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 s Q ? P ? u t Dynamic quantum frames 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. s Q ? P ? u t Dynamic quantum frames 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. s Q ? P ? u t Dynamic quantum frames ◮ 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 Dynamic Quantum Frame Piron Lattice Σ 1 L 1 f h Σ 2 L 2 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 F Dynamic Quantum Frame Piron Lattice Σ 1 L 1 f h Σ 2 L 2 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 ≃ Id DQF F ◦ G ≃ Id PL F Dynamic Quantum Frame Piron Lattice Σ 1 L 1 f h Σ 2 L 2 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 ≃ Id DQF F ◦ G ≃ Id PL F Dynamic Quantum Frame Piron Lattice Σ 1 P Σ 1 L 1 ⇐ ⇒ ⇐ ⇒ f − 1 f h Σ 2 P Σ 2 L 2 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 of 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 Introduction 1 Piron lattices 2 PL-morphisms Dynamic Quantum Frames 3 DQF-morphisms Duality 4 J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 5 / 19
Outline Introduction 1 Piron lattices 2 PL-morphisms Dynamic Quantum Frames 3 DQF-morphisms Duality 4 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. ⇒ � X ∈ L complete: 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. ⇒ � X ∈ L complete: 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. ⇒ � X ∈ L complete: X ⊆ L = atomic: for all p > 0 there exists an atom a such that a ≤ p orthomodular: there is an orthocomplementation ( · ) ′ such that ( p ′ ) ′ = p 1 p ∧ p ′ = 0 and p ∨ p ′ = 1 2 ⇒ q ′ ≤ p ′ p ≤ q = 3 ⇒ q ∧ ( q ′ ∨ p ) = p weakly modular: p ≤ q = 4 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. ⇒ � X ∈ L complete: X ⊆ L = atomic: for all p > 0 there exists an atom a such that a ≤ p orthomodular: there is an orthocomplementation ( · ) ′ such that ( p ′ ) ′ = p 1 p ∧ p ′ = 0 and p ∨ p ′ = 1 2 ⇒ q ′ ≤ p ′ p ≤ q = 3 ⇒ q ∧ ( q ′ ∨ p ) = p weakly modular: p ≤ q = 4 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. ⇒ � X ∈ L complete: X ⊆ L = atomic: for all p > 0 there exists an atom a such that a ≤ p orthomodular: there is an orthocomplementation ( · ) ′ such that ( p ′ ) ′ = p 1 p ∧ p ′ = 0 and p ∨ p ′ = 1 2 ⇒ q ′ ≤ p ′ p ≤ q = 3 ⇒ q ∧ ( q ′ ∨ p ) = p weakly modular: p ≤ q = 4 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 a 1 � = a 2 , then there exists an a 3 such that a 1 � = a 3 � = a 2 and a 1 ∨ a 2 = a 1 ∨ a 3 = a 2 ∨ a 3 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. ⇒ � X ∈ L complete: X ⊆ L = atomic: for all p > 0 there exists an atom a such that a ≤ p orthomodular: there is an orthocomplementation ( · ) ′ such that ( p ′ ) ′ = p 1 p ∧ p ′ = 0 and p ∨ p ′ = 1 2 ⇒ q ′ ≤ p ′ p ≤ q = 3 ⇒ q ∧ ( q ′ ∨ p ) = p weakly modular: p ≤ q = 4 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 a 1 � = a 2 , then there exists an a 3 such that a 1 � = a 3 � = a 2 and a 1 ∨ a 2 = a 1 ∨ a 3 = a 2 ∨ a 3 J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19
Piron Lattice morphism Given two Piron lattices L 1 = ( L 1 , ≤ 1 , ( · ) ′ ) and L 2 = ( L 2 , ≤ 2 , ( · ) ∗ ) , a PL-morphism is a function h : L 1 → L 2 such that h ( � S ) = � p ∈ S h ( s ) for all S ⊆ L 1 . h ( p ′ ) = h ( p ) ∗ for all atoms b ∈ L 2 there exists an atom a ∈ L 1 such that b ≤ h ( a ) . J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 8 / 19
Outline Introduction 1 Piron lattices 2 PL-morphisms Dynamic Quantum Frames 3 DQF-morphisms Duality 4 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 P ? → := � − → is non-orthogonality L 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 P ? → := � − → is non-orthogonality L ∼ 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 orthocomplement: if P ∈ L , then ∼ P ∈ L J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19
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