Big Toy Models: Representing Physical Systems as Chu Spaces

Big Toy Models Workshop on Informatic Penomena 2009 – 1

Big Toy Models:

Representing Physical Systems As Chu Spaces

Samson Abramsky

Oxford University Computing Laboratory

Introduction

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces

Representing PhysicalSystems

Characterizing ChuMorphisms onQuantum Chu Spaces

The RepresentationTheorem

Reducing The ValueSet

Discussion

Chu Spaces andCoalgebras

Primer on coalgebra

Basic Concepts

Representing PhysicalSystems AsCoalgebras

Comparison: A FirstTry

Semantics in OneCountryBig Toy Models

Themes

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts



Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 3

Themes

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.

Exemplifies one of the main thrusts of our group in Oxford:

methods and concepts which have been developed in TheoreticalComputer Science are ripe for use in Physics.

Themes

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts







• Models vs. Axioms. Examples: sheaves and toposes,domain-theoretic models of the λ-calculus.

Themes

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








• Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of

quantum states: A toy theory’.

Themes

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








• Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of

quantum states: A toy theory’.

• Big toy models.

Chu Spaces

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




Chu Spaces

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




We should understand Chu spaces as providing a very general (and, we

might reasonably say, rather simple) ‘logic of systems or structures’.

Indeed, they have been proposed by Barwise and Seligman as the

vehicle for a general logic of ‘distributed systems’ and information flow.

Chu Spaces

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








This logic of Chu spaces was in no way biassed in its conception towardsthe description of quantum mechanics or any other kind of physical

system.

Chu Spaces

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








This logic of Chu spaces was in no way biassed in its conception towardsthe description of quantum mechanics or any other kind of physical

system.

Just for this reason, it is interesting to see how much of

quantum-mechanical structure and concepts can be absorbed and

essentially determined by this more general systems logic.

Outline I

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




Outline I

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Chu spaces as a setting. We can find natural representations of

quantum (and other) systems as Chu spaces.

Outline I

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts






• The general ‘logic’ of Chu spaces and morphisms allow us to

‘rationally reconstruct’ many key quantum notions:

Outline I

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








• States as rays of Hilbert spaces fall out as the biextensional

collapse of the Chu spaces.

Outline I

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts










• Chu morphisms are automatically the unitaries and

antiunitaries — the physical symmetries of quantum systems.

Outline I

Introduction

• Themes

• Chu Spaces

• Outline I

• Outline II

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts










• Chu morphisms are automatically the unitaries and

antiunitaries — the physical symmetries of quantum systems.

• This leads to a full and faithful representation of the

groupoid of Hilbert spaces and their physical symmetries in

Chu spaces over the unit interval.

Outline II


Outline II


• This leads to a further question of conceptual interest: is this representation

preserved by collapsing the unit interval to finitely many values?

Outline II




• For the two canonical possibilistic collapses to two values, we showthat this fails .

Outline II





• However, the natural collapse to three values works! — A possible rolefor 3-valued logic in quantum foundations?

Outline II






• 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.

Outline II






• 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

Chu Spaces

Introduction

Chu Spaces

• Chu Spaces

• Definitions• Extensionality andSeparability

• BiextensionalCollapse





Discussion


Primer on coalgebra

Basic Concepts


Comparison: A FirstTryBig Toy Models

Chu Spaces


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





Chu spaces have several interesting aspects:

Chu Spaces






• They have a rich type structure, and in particular form models of Linear Logic

(Seely).

Chu Spaces







(Seely).

• They have a rich representation theory; many concrete categories of interest

can be fully embedded into Chu spaces (Lafont and Streicher, Pratt).

Chu Spaces







(Seely).



• There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an

interesting characterization of information transfer across Chu morphisms

(van Benthem).

Chu Spaces







(Seely).



• 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.

Definitions


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.

Definitions




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′).

Definitions




function.


f = (f∗ : X → X ′, f∗ : 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∗.

Definitions




function.


f = (f∗ : X → X ′, f∗ : 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 .

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





[∀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





[∀x ∈ X. e(x, a1) = e(x, a2)] ⇒ a1 = a2


[∀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).





[∀x ∈ X. e(x, a1) = e(x, a2)] ⇒ a1 = a2


[∀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.

Biextensional Collapse

Introduction

Chu Spaces

• Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts


Comparison: A FirstTryBig Toy Models Workshop on Informatic Penomena 2009 – 11

Proposition 1 If f : (X,A, e) → (X ′, A′, e′) is a Chu morphism, then

f∗ preserves ∼. That is, for all x1, x2 ∈ X ,

x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).


Introduction

Chu Spaces

• Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).

Proof For any a′ ∈ A′:

e′(f∗(x1), a′) = e(x1, f

∗(a′)) = e(x2, f∗(a′)) = e′(f∗(x2), a

′).

�


Introduction

Chu Spaces

• Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).


e′(f∗(x1), a′) = e(x1, f

∗(a′)) = e(x2, f∗(a′)) = e′(f∗(x2), a

′).

�

We shall write eChuK , sChuK and bChuK for the full subcategoriesof ChuK determined by the extensional, separated and biextensional

Chu spaces.


Introduction

Chu Spaces

• Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).


e′(f∗(x1), a′) = e(x1, f

∗(a′)) = e(x2, f∗(a′)) = e′(f∗(x2), a

′).

�

We shall write eChuK , sChuK and bChuK for the full subcategoriesof ChuK determined by the extensional, separated and biextensional

Chu spaces.

We shall mainly work with extensional and biextensional Chu spaces.

Obviously bChuK is a full sub-category of eChuK .

Proposition 2 The inclusion bChuK⊂ - eChuK has a left adjoint

Q, the biextensional collapse ..

Representing Physical Systems

Introduction

Chu Spaces


• The GeneralParadigm

• RepresentingQuantum Systems AsChu Spaces




Discussion


Primer on coalgebra

Basic Concepts



Semantics in OneCountryBig Toy Models

The General Paradigm




We take a system to be specified by its set of states S, and the set of questions Qwhich 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’ whenthe system is in state s.






evaluation function

e : S ×Q→ [0, 1]


This is a Chu space!






evaluation function

e : S ×Q→ [0, 1]



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.






evaluation function

e : S ×Q→ [0, 1]



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!

Representing Quantum Systems As Chu Spaces

Introduction

Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





Introduction

Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts




A quantum system with a Hilbert space H as its state space will be

represented as

(H◦, L(H), eH)

where


Introduction

Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





represented as

(H◦, L(H), eH)

where

• H◦ is the set of non-zero vectors of H


Introduction

Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





represented as

(H◦, L(H), eH)

where


• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions

of QM


Introduction

Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





represented as

(H◦, L(H), eH)

where



of QM

• The evaluation function eH is the ‘statistical algorithm’ giving the

basic predictive content of Quantum Mechanics:

eH(ψ, S) =〈ψ | PSψ〉

〈ψ | ψ〉=

〈PSψ | PSψ〉

〈ψ | ψ〉=

‖PSψ‖2

‖ψ‖2.


Introduction

Chu Spaces







Discussion


Primer on coalgebra

Basic Concepts





represented as

(H◦, L(H), eH)

where



of QM

• The evaluation function eH is the ‘statistical algorithm’ giving the

basic predictive content of Quantum Mechanics:

eH(ψ, S) =〈ψ | PSψ〉

〈ψ | ψ〉=

〈PSψ | PSψ〉

〈ψ | ψ〉=

‖PSψ‖2

‖ψ‖2.

We have thus directly transcribed the basic ingredients of the Dirac/von

Neumann-style formulation of Quantum Mechanics into the definition of

this Chu space.

Characterizing Chu Morphismson Quantum Chu Spaces

Introduction

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 PiecesTogether



Big Toy Models

Overview

Introduction

Chu Spaces



• Overview

• Biextensionaity









Overview

Introduction

Chu Spaces



• Overview

• Biextensionaity









We shall now see how the simple, discrete notions of Chu spaces suffice

to determine the appropriate notions of state equivalence, and to pick out

the physically significant symmetries on Hilbert space in a very striking

fashion. This leads to a full and faithful representation of the category of

quantum systems, with the groupoid structure of their physicalsymmetries, in the category of Chu spaces valued in the unit interval.

Overview

Introduction

Chu Spaces



• Overview

• Biextensionaity














The arguments here make use of Wigner’s theorem and the dualities of

projective geometry, in the modern form developed by Faure and

Frolicher, Modern Projective Geometry (2000).

Overview

Introduction

Chu Spaces



• Overview

• Biextensionaity














The arguments here make use of Wigner’s theorem and the dualities of

projective geometry, in the modern form developed by Faure and

Frolicher, Modern Projective Geometry (2000).

The surprising point is that unitarity/anitunitarity is essentially forced bythe mere requirement of being a Chu morphism. This even extends to

surjectivity, which here is derived rather than assumed.

Biextensionaity

Introduction

Chu Spaces



• Overview

• Biextensionaity









Biextensionaity

Introduction

Chu Spaces



• Overview

• Biextensionaity









Given a Hilbert space H, consider the Chu space (H◦, L(H), eH).

Biextensionaity

Introduction

Chu Spaces



• Overview

• Biextensionaity










A basic property of the evaluation.

Lemma 3 For ψ ∈ H◦ and S ∈ L(H):

ψ ∈ S ⇐⇒ eH(ψ, S) = 1.

Biextensionaity

Introduction

Chu Spaces



• Overview

• Biextensionaity












ψ ∈ S ⇐⇒ eH(ψ, S) = 1.

From this, we can prove:

Proposition 4 The Chu space (H◦, L(H), eH) is extensional but notseparable. The equivalence classes of the relation ∼ on states are

exactly the rays of H. That is:

φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.

Biextensionaity

Introduction

Chu Spaces



• Overview

• Biextensionaity












ψ ∈ S ⇐⇒ eH(ψ, S) = 1.

From this, we can prove:

Proposition 4 The Chu space (H◦, L(H), eH) is extensional but notseparable. The equivalence classes of the relation ∼ on states are

exactly the rays of H. That is:

φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.

Thus we have recovered the standard notion of pure states as the rays of

the Hilbert space from the general notion of state equivalence in Chu

spaces.

Projectivity = Biextensionality

Introduction

Chu Spaces



• Overview

• Biextensionaity










Introduction

Chu Spaces



• Overview

• Biextensionaity









We shall now use some notions and results from projective geometry.


Introduction

Chu Spaces



• Overview

• Biextensionaity










Given a vector ψ ∈ H◦, we write ψ = {λψ | λ ∈ C} for the ray which it

generates. The rays are the atoms in the lattice L(H).


Introduction

Chu Spaces



• Overview

• Biextensionaity












We write P(H) for the set of rays of H. By virtue of Proposition 4, we can

write the biextensional collapse of (H◦, L(H), eH) given by Proposition 2as

(P(H), L(H), eH)

where eH(ψ, S) = eH(ψ, S).


Introduction

Chu Spaces



• Overview

• Biextensionaity












We write P(H) for the set of rays of H. By virtue of Proposition 4, we can

write the biextensional collapse of (H◦, L(H), eH) given by Proposition 2as

(P(H), L(H), eH)

where eH(ψ, S) = eH(ψ, S).

We restate Lemma 3 for the biextensional case.


eH(ψ, S) = 1 ⇐⇒ ψ ⊆ S.

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).




(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.

�




(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).

Injectivity Assumption




Proposition 7 If f∗ is injective, then the following diagram commutes:

P(H)f∗

- P(K)

L(H)?

∩

�

f∗L(K)

?

∩

(1)

That is, for all ψ ∈ H◦:ψ = f∗(f∗(ψ)).



Proposition 7 If f∗ is injective, then the following diagram commutes:

P(H)f∗

- P(K)

L(H)?

∩

�

f∗L(K)

?

∩

(1)

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. �

Orthogonality is Preserved

Introduction

Chu Spaces



• Overview

• Biextensionaity









Another basic property of the evaluation.

Lemma 8 For any φ, ψ ∈ H◦:

eH(φ, ψ) = 0 ⇐⇒ φ⊥ψ.


Introduction

Chu Spaces



• Overview

• Biextensionaity











eH(φ, ψ) = 0 ⇐⇒ φ⊥ψ.

Proposition 9 If f∗ is injective, it preserves and reflectsorthogonality . That is, for all φ, ψ ∈ H◦:

φ⊥ψ ⇐⇒ f∗(φ)⊥ f∗(ψ).


Introduction

Chu Spaces



• Overview

• Biextensionaity











eH(φ, ψ) = 0 ⇐⇒ φ⊥ψ.

Proposition 9 If f∗ is injective, it preserves and reflectsorthogonality . That is, for all φ, ψ ∈ H◦:

φ⊥ψ ⇐⇒ f∗(φ)⊥ f∗(ψ).

Proof

φ⊥ψ ⇐⇒ eH(φ, ψ) = 0 Lemma 8

⇐⇒ eH(φ, f∗(f∗(ψ))) = 0 Proposition 7

⇐⇒ eK(f∗(φ), f∗(ψ)) = 0

⇐⇒ f∗(φ)⊥ f∗(ψ) Lemma 8.

Constructing the Left Adjoint

Introduction

Chu Spaces



• Overview

• Biextensionaity









We define a map f→ : L(H) → L(K):

f→(S) =∨

{f∗(ψ) | ψ ∈ S◦}

where S◦ = S \ {0}.


Introduction

Chu Spaces



• Overview

• Biextensionaity










f→(S) =∨

{f∗(ψ) | ψ ∈ S◦}

where S◦ = S \ {0}.

Lemma 10 The map f→ is left adjoint to f∗:

f→(S) ⊆ T ⇐⇒ S ⊆ f∗(T ).


Introduction

Chu Spaces



• Overview

• Biextensionaity










f→(S) =∨

{f∗(ψ) | ψ ∈ S◦}

where S◦ = S \ {0}.

Lemma 10 The map f→ is left adjoint to f∗:

f→(S) ⊆ T ⇐⇒ S ⊆ f∗(T ).

We can now extend the diagram (1):

P(H)f∗

- P(K)

L(H)?

∩

f→-

⊥�

f∗L(K)

?

∩

(2)

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 ofprojective 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).










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.










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!

Wigner’s Theorem

Introduction

Chu Spaces



• Overview

• Biextensionaity









Wigner’s Theorem

Introduction

Chu Spaces



• Overview

• Biextensionaity









Given a semilinear map g : V1 → V2, we define Pg : PV1 → PV2 by

P(g)(ψ) = g(ψ).

Wigner’s Theorem

Introduction

Chu Spaces



• Overview

• Biextensionaity









Given a semilinear map g : V1 → V2, we define Pg : PV1 → PV2 by

P(g)(ψ) = g(ψ).

We can now state Wigner’s Theorem in the form we shall use it.

Theorem 12 Let f : P(H) → P(K) be a total map of projective

geometries, where dimH > 2. If f preserves orthogonality, meaning

that

φ⊥ ψ ⇒ f(φ)⊥ f(ψ)

then there is a semilinear map g : H → K such that P(g) = f , and

〈g(φ) | g(ψ)〉 = σ(〈φ | ψ〉),

where σ is the homomorphism associated with g. Moreover, thishomomorphism is either the identity or complex conjugation, so g is either

linear or antilinear . The map g is unique up to a phase , i.e. a scalar of

modulus 1.

Remarks

Introduction

Chu Spaces



• Overview

• Biextensionaity









Remarks

Introduction

Chu Spaces



• Overview

• Biextensionaity









• Note that in our case, taking f∗ = f , Pg is just the action of the

biextensional collapse functor on Chu morphisms.

Remarks

Introduction

Chu Spaces



• Overview

• Biextensionaity











• Note that a total map of projective geometries must necessarily

come from an injective map g on the underlying vector spaces,

since P(g) maps rays to rays, and hence g must have trivial kernel.

Remarks

Introduction

Chu Spaces



• Overview

• Biextensionaity











• Note that a total map of projective geometries must necessarily

come from an injective map g on the underlying vector spaces,

since P(g) maps rays to rays, and hence g must have trivial kernel.

• For this reason, partial maps of projective geometries are

considered in the Faure-Frolicher approach. However, we are

simply following the ‘logic’ of Chu space morphisms here.

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 anautomorphism. 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.








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 + χwhere χ is a non-zero vector in Im g, which must exist since g is injective and Hhas positive dimension. This vector is not in Im g, nor is it orthogonal to Im g, sincee.g. 〈ψ + χ | χ〉 = 〈χ | χ〉 6= 0. This yields the required contradiction. �

Putting The Pieces Together

Introduction

Chu Spaces



• Overview

• Biextensionaity










Introduction

Chu Spaces



• Overview

• Biextensionaity









We say that a map U : H → K is semiunitary if it is either unitary or

antiunitary; that is, if it is a bijective map satisfying

U(φ+ψ) = Uφ+Uψ, U(λφ) = σ(λ)Uφ, 〈Uφ | Uψ〉 = σ(〈φ | ψ〉)

where σ is the identity if U is unitary, and complex conjugation if U is

antiunitary. Note that semiunitaries preserve norm, so if U and V are

semiunitaries and U = λV , then |λ| = 1.


Introduction

Chu Spaces



• Overview

• Biextensionaity









We say that a map U : H → K is semiunitary if it is either unitary or

antiunitary; that is, if it is a bijective map satisfying

U(φ+ψ) = Uφ+Uψ, U(λφ) = σ(λ)Uφ, 〈Uφ | Uψ〉 = σ(〈φ | ψ〉)

where σ is the identity if U is unitary, and complex conjugation if U is

antiunitary. Note that semiunitaries preserve norm, so if U and V are

semiunitaries and U = λV , then |λ| = 1.

Theorem 14 Let H, K be Hilbert spaces of dimension greater than 2.

Consider a Chu morphism

(f∗, f∗) : (P(H), L(H), eH) → (P(K), L(K), eK).

where f∗ is injective. Then there is a semiunitary U : H → K such that

f∗ = P(U). U is unique up to a phase.

The Representation Theorem

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors

• Not Quite Right Yet

• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid

• Jes’ Right

• PR is anembedding up to aphase


Discussion


Primer on coalgebra

Basic ConceptsBig Toy Models

The Big Picture


The Big Picture


We define a category SymmH as follows:

The Big Picture



• The objects are Hilbert spaces of dimension > 2.

The Big Picture




• Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps.

The Big Picture





• 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 twounitaries compose to form a unitary, while a unitary and an antiunitary

compose to form an antiunitary.

The Big Picture









This category is a groupoid, i.e. every arrow is an isomorphism.

The Big Picture









This category is a groupoid, i.e. every arrow is an isomorphism.

The seminunitaries are the physically significant symmetries of Hilbert spacefrom 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 .

Remarks


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.

Remarks







• Our results then show that if L can be represented as a Chu morphism, it

must in fact be semiunitary.

Remarks







• 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.

Functors

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra

Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 31

We define a functor R : SymmH → eChu[0,1]:

R : U : H → K 7−→ (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK)

where U◦ is the restriction of U to H◦.

As noted in Proposition 2, the inclusion bChu[0,1]⊂ - eChu[0,1] has

a left adjoint, which we name Q. By Proposition 4, this can be defined onthe image of R as follows:

Q : (H◦, L(H), eH) 7→ (PH, L(H), eH)

and for (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK),

Q : (U◦, U−1) 7−→ (PU,U−1).

Not Quite Right Yet

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra


Not Quite Right Yet

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra


We write emChu, bmChu for the subcategories of eChu[0,1] and

bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is

injective. The functors R and Q factor through these subcategories.

Not Quite Right Yet

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra





Proposition 15 Both

R : SymmH → emChu

and

Q : emChu → bmChu

are well-defined functors. R is faithful but not full; Q is full but not faithful.

Not Quite Right Yet

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra





Proposition 15 Both

R : SymmH → emChu

and

Q : emChu → bmChu


This involves verifying that unitaries and antiunitaries U : H → K doindeed yield Chu morphisms!

Not Quite Right Yet

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra





Proposition 15 Both

R : SymmH → emChu

and

Q : emChu → bmChu


This involves verifying that unitaries and antiunitaries U : H → K doindeed yield Chu morphisms!

The key property, for ψ ∈ H◦ and S ∈ L(H), is:

PS(Uψ) = U(PU−1(S)ψ).

Biextensionality and Scalar Factors

Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra



Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra


We can analyze the non-fullness of R more precisely as follows.

Proposition 16 Let (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK)

be a Chu morphism in the image of R. Given an arbitrary function

f : H → C \ {0}, define fU : H◦ → K◦ by:

fU(ψ) = f(ψ)U(ψ).

Then (fU, U−1) ∼ (U◦, U−1). Moreover, the ∼-equivalence class of U

is exactly the set of functions of this form.


Introduction

Chu Spaces




• The Big Picture

• Remarks

• Functors



• Jes’ Right



Discussion


Primer on coalgebra


We can analyze the non-fullness of R more precisely as follows.

Proposition 16 Let (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK)

be a Chu morphism in the image of R. Given an arbitrary function

f : H → C \ {0}, define fU : H◦ → K◦ by:

fU(ψ) = f(ψ)U(ψ).

Then (fU, U−1) ∼ (U◦, U−1). Moreover, the ∼-equivalence class of U

is exactly the set of functions of this form.

Thus before biextensional collapse, Chu morphisms can introduce

arbitrary scalar factors pointwise. Once we move to the biextensional

category, we know by Theorem 14 that our representation will be full, and

essentially faithful — up to a global phase. This points to the need for aprojective version of the symmetry groupoid.

Projectivising The Symmetry Groupoid




The mathematical object underlying phases is the circle group U(1):

U(1) = {λ ∈ C | |λ| = 1} = {eiθ | θ ∈ R}

Since λ has modulus 1 if and only if λλ = 1, U(1) is the unitary group on the

one-dimensional Hilbert space.




U(1) = {λ ∈ C | |λ| = 1} = {eiθ | θ ∈ R}


one-dimensional Hilbert space.The circle group acts on the symmetry groupoid SymmH by scalar multiplication.

For Hilbert spaces H, K we can define

U(1) × SymmH(H,K) → SymmH(H,K) :: (λ, U) 7→ λU.

Moreover, this is a category action, meaning that

(λU) ◦ V = U ◦ (λV ) = λ(U ◦ V ).




U(1) = {λ ∈ C | |λ| = 1} = {eiθ | θ ∈ R}


one-dimensional Hilbert space.The circle group acts on the symmetry groupoid SymmH by scalar multiplication.

For Hilbert spaces H, K we can define

U(1) × SymmH(H,K) → SymmH(H,K) :: (λ, U) 7→ λU.

Moreover, this is a category action, meaning that

(λU) ◦ V = U ◦ (λV ) = λ(U ◦ V ).

It follows that we can form a quotient category (in fact again a groupoid) with the

same objects as SymmH, and in which the morphisms will be the orbits of this

group action:

U ∼ V ⇔ ∃λ ∈ U(1). U = λV.

Jes’ Right


Jes’ Right


We call the resulting category PSymmH, the projective quantum symmetrygroupoid . It is a natural generalization of the standard notion of the projectiveunitary group on Hilbert space.

Jes’ Right



There is a quotient functor P : SymmH → PSymmH, and by virtue of

Theorem 14, we can factor Q ◦R through this quotient to obtain a functor

PR : PSymmH → bmChu.

Jes’ Right






The situation can be summarized by the following diagram:

SymmH >R

> emChu

PSymmH

P∨∨

>PR

>> bmChu

Q∨∨

Jes’ Right






The situation can be summarized by the following diagram:

SymmH >R

> emChu

PSymmH

P∨∨

>PR

>> bmChu

Q∨∨

Theorem 17 The functor PR : PSymmH → bmChu is full and faithful.

PR is an embedding up to a phase




• To see that PR is essentially injective on objects, we can use therepresentation theorems of Piron and Soler, which tell us that we can

reconstruct H as a Hilbert space from L(H).





• This reconstruction will give us a Hilbert space H′ such that L(H) ∼= L(H′),

and P(H) ∼= P(H′).






and P(H) ∼= P(H′).

• We can apply Wigner’s theorem to this isomorphism to obtain a semiunitary

U : H ∼= H′ from which we can recover the Hilbert space structure on H.






and P(H) ∼= P(H′).

• We can apply Wigner’s theorem to this isomorphism to obtain a semiunitary

U : H ∼= H′ from which we can recover the Hilbert space structure on H.

• This means that we have recovered H uniquely to within the coset of idH inPSymmH.

Reducing The Value Set

Introduction

Chu Spaces





• Generalities

• The Question

• Two Values?• The CanonicalPossibilistic Reductions

• Two is Too Few

• Other Case

• Analysis

• Three ValuesSuffice!

Discussion


Primer on coalgebra

Basic Concepts

Representing PhysicalSystems AsBig Toy Models

Generalities

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts

Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 38

Generalities

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts


We now return to the issue of whether it is necessary to use the full unit

interval as the value set for our Chu spaces.

Generalities

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts




We begin with some generalities. Given a function v : K → L, we define

a functor Fv : ChuK → ChuL:

Fv : (X,A, e) 7→ (X,A, v ◦ e)

and Fvf = f for Chu morphisms f .

Generalities

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts




We begin with some generalities. Given a function v : K → L, we define

a functor Fv : ChuK → ChuL:

Fv : (X,A, e) 7→ (X,A, v ◦ e)

and Fvf = f for Chu morphisms f .

Proposition 18 Fv is a faithful functor. If v is injective, it is full.

The Question


The Question


We can now state the question we wish to pose more precisely:

Is there a mapping v : [0, 1] → K from the unit interval to some

finite set K such that the restriction of the functor Fv to the image ofPR is full, and thus the composition

Fv ◦ PR : PSymmH → bmChuK

is a representation?

The Question


We can now state the question we wish to pose more precisely:

Is there a mapping v : [0, 1] → K from the unit interval to some

finite set K such that the restriction of the functor Fv to the image ofPR is full, and thus the composition

Fv ◦ PR : PSymmH → bmChuK

is a representation?

There is no general reason to suppose that this is possible; in fact, we shall showthat it is, although not quite in the obvious fashion.

Two Values?

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts


Two Values?

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts


We shall write n = {0, . . . , n− 1} for the finite ordinals. The most

popular choice of value set for Chu spaces, by far, has been 2, and

indeed many interesting categories can be strictly (and even concretely)

represented in Chu2.

Two Values?

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts






This makes the following question natural:

Question 19 Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full

and faithful?

Two Values?

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts






This makes the following question natural:

Question 19 Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full

and faithful?

What we can show is that the most plausible candidates for suchfunctions, yielding the two canonical forms of possibilistic semantics as

a coarse-graining of probabilistic semantics, both in fact fail .

The Canonical Possibilistic Reductions

Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts



Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts


Note that any function v : [0, 1] → {0, 1} must lose information either on

0 or on 1 — or both.


Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts




In this sense, the two ‘sharpest’ mappings will be:

v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1.


Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts





v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1.

These are the two canonical reductions of probabilistic to possibilisticinformation: the first maps ‘definitely not’ to ‘no’, and anything else to

‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitelyyes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’.


Introduction

Chu Spaces





• Generalities

• The Question


• Two is Too Few

• Other Case

• Analysis


Discussion


Primer on coalgebra

Basic Concepts





v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1.

These are the two canonical reductions of probabilistic to possibilisticinformation: the first maps ‘definitely not’ to ‘no’, and anything else to

‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitelyyes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’.

Note that, under the first of these, we no longer have

eH(ψ, S) = 1 ⇐⇒ ψ ∈ S

while under the second, we no longer have

eH(ψ, S) = 0 ⇐⇒ ψ⊥S.

Two is Too Few


Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full.

Two is Too Few



Let H be a Hilbert space with 2 < dimH <∞, and let (g, σ) be any semilinear

automorphism of H, where σ can be any automorphism of the complex field. (We

can extend the argument to infinite-dimensional Hilbert space by requiring g to becontinuous.) For each of the above two mappings of the unit interval to 2, we shall

construct a Chu2 endomorphism f : Fv ◦ PR(H) → Fv ◦ PR(H) with

f∗ = P(g). This will show the non-fullness of Fv .

Two is Too Few



Let H be a Hilbert space with 2 < dimH <∞, and let (g, σ) be any semilinear

automorphism of H, where σ can be any automorphism of the complex field. (We

can extend the argument to infinite-dimensional Hilbert space by requiring g to becontinuous.) For each of the above two mappings of the unit interval to 2, we shall

construct a Chu2 endomorphism f : Fv ◦ PR(H) → Fv ◦ PR(H) with

f∗ = P(g). This will show the non-fullness of Fv .

Case 1 Here we consider the mapping v1 which sends [0, 1) to 0 and fixes 1. In this

case:eH(ψ, S) = 1 ⇐⇒ ψ ∈ S

and hence the Chu morphism condition on (f∗, f∗), where f∗ = P(g), is:

ψ ∈ f∗(S) ⇐⇒ g(ψ) ∈ S.

Taking f∗ = g−1 obviously fulfills this condition. Note that, since g is a semilinear

automorphism, and H is finite-dimensional, g−1 : L(H) → L(H) is well-defined.

Other Case


Case 2 Now consider the mapping v0 keeping 0 fixed and sending (0, 1] to 1. In

this case:

eH(ψ, S) = 0 ⇐⇒ ψ⊥S


ψ⊥ f∗(S) ⇐⇒ g(ψ)⊥S.

Other Case



this case:

eH(ψ, S) = 0 ⇐⇒ ψ⊥S


ψ⊥ f∗(S) ⇐⇒ g(ψ)⊥S.

We define f∗(S) = g−1(S⊥)⊥. Note that f∗ : L(H) → L(H) is well defined, and

also that g−1(S⊥) is a subspace of H; hence g−1(S⊥)⊥⊥ = g−1(S⊥).

Other Case



this case:

eH(ψ, S) = 0 ⇐⇒ ψ⊥S


ψ⊥ f∗(S) ⇐⇒ g(ψ)⊥S.

We define f∗(S) = g−1(S⊥)⊥. Note that f∗ : L(H) → L(H) is well defined, and

also that g−1(S⊥) is a subspace of H; hence g−1(S⊥)⊥⊥ = g−1(S⊥).

ψ⊥ f∗S ⇐⇒ ψ ∈ g−1(S⊥)⊥⊥ = g−1(S⊥)

⇐⇒ g(ψ) ∈ S⊥

⇐⇒ g(ψ)⊥S.

and hence (f∗, f∗) is a Chu morphism as required.

Analysis


Analysis


However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:

v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1

Thus we lose information only on the probabilities strictly between 0 and 1, which

are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.

Analysis



v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1



Why is this adequate? Given a vector ψ and a subspace S, we can write ψ uniquely

as θ + χ, where θ ∈ S and χ ∈ S⊥. For non-zero ψ, there are only three

possibilities:

Analysis



v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1





possibilities:

• θ = 0 and χ 6= 0, so eH(φ, S) = 0

Analysis



v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1





possibilities:

• θ = 0 and χ 6= 0, so eH(φ, S) = 0

• θ 6= 0 and χ = 0, so eH(φ, S) = 1

Analysis



v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1





possibilities:

• θ = 0 and χ 6= 0, so eH(φ, S) = 0

• θ 6= 0 and χ = 0, so eH(φ, S) = 1

• θ 6= 0 and χ 6= 0, so eH(ψ, S) ∈ (0, 1), and hence v ◦ eH(ψ, S) = 2.

Analysis



v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1





possibilities:

• θ = 0 and χ 6= 0, so eH(φ, S) = 0

• θ 6= 0 and χ = 0, so eH(φ, S) = 1

• θ 6= 0 and χ 6= 0, so eH(ψ, S) ∈ (0, 1), and hence v ◦ eH(ψ, S) = 2.

These are the only case discriminations which are used in pro ving theRepresentation Theorem .

Three Values Suffice!




Hence we have:

Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is arepresentation.



Hence we have:


We note that Chu3 has found some uses in concurrency and verification (Pratt03,

Ivanov08), under a temporal interpretation: the three values are read as ‘before’,

‘during’ and ‘after’, whereas in our setting the three values represent ‘definitely yes’,

‘definitely no’ and ‘maybe’.



Hence we have:


We note that Chu3 has found some uses in concurrency and verification (Pratt03,

Ivanov08), under a temporal interpretation: the three values are read as ‘before’,

‘during’ and ‘after’, whereas in our setting the three values represent ‘definitely yes’,

‘definitely no’ and ‘maybe’.

Theorem 21 may suggest some interesting uses for 3-valued ‘local logics’ in the

sense of Jon Barwise.

Discussion

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts



Semantics in OneCountry

ExternalisingContravariance AsIndexingBig Toy Models

Where Next?

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts




ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47

Where Next?

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts





• Connections and contrasts between Chu spaces and coalgebras .

Where Next?

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts






• Mixed states — handled generally at the level of Chu spaces.

Where Next?

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts







• Universal Chu spaces.

Where Next?

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts








• Linear and other type theories.

Where Next?

Introduction

Chu Spaces





Discussion

• Where Next?


Primer on coalgebra

Basic Concepts








• Linear and other type theories.

• Local logics.

Chu Spaces and Coalgebras

Introduction

Chu Spaces





Discussion


• Chu Spaces andCoalgebras

Primer on coalgebra

Basic Concepts




ExternalisingContravariance AsBig Toy Models


Introduction

Chu Spaces





Discussion



Primer on coalgebra

Basic Concepts




ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49


Introduction

Chu Spaces





Discussion



Primer on coalgebra

Basic Concepts





• Coalgebras over Set; ‘universal coalgebra’.


Introduction

Chu Spaces





Discussion



Primer on coalgebra

Basic Concepts






• Each of these general systems models has been studied

extensively.


Introduction

Chu Spaces





Discussion



Primer on coalgebra

Basic Concepts







extensively.Their connections have not been studied at all.


Introduction

Chu Spaces





Discussion



Primer on coalgebra

Basic Concepts








• They have complementary merits and deficiencies.

• Chu spaces have , coalgebras lack : contravariance.

• Coalgebras have , Chu spaces lack : extension in time.

• Symmetry vs. rigidity.


Introduction

Chu Spaces





Discussion



Primer on coalgebra

Basic Concepts








• They have complementary merits and deficiencies.

• Chu spaces have , coalgebras lack : contravariance.

• Coalgebras have , Chu spaces lack : extension in time.

• Symmetry vs. rigidity.

• Interesting formal consequences:

• Indexed structure (‘externalising contravariance’)

• Grothendieck construction: new description of Chu spaces.

• Truncation functors.

Primer on coalgebra

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts




ExternalisingContravariance AsIndexingBig Toy Models

Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts





Category theory allows us to dualize algebras to obtain a notion of

coalgebras of an endofunctor . However, while algebras abstract a

familiar set of notions, coalgebras open up a new and rather unexpected

territory, and provides an effective abstraction and mathematical theory

for a central class of computational phenomena:

Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts










• Programming over infinite data structures : streams, lazy lists,

infinite trees . . .

Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts












• A novel notion of coinduction

Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts













• Modelling state-based computations of all kinds

Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts














• The key notion of bisimulation equivalence between processes.

Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

• Coalgebras

Basic Concepts














• The key notion of bisimulation equivalence between processes.

• A general coalgebraic logic can be read off from the functor, andused to specify and reason about properties of systems.

Basic Concepts

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras

• Final F -coalgebras

• Labelled TransitionSystems

• Transition Graphs asCoalgebras

• The Final Coalgebra



Big Toy Models

F -Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras








Let F : C −→ C be a functor.An F -coalgebra is a pair (A, γ : A −→ FA) for some object A of C.

We say that A is the carrier of the coalgebra, while γ is the behaviourmap .

An F -coalgebra homomorphism from (A, γ : A −→ FA) to

(B, δ : B −→ FB) is an arrow h : A −→ B such that

Aγ

- FA

B

h

?

δ- FB

Fh

?

F -coalgebras and their homomorphisms form a category F−Coalg.

Final F -coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras








An F -coalgebra (C, γ) is final if for every F -coalgebra (A,α) there is a

unique homomorphism from (A,α) to (C, γ).

Proposition 22 If a final F -coalgebra exists, it is unique up to

isomorphism.

Proposition 23 (Lambek Lemma) If γ : A −→ FA is final, it is an

isomorphism

Labelled Transition Systems

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras








Let A be a set of actions . A labelled transition system over A is a

coalgebra for the functor

LTA : Set −→ Set :: X 7→ Pf(A×X).

Such a coalgebra

γ : S −→ Pf(A× S)

can be understood operationally as follows:

• S is a set of states

• For each state s ∈ S, γ(s) specifies the possible transitions from

that state. We write sa

−→ s′ if (a, s′) ∈ γ(s). We think of such a

transition as consisting of the system performing the action a, and

changing state from s to s′. Note that we regard actions as directlyobservable , while states are not.

Transition Graphs as Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras








Note that any labelled transition graph (or “state machine”) with labels in

A is a coalgebra for LTA.

Examples 1.

1 2b ca

This corresponds to the coalgebra ({1, 2}, γ)

γ : 1 7→ {(a, 1), (b, 2)}, 2 7→ {(c, 2)}

2.

1 2 3b

a

ac

1 7→ {(b, 2), (c, 1)}, 2 7→ {(a, 1), (a, 3)}, 3 7→ ∅

The Final Coalgebra

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras








The key point is this.

Proposition 24 For any set A of actions, there is a final LTA-coalgebra

(ProcA, out).

The Final Coalgebra

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras










(ProcA, out).

We think of elements of the final coalgebra as processes . The final

coalgebra provides a “universal semantics” for transition systems, which

is “fully abstract” with respect to observable behaviour.

The Final Coalgebra

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts

• F -Coalgebras










(ProcA, out).

We think of elements of the final coalgebra as processes . The final

coalgebra provides a “universal semantics” for transition systems, which

is “fully abstract” with respect to observable behaviour.

All of this generalizes to a large class of endofunctors.

Representing Physical SystemsAs Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts


• Coalgebras asModels of PhysicalSystems



ExternalisingBig Toy Models

Coalgebras as Models of Physical Systems

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





ExternalisingBig Toy Models Workshop on Informatic Penomena 2009 – 59


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts






Recall our basic setup: systems are modelled by a set of states S, of

questions Q, and an evaluation

e : S ×Q→ [0, 1].


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








e : S ×Q→ [0, 1].

Problems:


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








e : S ×Q→ [0, 1].

Problems:

• In Chu spaces, we get to specify Q as well as S. How do we do

this with coalgebras?


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








e : S ×Q→ [0, 1].

Problems:

• In Chu spaces, we get to specify Q as well as S. How do we do

this with coalgebras?

• Q is contravariant (the maps f∗ go backwards.). Coalgebras are

based on covariant functors. (We could work with domains, but

there are drawbacks).

Comparison: A First Try

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts



• First Approximation

• Comparison

• Discussion: Critiqueof Coalgebras

• Discussion: In Praiseof Coalgebras

• Extension in Time

Big Toy Models

First Approximation


Fix a set K. We can define a functor on Set:

FK : X 7→ KPX .

First Approximation



FK : X 7→ KPX .

If we use the contravariant powerset functor, F will be covariant. Explicitly, for

f : X → Y :

FKf(g)(S) = g(f−1(S)),

where g ∈ KPX and S ∈ PY .

First Approximation



FK : X 7→ KPX .


f : X → Y :

FKf(g)(S) = g(f−1(S)),


A coalgebra for this functor will be a map of the form

α : X → KPX .

First Approximation



FK : X 7→ KPX .


f : X → Y :

FKf(g)(S) = g(f−1(S)),


A coalgebra for this functor will be a map of the form

α : X → KPX .

Consider a Chu space C = (X,A, e) over K. We suppose that this Chu space is

normal , meaning that A = PX . We can define an FK -coalgebra on X by

α(x)(S) = e(x, S).

We write GC = (X,α).

Comparison


Proposition 25 Suppose we are given a Chu morphism f : C → C ′, where Cand C ′ are normal Chu spaces, such that f∗ = f−1

∗ . Then f∗ : GC → GC ′ is an

FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism

f : GC → GC ′, then (f, f−1) : C → C ′ is a Chu morphism.

Comparison






Let NChuK be the category of normal Chu spaces and Chu morphisms of the

form (f, f−1). Then by the Proposition, G extends to a functorG : NChuK → FK−Coalg, with G(f, f−1) = f .

Comparison








There is an evident inverse to this functor.

Comparison








There is an evident inverse to this functor.

Proposition 26 NChuK and FK−Coalg are isomorphic categories.

Discussion: Critique of Coalgebras




• Assuming Chu spaces are normal is overly restrictive.




The use of powersets, full or not, to represent ‘questions’ is fairly crude and ad

hoc. The degree of freedom afforded by Chu spaces to choose both the

states and the questions appropriately is a major benefit to conceptually

natural and formally adequate modelling of a wide range of situations.








• The functors FK are somewhat problematic from the point of view of

coalgebra — they fail to preserve weak pullbacks.








• The functors FK are somewhat problematic from the point of view of

coalgebra — they fail to preserve weak pullbacks.

• They will also fail to have final coalgebras. However, this can be fixed easily

enough by using bounded powerset and bounded partial functions.

Discussion: In Praise of Coalgebras

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison






Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison





• The coalgebraic point of view can be described as state-based , but

in a way that emphasizes that the meaning of states lies in their

observable behaviour . Indeed, in the “universal model” we shallconstruct, the states are determined exactly as the possible

observable behaviours — we actually find a canonical solution for

what the state space should be in these terms. States are

identified exactly if they have the same observable behaviour.


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison











We can see this as a kind of reconciliation between the ontic and

epistemic standpoints.


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison











We can see this as a kind of reconciliation between the ontic and

epistemic standpoints.

• Coalgebras allow us to capture the ‘dynamics of measurement’ —

what happens after a measurement — in a way that Chu spaces

don’t. They have ‘extension in time’.

Extension in Time

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison





Extension in Time

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison





Consider a coalgebraic representation of stochastic transducers :

F : X 7→ Prob(O ×X)I

where I is a fixed set of inputs , O a fixed set of outputs , and Prob(S) is

the set of probability distributions on S.

Extension in Time

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison









We can think of I as a set of questions , and O as a set of answers(which we could standardize by only considering yes/no questions).

Extension in Time

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• Comparison









We can think of I as a set of questions , and O as a set of answers(which we could standardize by only considering yes/no questions).

What we learn from this is that

QM is less nondeterministic/probabilistic than stochastic transducers

since in QM if we know the preparation and the outcome of the

measurement, we know (by the projection postulate) exactly what theresulting quantum state will be.

Semantics in One Country

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




• CoalgebraicSemantics For OneSystem

• Well BehavedFunctorsBig Toy Models

Coalgebraic Semantics For One System




We fix attention on a single Hilbert space H. This determines a set of question

Q = L(H).




Q = L(H).

We now define an endofunctor on Set:

FQ : X 7→ ({0} + (0, 1] ×X)Q.




Q = L(H).

We now define an endofunctor on Set:

FQ : X 7→ ({0} + (0, 1] ×X)Q.

A coalgebra for this functor is then a map

α : X → ({0} + (0, 1] ×X)Q

The interpretation is that X is a set of states; the coalgebra map sends its state to

its behaviour, which is a function from questions in Q to the probability that the

answer is ‘yes’; and, if the probability is not 0 , to the successor state following a

‘yes’ answer.

Well Behaved Functors


Unlike the functors FK , the functors FQ are very well-behaved from the point of

view of coalgebra (they are in fact polynomial functors ). They preserve weak

pull-backs, which guarantees a number of nice properties, and they are bounded

and admit final coalgebras

γQ : UQ → ({0} + (0, 1] × UQ)Q.

Well Behaved Functors


Unlike the functors FK , the functors FQ are very well-behaved from the point of

view of coalgebra (they are in fact polynomial functors ). They preserve weak

pull-backs, which guarantees a number of nice properties, and they are bounded

and admit final coalgebras

γQ : UQ → ({0} + (0, 1] × UQ)Q.

The elements of UQ can be visualized as ‘Q-branching trees’ with the arcs labelledby probabilities.

Representing One Quantum System As A Coalgebra




The FQ-coalgebra which is of primary interest to us is the map

aH : H◦ → ({0} + (0, 1] ×H◦)Q

defined by:

aH(ψ)(S) =

0, eH(ψ, S) = 0

(r, PSψ), eH(ψ, S) = r > 0



The FQ-coalgebra which is of primary interest to us is the map

aH : H◦ → ({0} + (0, 1] ×H◦)Q

defined by:

aH(ψ)(S) =

0, eH(ψ, S) = 0

(r, PSψ), eH(ψ, S) = r > 0

The new ingredient compared with the Chu space representation of H is the statewhich results in the case of a ‘yes’ answer to the question, which is computed

according to the Luders rule .

Externalising Contravariance AsIndexing

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts




ExternalisingContravariance AsIndexing

• The IndexedBig Toy Models

The Indexed Category

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 71

We define a functor

F : Setop → CAT

with

Q 7→ FQ−Coalg

and for f : Q′ → Q:

tf : FQ → FQ′

:: Θ 7→ Θ ◦ f

is a natural transformation, and

F(f) = f∗ : Coalg−FQ → Coalg−FQ′

f∗ : (A,α) 7→ (A, tfA ◦ α)

is a functor.

The Indexed Category

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts






We define a functor

F : Setop → CAT

with

Q 7→ FQ−Coalg

and for f : Q′ → Q:

tf : FQ → FQ′

:: Θ 7→ Θ ◦ f

is a natural transformation, and

F(f) = f∗ : Coalg−FQ → Coalg−FQ′

f∗ : (A,α) 7→ (A, tfA ◦ α)

is a functor.

Thus we get a strict indexed category of coalgebra categories, with

contravariant indexing.

The Grothendieck Construction

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts






Where we have an indexed category, we can apply the Grothendieckconstruction to glue all the fibres together (and get a fibration).


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts







Given a functorI : C

op → CAT

define∫

I with objects (A, a), where A is an object of C and a is an

object of I(A).


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








op → CAT

define∫


object of I(A).

Arrows are (G, g) : (A, a) → (B, b), where G : B → A and

g : I(G)(a) → b.


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts








op → CAT

define∫


object of I(A).

Arrows are (G, g) : (A, a) → (B, b), where G : B → A and

g : I(G)(a) → b.

Composition of (G, g) : (A, a) → (B, b) and (H,h) : (B, b) → (C, c)is given by

(G ◦H,h ◦ I(G)(g)) : (A, a) → (C, c).

Indexed Comparison With ChuSpaces

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison

Big Toy Models

Slicing and Dicing Chu


For each Q, we define ChuQK to be the subcategory of ChuK of Chu spaces

(X,Q, e) and morphisms of the form (f∗, idQ).





This doesn’t look too exciting. In fact, it is just the comma category

(−×Q, K)

where K : 1 → Set picks out the object K.






(−×Q, K)


Given f : Q′ → Q, we define a functor

f∗ : ChuQK → Chu

Q′

K :: (X,Q, e) 7→ (X,Q′, e ◦ (1 × f))

and which is the identity on morphisms.






(−×Q, K)


Given f : Q′ → Q, we define a functor

f∗ : ChuQK → Chu

Q′

K :: (X,Q, e) 7→ (X,Q′, e ◦ (1 × f))

and which is the identity on morphisms.

This gives an indexed category

Chu : Setop → CAT

Grothendieck puts Chu back together again

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison


Grothendieck puts Chu back together again

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison


Proposition 27∫

Chu ∼= ChuK .

The Truncation Functor

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison



Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison


The relationship between coalgebras and Chu spaces is further clarified

by an indexed truncation functor T : F → Chu.


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison




For each set Q there is a functor

TQ : FQ−Coalg → Chu

QK

TQ(X,α) = (X,Q, e)

where

e(x, q) =

0, α(x)(q) = 0

r, α(x)(q) = (r, x′)


Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison




For each set Q there is a functor

TQ : FQ−Coalg → Chu

QK

TQ(X,α) = (X,Q, e)

where

e(x, q) =

0, α(x)(q) = 0

r, α(x)(q) = (r, x′)

For f : Q′ → Q there is a natural transformation

τ f : TQ → TQ′

τ f

(X,α)= (idX , f) : TQ(X,α) → TQ′

(X,α).

A Universal Model

Introduction

Chu Spaces





Discussion


Primer on coalgebra

Basic Concepts





Indexed Comparison

Big Toy Models

A Universal Model


A Universal Model


We can now define a single coalgebra which is universal for quantum systems in

the following sense:

A Universal Model




• Fix a countably-infinite dimensional Hilbert space, e.g. H = ℓ2(N). Take

Q = L(H). Let (U, γ) be the final coalgebra for FQ.

A Universal Model






• Any quantum system is described by a separable Hilbert space K, say with a

preferred basis. This basis will induce an isometric embedding i : K- - H.

Taking Q′ = L(K), this induces a map f = i−1 : Q→ Q′. This in turninduces a functor f∗ : FQ′

−Coalg → FQ−Coalg.

A Universal Model






• Any quantum system is described by a separable Hilbert space K, say with a

preferred basis. This basis will induce an isometric embedding i : K- - H.

Taking Q′ = L(K), this induces a map f = i−1 : Q→ Q′. This in turninduces a functor f∗ : FQ′

−Coalg → FQ−Coalg.

• This functor can be applied to the coalgebra (K◦, α) corresponding to theHilbert space K to yield a coalgebra in FQ−Coalg.

Universality


Universality


• Since (U, γ) is the final coalgebra in FQ−Coalg, there is a unique

coalgebra homomorphism h : f∗(K◦, α) → (U, γ).

Universality




• This homomorphism maps the quantum system (K◦, α) into (U, γ) in a fullyabstract fashion , i.e. identifying states precisely according to observational

equivalence.

Universality





equivalence.

• This homomorphism is an arrow in the Grothendieck category.

Universality





equivalence.


• This works for all quantum systems, with respect to a single coalgebra.

Universality





equivalence.


• This works for all quantum systems, with respect to a single coalgebra.

This is truly a Big Toy Model!

Big Toy Models: Representing Physical Systems as Chu Spaces

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Transcript of Big Toy Models: Representing Physical Systems as Chu Spaces