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Duality for the logic of quantum actions

Jort Martinus Bergfeld

Joint work with: A. Baltag, K. Kishida, J. Sack, S. Smets, S. Zhong Institute for logic, language and computation Universiteit van Amsterdam

Saturday 30 November 2013 Whither Quantum Structures?

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

Two approaches

Piron lattices Dynamic quantum frames s t u P? Q?

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 2 / 19

Two approaches

Piron lattices

◮ Algebraic approach ◮ Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.

Dynamic quantum frames s t u P? Q?

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 2 / 19

Two approaches

Piron lattices

◮ Algebraic approach ◮ Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.

Dynamic quantum frames s t u P? Q?

◮ Spatial approach ◮ Based on Propositional Dynamic Logic J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 2 / 19

Duality

A category consists of objects and morphisms Σ1 Σ2 Dynamic Quantum Frame

f

L1 L2 Piron Lattice

h

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

Duality

A category consists of objects and morphisms A functor acts on objects and morphisms Σ1 Σ2 Dynamic Quantum Frame

f

L1 L2 Piron Lattice

h F G

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

Duality

A category consists of objects and morphisms A functor acts on objects and morphisms A duality means: G ◦ F ≃ IdDQF F ◦ G ≃ IdPL Σ1 Σ2 Dynamic Quantum Frame

f

L1 L2 Piron Lattice

h F G

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

Duality

A category consists of objects and morphisms A functor acts on objects and morphisms A 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

⇐ ⇒ ⇐ ⇒

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

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 Complete

Lattices”, Int. J. Theor. Phys. 39, 601–610. C.-A. Faure and A. Frölicher (1995), “Dualities for Infinite- Dimensional Projective Geometries”, Geom. Ded. 56, 225–236.

• D. Moore (1995), “Categories of Representations of Physical Systems”, Helvetia

Physica Acta 68, 658–678.

• C. Piron (1976), Foundations of Quantum Physics, W. A. Benjamin.
• I. Stubbe and B. Van Steirteghem (2007), “Propositional Systems, Hilbert

Lattices and Generalized Hilbert Spaces”, in K. Engesser, et al., eds., Handbook

• f Quantum Logic and Quantum Structures: Quantum Structures, Elsevier, pp.

477–524.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 4 / 19

Outline

1

Introduction

2

Piron lattices PL-morphisms

3

Dynamic Quantum Frames DQF-morphisms

4

Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 5 / 19

Outline

1

Introduction

2

Piron lattices PL-morphisms

3

Dynamic Quantum Frames DQF-morphisms

4

Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 6 / 19

Piron lattices

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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron lattices

A Piron lattice L = (L, ≤, (·)′, 0, 1) is a complete, orthomodular, atomic and irreducible lattice satisfying the covering law. complete: X ⊆ L = ⇒ X ∈ L

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron lattices

A Piron lattice L = (L, ≤, (·)′, 0, 1) is a complete, orthomodular, atomic and irreducible lattice satisfying the covering law. complete: X ⊆ L = ⇒ X ∈ L atomic: for all p > 0 there exists an atom a such that a ≤ p

Definition

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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron lattices

A Piron lattice L = (L, ≤, (·)′, 0, 1) is a complete, orthomodular, atomic and irreducible lattice satisfying the covering law. complete: X ⊆ L = ⇒ X ∈ L atomic: for all p > 0 there exists an atom a such that a ≤ p

• rthomodular: there is an orthocomplementation (·)′ such that

1

(p′)′ = p

2

p ∧ p′ = 0 and p ∨ p′ = 1

3

p ≤ q = ⇒ q′ ≤ p′

4

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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron lattices

A Piron lattice L = (L, ≤, (·)′, 0, 1) is a complete, orthomodular, atomic and irreducible lattice satisfying the covering law. complete: X ⊆ L = ⇒ X ∈ L atomic: for all p > 0 there exists an atom a such that a ≤ p

• rthomodular: there is an orthocomplementation (·)′ such that

1

(p′)′ = p

2

p ∧ p′ = 0 and p ∨ p′ = 1

3

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 that a ∧ p = 0, then (a ∨ p) ∧ p′ is an atom

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron lattices

A Piron lattice L = (L, ≤, (·)′, 0, 1) is a complete, orthomodular, atomic and irreducible lattice satisfying the covering law. complete: X ⊆ L = ⇒ X ∈ L atomic: for all p > 0 there exists an atom a such that a ≤ p

• rthomodular: there is an orthocomplementation (·)′ such that

1

(p′)′ = p

2

p ∧ p′ = 0 and p ∨ p′ = 1

3

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 that a ∧ p = 0, then (a ∨ p) ∧ p′ is an atom irreducible if a1 = a2, then there exists an a3 such that a1 = a3 = a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron lattices

A Piron lattice L = (L, ≤, (·)′, 0, 1) is a complete, orthomodular, atomic and irreducible lattice satisfying the covering law. complete: X ⊆ L = ⇒ X ∈ L atomic: for all p > 0 there exists an atom a such that a ≤ p

• rthomodular: there is an orthocomplementation (·)′ such that

1

(p′)′ = p

2

p ∧ p′ = 0 and p ∨ p′ = 1

3

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 that a ∧ p = 0, then (a ∨ p) ∧ p′ is an atom irreducible if a1 = a2, then there exists an a3 such that a1 = a3 = a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron Lattice morphism

Given two Piron lattices L1 = (L1, ≤1, (·)′) and L2 = (L2, ≤2, (·)∗), a PL-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).

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 8 / 19

Outline

1

Introduction

2

Piron lattices PL-morphisms

3

Dynamic Quantum Frames DQF-morphisms

4

Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 9 / 19

Dynamic quantum frame

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

P?

− → represent projections that satisfies the following properties:

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 10 / 19

Dynamic quantum frame

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

P?

− → represent projections →:=

L P?

− → is non-orthogonality that satisfies the following properties:

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 10 / 19

Dynamic quantum frame

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

P?

− → represent projections →:=

L P?

− → is non-orthogonality ∼P := {s ∈ Σ : s t for all t ∈ P} that satisfies the following properties:

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 10 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: if P ∈ L, then ∼P ∈ L

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: if P ∈ L, then ∼P ∈ L

atomicity: {s} ∈ L for all s ∈ Σ

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: if P ∈ L, then ∼P ∈ L

atomicity: {s} ∈ L for all s ∈ Σ adequacy: if s ∈ P, then s P? − → s

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: 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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: 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 that t

P?

− → w → s

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: 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 that t

P?

− → w → s covering law: if s P? − → t = v ∈ P, then there is a w ∈ P such that v → w s

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then X ∈ L

• rthocomplement: 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 that t

P?

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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

DQF-morphisms

Given two DQFs F1 = (Σ1, { P? − →}P∈L1) and F2 = (Σ2, { P? − →}P∈L2), a DQF-morphism is a function f : Σ1 → Σ2 such that f is a bounded morpism:

◮ if s → t, then f(s) → f(t) ◮ if f(s) → w, then there exists a t ∈ Σ1 such that s → t and f(t) = w. J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 12 / 19

Outline

1

Introduction

2

Piron lattices PL-morphisms

3

Dynamic Quantum Frames DQF-morphisms

4

Duality

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

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible symmetry of →

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible symmetry of → partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible symmetry of → partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible symmetry of → partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF → PL

Theorem

The set of testable properties L is a Piron lattice with ⊆ as partial order and ∼ as the orthocomplement. intersection

• rthocomplement

atomicity adequacy repeatability self-adjointness covering law proper superposition complete atomic

• rthomodular

covering law irreducible symmetry of → partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

PL → DQF

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

p?

− → b iff b = (a ∨ p′) ∧ p

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 15 / 19

PL → DQF

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

p?

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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 15 / 19

PL → DQF

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

p?

− → 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.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 15 / 19

DQF- → PL-Morphisms

PL-morphism

h( S) =

p∈S h(s) for all S ⊆ L2.

h(p′) = h(p)∗ for all atoms b ∈ L1 there exists an atom a ∈ L2 such that b ≤ h(a). Let f : Σ1 → Σ2 be a DQF-morphism.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

DQF- → PL-Morphisms

PL-morphism

h( S) =

p∈S h(s) for all S ⊆ L2.

h(p′) = h(p)∗ for all atoms b ∈ L1 there exists an atom a ∈ L2 such that b ≤ h(a). Let f : Σ1 → Σ2 be a DQF-morphism. f −1 preserves = , and ¬.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

DQF- → PL-Morphisms

PL-morphism

h( S) =

p∈S h(s) for all S ⊆ L2.

h(p′) = h(p)∗ for all atoms b ∈ L1 there exists an atom a ∈ L2 such that 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).

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

DQF- → PL-Morphisms

PL-morphism

h( S) =

p∈S h(s) for all S ⊆ L2.

h(p′) = h(p)∗ for all atoms b ∈ L1 there exists an atom a ∈ L2 such that 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 ∼.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

Preservation of ∼

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

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 17 / 19

Preservation of ∼

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

Theorem

f is a bounded morphism iff f −1 preserves .

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 17 / 19

Preservation of ∼

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

Theorem

f is a bounded morphism iff f −1 preserves . ∼A = ◦ ¬A

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 17 / 19

PL- → DQF-morphism

Let h : L1 → L2 be a PL-morphism. Define ℓ by ℓ(x) =

• x≤h(y)

y. ℓ is a left adjoin of h: ℓ(x) ≤ y ⇔ x ≤ h(y).

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

PL- → DQF-morphism

Let h : L1 → L2 be a PL-morphism. Define ℓ by ℓ(x) =

• x≤h(y)

y. ℓ is a left adjoin of h: ℓ(x) ≤ y ⇔ x ≤ h(y). ℓ sends atoms to atoms.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

PL- → DQF-morphism

Let h : L1 → L2 be a PL-morphism. Define ℓ by ℓ(x) =

• x≤h(y)

y. ℓ is a left adjoin of h: ℓ(x) ≤ y ⇔ x ≤ h(y). ℓ sends atoms to atoms. f = ℓ ↾ (atoms of L1)

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

PL- → DQF-morphism

Let h : L1 → L2 be a PL-morphism. Define ℓ by ℓ(x) =

• x≤h(y)

y. ℓ is a left adjoin of h: ℓ(x) ≤ y ⇔ x ≤ h(y). ℓ sends atoms to atoms. f = ℓ ↾ (atoms of L1) f −1 preserves , because = ∼ ◦ ¬.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

Thank you for your attention.

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