duality for the logic of quantum actions · duality for the logic of quantum actions jort martinus...

Duality for the logic of quantum actions

Jort Martinus Bergfeld

Joint work with: A. Baltag, K. Kishida, J. Sack, S. Smets, S. ZhongInstitute for logic, language and computation

Universiteit van Amsterdam

Saturday 30 November 2013Whither Quantum Structures?

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 1 / 19

Two approaches

Piron lattices

I Algebraic approachI Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.

Dynamic quantum frames

s

t u

P? Q?

I Spatial approachI Based on Propositional Dynamic Logic


Two approaches

Piron latticesI Algebraic approachI Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.


s

t u

P? Q?

I Spatial approachI Based on Propositional Dynamic Logic


DualityA category consists of objects and morphisms

A functor acts on objects and morphismsA duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2

Dynamic Quantum Frame

f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒


DualityA category consists of objects and morphismsA functor acts on objects and morphisms

A duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2


f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒


DualityA category consists of objects and morphismsA functor acts on objects and morphismsA duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2


f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒


References

A. Baltag and S. Smets (2005), “Complete Axiomatization for Quantum Actions”,Int. J. Theor. Phys. 44, 2267–2282.B. Coecke and D. Moore (2000), “Operational Galois Adjunction”, in B. Coecke,et al., eds., Current Research in Operational Quantum Logic, pp. 195–218,Kluwer.B. Coecke and I. Stubbe (2000), “State Transitions as Morphisms for CompleteLattices”, Int. J. Theor. Phys. 39, 601–610.C.-A. Faure and A. Frölicher (1995), “Dualities for Infinite- Dimensional ProjectiveGeometries”, Geom. Ded. 56, 225–236.D. Moore (1995), “Categories of Representations of Physical Systems”, HelvetiaPhysica Acta 68, 658–678.C. Piron (1976), Foundations of Quantum Physics, W. A. Benjamin.I. Stubbe and B. Van Steirteghem (2007), “Propositional Systems, HilbertLattices and Generalized Hilbert Spaces”, in K. Engesser, et al., eds., Handbookof Quantum Logic and Quantum Structures: Quantum Structures, Elsevier, pp.477–524.


Outline

1 Introduction

2 Piron latticesPL-morphisms

3 Dynamic Quantum FramesDQF-morphisms

4 Duality


Outline

1 Introduction



4 Duality


Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.




X ∈ L

atomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′







X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ p

orthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′








1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′








1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′


covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atom

irreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3






1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′





Piron Lattice morphism

Given two Piron lattices L1 = (L1,≤1, (·)′) and L2 = (L2,≤2, (·)∗), aPL-morphism is a function h : L1 → L2 such that

h(∧

S) =∧

p∈S h(s) for all S ⊆ L1.h(p′) = h(p)∗

for all atoms b ∈ L2 there exists an atom a ∈ L1 such that

b ≤ h(a).


Outline

1 Introduction



4 Duality


Dynamic quantum frame

A dynamic quantum frame is a tuple F = (Σ, { P?−→}P∈L) with:Σ a set of statesL ⊆ P(Σ) a set of testable propertiesP?−→ represent projections

→:=⋃L

P?−→ is non-orthogonality∼P := {s ∈ Σ : s 9 t for all t ∈ P}

that satisfies the following properties:




→:=⋃L

P?−→ is non-orthogonality

∼P := {s ∈ Σ : s 9 t for all t ∈ P}





→:=⋃L

P?−→ is non-orthogonality∼P := {s ∈ Σ : s 9 t for all t ∈ P}




intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ L

atomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t




X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ










self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → s

covering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t







self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 s

proper superposition: for all s, t there is w such that s → w → t


DQF-morphisms

Given two DQFs F1 = (Σ1, {P?−→}P∈L1) and F2 = (Σ2, {

P?−→}P∈L2), aDQF-morphism is a function f : Σ1 → Σ2 such that

f is a bounded morpism:I if s → t , then f (s)→ f (t)I if f (s)→ w , then there exists a t ∈ Σ1 such that s → t and f (t) = w .


Outline

1 Introduction



4 Duality


DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality


PL→ DQF

Let L = (L,≤, (·)′) be a Piron lattice.Σ = atoms of LL = {atoms[p] | p ∈ L}

ap?−→ b iff b = (a ∨ p′) ∧ p

Note that:if a ∧ p′ = 0, then (a ∨ p′) ∧ p is an atom.

if b = (a ∨ p′) ∧ p, then b ∧ p = b.


PL→ DQF

Let L = (L,≤, (·)′) be a Piron lattice.Σ = atoms of LL = {atoms[p] | p ∈ L}

ap?−→ b iff b = (a ∨ p′) ∧ p

Note that:if a ∧ p′ = 0, then (a ∨ p′) ∧ p is an atom.if b = (a ∨ p′) ∧ p, then b ∧ p = b.


DQF-→ PL-Morphisms

PL-morphismh(∧

S) =∧



b ≤ h(a).

Let f : Σ1 → Σ2 be a DQF-morphism.

f−1 preserves⋂

=∧,⋃

and ¬.We have s ∈ f−1[{t}], for t = f (s).Remains to show: f−1 preserves ∼.



PL-morphismh(∧

S) =∧



b ≤ h(a).


f−1 preserves⋂

=∧,⋃

and ¬.

We have s ∈ f−1[{t}], for t = f (s).Remains to show: f−1 preserves ∼.



PL-morphismh(∧

S) =∧



b ≤ h(a).


f−1 preserves⋂

=∧,⋃

and ¬.We have s ∈ f−1[{t}], for t = f (s).

Remains to show: f−1 preserves ∼.



PL-morphismh(∧

S) =∧



b ≤ h(a).


f−1 preserves⋂

=∧,⋃

and ¬.We have s ∈ f−1[{t}], for t = f (s).Remains to show: f−1 preserves ∼.


Preservation of ∼

Let �A = {s ∈ Σ | t ∈ A whenever s → t}

Theoremf is a bounded morphism iff f−1 preserves �.

∼A = � ◦ ¬A


PL-→ DQF-morphismLet h : L1 → L2 be a PL-morphism. Define ` by

`(x) =∧

x≤h(y)

y .

` is a left adjoin of h:

`(x) ≤ y ⇔ x ≤ h(y).

` sends atoms to atoms.

f = ` � (atoms of L1)

f−1 preserves �, because

� = ∼ ◦ ¬.


Thank you for your attention.


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