On a Connection between Piron Lattices and Kripke Frames Shengyang - - PowerPoint PPT Presentation

Introduction Quantum Kripke Frames Directions for Future Work

On a Connection between Piron Lattices and Kripke Frames

Shengyang Zhong

(zhongshengyang@163.com)

Institute for Logic, Language and Computation, University of Amsterdam

November 30th, 2013 Whither Quantum Structures in the XXIth Century? Brussels, Belgium

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Orthogonality Relation in Quantum Theory

Consider an isolated quantum system described by some Hilbert space H over complex numbers. Two non-zero vectors |ψ and |φ are said to be orthogonal, denoted as |ψ ⊥ |φ, if the inner product ψ|φ is 0. This binary relation on vectors induces a binary relation on

• ne-dimensional subspaces of H and thus on states of the

quantum system, which is also called orthogonality relation. By studying this relation, we get many representation theorems for lattices emerging from quantum logic via Kripke frames.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Ortholattices and Orthogonality Spaces [Goldblatt, 1974]

‘That the ⊥-closed subsets of an orthogonality space form an

• rtholattice under the partial ordering of set inclusion is a

result of long standing (cf. Birkhoff,[1]§V.7).’ ‘Every ortholattice is, within isomorphism, a subortholattice of the lattice of ⊥-closed subsets of some orthogonality space.’ Ortholattice: an orthocomplemented lattice Orthogonality space: a Kripke frame in which the binary relation is irreflexive and symmetric.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Property Lattices and State Spaces [Moore, 1995]

Property lattice: a complete atomistic orthocomplemented lattice. State space: a Kripke frame (Σ, ⊥) in which the binary relation ⊥ is irreflexive, symmetric and separated in the following sense: there is w ∈ Σ such that w ⊥ s and w ⊥ t, for any s, t ∈ Σ such that s = t. The main result in this paper is a duality between

a category with property lattices as objects, and a category with state spaces as objects.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

What about Piron Lattices?

Compared to lattices emerging from quantum theory, i.e. lattices of closed linear subspaces of Hilbert spaces, both

• rtholattices and property lattices are too general.

Piron lattice: an irreducible, complete, atomistic,

• rthocomplemented lattices satisfying weak modularity and

the Covering Law. They are also called irreducible propositional systems.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Piron Lattices and Hilbert Spaces

Piron’s Theorem (1964) The lattice of bi-orthogonally closed subspaces of a generalized Hilbert space is always a Piron lattice; and every Piron lattice of rank at least 4 is isomorphic to such a lattice. A Corollary of the Amemiya-Araki-Piron Theorem Generalized Hilbert spaces over the real numbers, the complex numbers and the quaternions are Hilbert spaces over these ∗-fields, in such a way that bi-orthogonally closed subspaces are exactly closed linear subspaces.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Piron Lattices and Quantum Dynamic Frames

In [Baltag and Smets, 2005], the authors give a representation theorem for Piron lattices (satisfying Mayet’s condition) using quantum dynamic frames. A quantum dynamic frame is a tuple (Σ, {P? →}P∈L), where Σ is a non-empty set, L is a subset of the power set of Σ and P? → is a binary relation on Σ for all P ∈ L. The orthogonality relation, denoted as ⊥, is defined as follows: s ⊥ t ⇐ ⇒ there is no P ∈ L such that s P? → t.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

The Work in this Talk

I will define a kind of Kripke frames, and use them to give a representation theorem for Piron lattices. This work inspired by Baltag and Smets’ work, provides an alternative way of defining quantum dynamic frame; continues the logical study of the orthogonality relation extending Moore’s result and thus Goldblatt’s result.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Some Terminologies of Kripke Frames

Kripke Frame A Kripke frame F is a tuple (Σ, →), where Σ is a non-empty set and → ⊆ Σ × Σ. Write s → t for (s, t) ∈ →. Given P ⊆ Σ, the orthocomplement of P (w.r.t. →) is defined as follows: ∼P def = {s ∈ Σ | s → t, for every t ∈ P} P is bi-orthogonally closed, if P = ∼∼P.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Kripke Frames

Definition (Quantum Kripke Frame) A quantum Kripke frame (QKF) F is a Kripke frame (Σ, →) satisfying the following conditions: (i) → is reflexive and symmetric. (ii) (Existence of Good Approximation) if s ∈ ∼P and ∼∼P = P, then there is t ∈ P such that s → u if and only if t → u for each u ∈ P; (iii) (Separation) if s = t, then there is w ∈ Σ such that w → s and w → t; (iv) (Superposition) for any s, t ∈ Σ, there is w ∈ Σ such that w → s and w → t.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Good Approximations are the Best

Consider s ∈ Σ and P ⊆ Σ such that good approximation of s in P exists according to condition (ii). There is t ∈ P such that s → u ⇔ t → u, for every u ∈ P. Condition (iii), i.e. Separation, guarantees that the t with this property is unique. This t will be called the best approxiamtion of s in P. Given a bi-orthogonally closed P ⊆ Σ, define a partial function P?(·) : Σ Σ as follows: P?(s) def = the best approxiamtion t of s in P, if s ∈ ∼P undefined,

• therwise

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Main Results

Theorem 1 For any quantum Kripke frame F = (Σ, →), (LF, ⊆, ∼(·)) is a Piron lattice, where LF = {P ⊆ Σ | ∼∼P = P} and ∼(·) is the

• rthocomplement operation (w.r.t. →).

Theorem 2 Every Piron lattice L is isomorphic to (LF, ⊆, ∼(·)) for some quantum Kripke frame F.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Kripke Frames and State Spaces

Quantum Kripke frames are a special kind of Moore’s state spaces, because conditions (i) and (iii) are equivalent to the conditions on state spaces, despite the fact that quantum Kripke frames take as primitive the non-orthogonality relation instead of the

• rthogonality relation.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Relations with Some More Structures

Given the close relations between quantum Kripke frames and Piron lattices, between Piron lattices and quantum dynamic frames, between Piron lattices and irreducible Hilbert geometries, we can conceive of using quantum Kripke frames: a representation theorem for quantum dynamic frames, a representation theorem for irreducible Hilbert geometries.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Kripke Frames and Quantum Dynamic Frames

Proposition Given a quantum Kripke frame F = (Σ, →), let LF denote {P ⊆ Σ | P = ∼∼P}, and for each P ∈ LF, define P? → ⊆ Σ × Σ such that: s P? → t ⇐ ⇒ s ∈ ∼P and t = P?(s). Then (Σ, {P? →}P∈LF) is a quantum dynamic frame. Proposition Every quantum dynamic frame is isomorphic to (Σ, {P? →}P∈LF) for some quantum Kripke frame F = (Σ, →).

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Hilbert Geometries

Hilbert Geometry A Hilbert geometry is a tuple (G, l, ⊥), where (G, l) is a projective geometry, i.e. G is a non-empty set and l ⊆ G × G × G such that: (G1) l(a, b, a); (G2) if l(a, p, q), l(b, p, q) and p = q, then l(a, b, p); (G3) if l(p, a, b) and l(p, c, d), then there exists q ∈ G such that l(q, a, c) and l(q, b, d); moreover, ⊥ ⊆ G × G satisfies the following: (O1) if a ⊥ b, then a = b; (O2) if a ⊥ b, then b ⊥ a; (O3) if a = b, a ⊥ p, b ⊥ p and l(c, a, b), then c ⊥ p; (O4) if a = b, then there is q ∈ G such that l(q, a, b) and q ⊥ a; (O5) if S ⊆ G is a subspace such that S⊥⊥ = S, then S ∨ S⊥ = G.

This is Definition 51 on page 499 of [Stubbe and van Steirteghem, 2007].

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Kripke Frames and Irreducible Hilbert Geometries

Proposition Let F = (Σ, →) be a quantum Kripke frame. Define a relation lF ⊆ Σ × Σ × Σ such that for any u, v, w ∈ Σ, (u, v, w) ∈ lF, if and only if one of the following holds: v = w; s → u implies that s → v or s → w, for every s ∈ Σ. Then (Σ, lF, →) is an irreducible Hilbert geometry. Proposition Every irreducible Hilbert geometry (G, l, ⊥) is isomorphic to (Σ, lF, →) for some quantum Kripke frame F = (Σ, →).

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Kripke Frames and Classical Frames

Definition (Classical Frame) A classical frame F is a Kripke frame (Σ, →) in which → is the identity relation, i.e. → = {(s, t) ∈ Σ × Σ | s = t}. Bi-orthogonally closed subsets of a classical frame form a Boolean lattice.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Kripke Frames and Classical Frames

Definition (Classical Frame) A classical frame F is a Kripke frame (Σ, →) in which → is the identity relation, i.e. → = {(s, t) ∈ Σ × Σ | s = t}. Bi-orthogonally closed subsets of a classical frame form a Boolean lattice. Proposition Let F = (Σ, →) be a Kripke frame satisfying conditions (i) to (iii) in the definition of quantum Kripke frames but not condition (iv), i.e. superposition. Then F is a quantum Kripke frame, iff superposition holds; F is a classical frame, iff → is transitive. Moreover, if Σ has at least 2 elements, then superposition and transitivity of → can not hold simultaneously.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Minimal Requirement of Probabilistic QKFs

A probabilistic quantum Kripke frame should consist of a quantum Kripke frame F = (Σ, →) and a function ρ : Σ × Σ → [0, 1]. Given such a pair and s ∈ Σ, define µs : LF → [0, 1] such that µs(P) = if s ∈ ∼P, ρ(s, P?(s))

• therwise.

Minimal Requirement of Probabilistic Quantum Kripke Frames For every s ∈ Σ, µs defined in the above way is a quantum probability measure on the Piron lattice (LF, ⊆, ∼(·)).

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Quantum Probability Measure

Quantum Probability Measure A quantum probability measure is a function p from a Piron lattice L = (L, ≤, (·)′) to [0, 1] such that: p(I) = 1;

• i∈A p(bi) exists and is equal to p(

i∈A bi),

for every {bi | i ∈ A} ⊆ L with A at most countable and bi ≤ b′

j when i = j.

p(b) = p(c) = 0 implies that p(b ∨ c) = 0, for every b, c ∈ L.

This definition is adapted from Definition (4.38) on page 82 of [Piron, 1976].

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

Probabilistic Quantum Kripke Frames

Probabilistic Quantum Kripke Frame A probabilistic quantum Kripke frame FP is a tuple (F, ρ), where F = (Σ, →) is a quantum Kripke frame and ρ is a function from Σ × Σ to [0, 1] satisfying the following:

1 ρ(s, t) = ρ(t, s); 2 ρ(s, t) = 0, if and only if (s, t) ∈ →; 3 if {ti | i ∈ I} ⊆ Σ satisfies that I is at most countable and

ti ⊥ tj whenever i = j, then

i∈I ρ(s, ti) ≤ 1;

and equality holds if and only if s ∈ ∼∼ {ti | i ∈ I};

4 if P ∈ LF, s ∈ ∼P and t ∈ P, then

ρ(s, t) = ρ(s, P?(s)) · ρ(P?(s), t). Proposition This definition satisfies the minimal requirement.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Outline

1

Introduction

2

Quantum Kripke Frames Definition and Main Result Relations with Other Structures Probabilistic Quantum Kripke Frames

3

Directions for Future Work

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Simplifying the Definition of Quantum Kripke Frames

Condition (ii), i.e. existence of good approximation, looks complicated, because it involves quantification over subsets of Σ. Theorem in [Goldblatt, 1984] There is no first-order formula ϕ in the language with one binary relation symbol such that, for any pre-Hilbert space P, the following are equivalent:

(P, ⊥) | = ϕ;

• rthomodularity holds in the lattice of orthoclosed subspaces of P.

It’s interesting to see whether condition (ii) can be simplified under some specific constraints, e.g. those on ‘dimension’.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Axiomatizing Quantum Kripke Frames

Kripke frames with various properties are often described by the modal propositional language with one unary modality . Try to find a proof system in this language, which is sound and complete w.r.t. the class of quantum Kripke frames. This logic will have classical negation ¬ and orthocomplement (quantum negation) ∼ can be defined as ¬. One of the challenges is that conditions (ii) and (iii) involve saying that a state can not access another state, which is a characteristic feature of undefinable properties of modal language.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

More Work on Probabilistic Quantum Kripke Frames

Characterize quantum Kripke frames that are induced by Hilbert spaces with some conditions involving probability. Capture the notions of quantum probability measure and mixed states in this framework.

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work Baltag, A. and Smets, S. (2005). Complete Axiomatizations for Quantum Actions. International Journal of Theoretical Physics, 44(12):2267–2282. Goldblatt, R. (1974). Semantics Analysis of Orthologic. Journal of Philosophical Logic, 3:19–35. Goldblatt, R. (1984). Orthomodularity Is Not Elementary. Journal of Symbolic Logic, 49:401–404. Mayet, R. (1998). Some Characterizations of the Underlying Division Ring of a Hilbert Lattice by Automorphisms . International Journal of Theoretical Physics, 37:109 – 114. Moore, D. (1995). Categories of Representations of Physical Systems. Helv Phys Acta, 68:658 – 678. Piron, C. (1976). Foundations of Quantum Physics. W.A. Benjamin Inc. Sol` er, M. (1995). Characterization of Hilbert Spaces with Orthomodularity Spaces . Communications in Algebra, 23:219 – 243. Stubbe, I. and van Steirteghem, B. (2007). Propositional Systems, Hilbert Lattices and Generalized Hilbert Spaces. In Engesser, K., Gabbay, D. M., and Lehmann, D., editors, Handbook of Quantum Logic and Quantum Structures: Quantum Structures, pages 477–523. Elsevier B.V. Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames

Introduction Quantum Kripke Frames Directions for Future Work

Thank you very much!

Shengyang Zhong (zhongshengyang@163.com) On a Connection between Piron Lattices and Kripke Frames