Chu Spaces
to determine the appropriate notions of state equivalence, and to pick out
Representing Physical
Systems the physically significant symmetries on Hilbert space in a very striking
Characterizing Chu fashion. This leads to a full and faithful representation of the category of
Morphisms on
Quantum 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 of
Morphisms
• Injectivity
projective geometry, in the modern form developed by Faure and
Assumption ¨
Frolicher, Modern Projective Geometry (2000).
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
The surprising point is that unitarity/anitunitarity is essentially forced by
• Using Projective the mere requirement of being a Chu morphism. This even extends to
Duality
• Wigner’s Theorem surjectivity, which here is derived rather than assumed.
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 16

65. Biextensionaity
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 17

66. Biextensionaity
Introduction
Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 17

67. Biextensionaity
Introduction
Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).
Chu Spaces
Representing Physical A basic property of the evaluation.
Systems
Characterizing Chu
Morphisms 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
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 17

68. Biextensionaity
Introduction
Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).
Chu Spaces
Representing Physical A basic property of the evaluation.
Systems
Characterizing Chu
Morphisms 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
• Injectivity
Assumption
• Orthogonality is
Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but not
Preserved
• Constructing the Left
separable. The equivalence classes of the relation ∼ on states are
Adjoint exactly the rays of H. That is:
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 17

69. Biextensionaity
Introduction
Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).
Chu Spaces
Representing Physical A basic property of the evaluation.
Systems
Characterizing Chu
Morphisms 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
• Injectivity
Assumption
• Orthogonality is
Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but not
Preserved
• Constructing the Left
separable. The equivalence classes of the relation ∼ on states are
Adjoint exactly the rays of H. That is:
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.
• A Surprise:
Surjectivity Comes for
Free! Thus we have recovered the standard notion of pure states as the rays of
• Putting The Pieces
Together the Hilbert space from the general notion of state equivalence in Chu
The Representation spaces.
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 17

70. Projectivity = Biextensionality
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 18

71. Projectivity = Biextensionality
Introduction
We shall now use some notions and results from projective geometry.
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 18

72. Projectivity = Biextensionality
Introduction
We shall now use some notions and results from projective geometry.
Chu Spaces
Representing Physical ¯
Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which it
Systems
Characterizing Chu
generates. The rays are the atoms in the lattice L(H).
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 18

73. Projectivity = Biextensionality
Introduction
We shall now use some notions and results from projective geometry.
Chu Spaces
Representing Physical ¯
Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which it
Systems
Characterizing Chu
generates. The rays are the atoms in the lattice L(H).
Morphisms on
Quantum 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 Chu
Morphisms
(P(H), L(H), eH)
¯
• Injectivity
Assumption
• Orthogonality is
¯ ¯
where eH (ψ, S) = eH (ψ, S).
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 18

74. Projectivity = Biextensionality
Introduction
We shall now use some notions and results from projective geometry.
Chu Spaces
Representing Physical ¯
Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which it
Systems
Characterizing Chu
generates. The rays are the atoms in the lattice L(H).
Morphisms on
Quantum 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 Chu
Morphisms
(P(H), L(H), eH)
¯
• Injectivity
Assumption
• Orthogonality is
¯ ¯
where eH (ψ, S) = eH (ψ, S).
Preserved
• Constructing the Left
Adjoint
We restate Lemma 3 for the biextensional case.
• Using Projective
Duality
• 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 Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 18

75. Characterizing Chu Morphisms
Big Toy Models Workshop on Informatic Penomena 2009 – 19

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. 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. 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. Injectivity Assumption
Big Toy Models Workshop on Informatic Penomena 2009 – 20

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. 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. Orthogonality is Preserved
Another basic property of the evaluation.
Introduction
Chu Spaces Lemma 8 For any φ, ψ ∈ H◦ :
Representing Physical
Systems
¯ ¯ ¯
eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 21

83. Orthogonality is Preserved
Another basic property of the evaluation.
Introduction
Chu Spaces Lemma 8 For any φ, ψ ∈ H◦ :
Representing Physical
Systems
¯ ¯ ¯
eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.
Characterizing Chu
Morphisms on
Quantum 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 Chu
Morphisms
• Injectivity ¯ ¯
φ ⊥ ψ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ).
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 21

84. Orthogonality is Preserved
Another basic property of the evaluation.
Introduction
Chu Spaces Lemma 8 For any φ, ψ ∈ H◦ :
Representing Physical
Systems
¯ ¯ ¯
eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.
Characterizing Chu
Morphisms on
Quantum 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 Chu
Morphisms
• Injectivity ¯ ¯
φ ⊥ ψ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ).
Assumption
• Orthogonality is
Preserved
• Constructing the Left Proof
Adjoint
• 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 7
Free!
• Putting The Pieces ¯ ¯
⇐⇒ eK (f∗ (φ), f∗ (ψ)) = 0
¯
Together
The Representation
Theorem
¯ ¯
⇐⇒ f∗ (φ) ⊥ f∗ (ψ) Lemma 8.
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 21

85. Constructing the Left Adjoint
Introduction
We define a map f → : L(H) → L(K):
Chu Spaces
Representing Physical
Systems f → (S) = ¯
{f∗ (ψ) | ψ ∈ S◦ }
Characterizing Chu
Morphisms on
Quantum Chu Spaces where S◦ = S {0}.
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 22

86. Constructing the Left Adjoint
Introduction
We define a map f → : L(H) → L(K):
Chu Spaces
Representing Physical
Systems f → (S) = ¯
{f∗ (ψ) | ψ ∈ S◦ }
Characterizing Chu
Morphisms on
Quantum Chu Spaces where S◦ = S {0}.
• Overview
• Biextensionaity
• Projectivity = Lemma 10 The map f → is left adjoint to f ∗ :
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ).
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 22

87. Constructing the Left Adjoint
Introduction
We define a map f → : L(H) → L(K):
Chu Spaces
Representing Physical
Systems f → (S) = ¯
{f∗ (ψ) | ψ ∈ S◦ }
Characterizing Chu
Morphisms on
Quantum Chu Spaces where S◦ = S {0}.
• Overview
• Biextensionaity
• Projectivity = Lemma 10 The map f → is left adjoint to f ∗ :
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ).
Assumption
• Orthogonality is
Preserved We can now extend the diagram (1):
• Constructing the Left
Adjoint
• Using Projective
f∗
Duality
• Wigner’s Theorem
P(H) - P(K)
∩ ∩
• Remarks
• A Surprise:
Surjectivity Comes for (2)
Free!
• Putting The Pieces
Together
? f→
- ?
The Representation L(H) ⊥ L(K)
Theorem
Big Toy Models
Reducing The Value
f∗ Workshop on Informatic Penomena 2009 – 22

88. Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23

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. 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. 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. 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. 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. 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. Wigner’s Theorem
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 24

96. Wigner’s Theorem
Introduction
Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 by
Chu Spaces
Representing Physical
Systems
¯
P(g)(ψ) = g(ψ).
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 24

97. Wigner’s Theorem
Introduction
Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 by
Chu Spaces
Representing Physical
Systems
¯
P(g)(ψ) = g(ψ).
Characterizing Chu
Morphisms on
Quantum 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 Chu
Morphisms that
• Injectivity
Assumption
¯ ¯ ¯ ¯
φ ⊥ ψ ⇒ f (φ) ⊥ f (ψ)
• Orthogonality is
Preserved
• Constructing the Left then there is a semilinear map g : H → K such that P(g) = f , and
Adjoint
• Using Projective
Duality 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, this
Free!
• Putting The Pieces
homomorphism is either the identity or complex conjugation, so g is either
Together
linear or antilinear. The map g is unique up to a phase, i.e. a scalar of
The Representation
Theorem modulus 1.
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 24

98. Remarks
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 25

99. Remarks
Introduction
Chu Spaces
• Note that in our case, taking f∗ = f , Pg is just the action of the
Representing Physical biextensional collapse functor on Chu morphisms.
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 25

100. Remarks
Introduction
Chu Spaces
• Note that in our case, taking f∗ = f , Pg is just the action of the
Representing Physical biextensional collapse functor on Chu morphisms.
Systems
Characterizing Chu
Morphisms on
• Note that a total map of projective geometries must necessarily
Quantum 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
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 25

101. Remarks
Introduction
Chu Spaces
• Note that in our case, taking f∗ = f , Pg is just the action of the
Representing Physical biextensional collapse functor on Chu morphisms.
Systems
Characterizing Chu
Morphisms on
• Note that a total map of projective geometries must necessarily
Quantum 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 are
Morphisms
• Injectivity ¨
considered in the Faure-Frolicher approach. However, we are
Assumption
• Orthogonality is simply following the ‘logic’ of Chu space morphisms here.
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 25

102. A Surprise: Surjectivity Comes for Free!
Big Toy Models Workshop on Informatic Penomena 2009 – 26

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. 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. 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. 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. Putting The Pieces Together
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =
Biextensionality
• Characterizing Chu
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 27

108. Putting The Pieces Together
Introduction
We say that a map U : H → K is semiunitary if it is either unitary or
Chu Spaces
antiunitary; that is, if it is a bijective map satisfying
Representing Physical
Systems
Characterizing Chu U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, U φ | U ψ = σ( φ | ψ )
Morphisms on
Quantum 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 are
Biextensionality
• Characterizing Chu semiunitaries and U = λV , then |λ| = 1.
Morphisms
• Injectivity
Assumption
• Orthogonality is
Preserved
• Constructing the Left
Adjoint
• Using Projective
Duality
• Wigner’s Theorem
• Remarks
• A Surprise:
Surjectivity Comes for
Free!
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 27

109. Putting The Pieces Together
Introduction
We say that a map U : H → K is semiunitary if it is either unitary or
Chu Spaces
antiunitary; that is, if it is a bijective map satisfying
Representing Physical
Systems
Characterizing Chu U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, U φ | U ψ = σ( φ | ψ )
Morphisms on
Quantum 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 are
Biextensionality
• Characterizing Chu semiunitaries and U = λV , then |λ| = 1.
Morphisms
• Injectivity
Assumption
• Orthogonality is
Theorem 14 Let H, K be Hilbert spaces of dimension greater than 2.
Preserved
• Constructing the Left
Consider a Chu morphism
Adjoint
(f∗ , f ∗ ) : (P(H), L(H), eH) → (P(K), L(K), eK ).
• Using Projective
Duality ¯ ¯
• Wigner’s Theorem
• Remarks
• A Surprise: where f∗ is injective. Then there is a semiunitary U : H → K such that
Surjectivity Comes for
Free! f∗ = P(U ). U is unique up to a phase.
• Putting The Pieces
Together
The Representation
Theorem
Big Toy Models
Reducing The Value Workshop on Informatic Penomena 2009 – 27

110. Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
• The Big Picture
• Remarks
The Representation Theorem
• Functors
• Not Quite Right Yet
• Biextensionality and
Scalar Factors
• Projectivising The
Symmetry Groupoid
• Jes’ Right
• PR is an
embedding up to a
phase
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Big Toy Models
Basic Concepts

111. The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29

112. The Big Picture
We define a category SymmH as follows:
Big Toy Models Workshop on Informatic Penomena 2009 – 29

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. 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. 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. 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. 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. Remarks
Big Toy Models Workshop on Informatic Penomena 2009 – 30

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. 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

121. 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.
• This delineation of the physically significant symmetries by the logic of Chu
morphisms should be seen as a strong point in favour of this representation by
Chu spaces.
Big Toy Models Workshop on Informatic Penomena 2009 – 30

122. Functors
Introduction
We define a functor R : SymmH → eChu[0,1] :
Chu Spaces
R : U : H → K −→ (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK )
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces where U◦ is the restriction of U to H◦ .
The Representation As noted in Proposition 2, the inclusion bChu[0,1] ⊂ - eChu[0,1] has
Theorem
• The Big Picture a left adjoint, which we name Q. By Proposition 4, this can be defined on
• Remarks
the image of R as follows:
• Functors
• Not Quite Right Yet
• Biextensionality and
Scalar Factors
Q : (H◦ , L(H), eH ) → (PH, L(H), eH )
¯
• Projectivising The
Symmetry Groupoid
• Jes’ Right and for (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK ),
• PR is an
embedding up to a
phase Q : (U◦ , U −1 ) −→ (PU, U −1 ).
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Big Toy Models Workshop on Informatic Penomena 2009 – 31
Basic Concepts

123. Not Quite Right Yet
Introduction
Chu Spaces
Representing Physical
Systems
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality and
Scalar Factors
• Projectivising The
Symmetry Groupoid
• Jes’ Right
• PR is an
embedding up to a
phase
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Big Toy Models Workshop on Informatic Penomena 2009 – 32
Basic Concepts

124. Not Quite Right Yet
Introduction
We write emChu, bmChu for the subcategories of eChu[0,1] and
Chu Spaces
bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is
Representing Physical
Systems injective. The functors R and Q factor through these subcategories.
Characterizing Chu
Morphisms on
Quantum Chu Spaces
The Representation
Theorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality and
Scalar Factors
• Projectivising The
Symmetry Groupoid
• Jes’ Right
• PR is an
embedding up to a
phase
Reducing The Value
Set
Discussion
Chu Spaces and
Coalgebras
Primer on coalgebra
Big Toy Models Workshop on Informatic Penomena 2009 – 32
Basic Concepts