On a Connection between Piron Lattices and Kripke Frames · 2013. 12. 18. · Introduction Quantum...

IntroductionQuantum Kripke Frames

Directions for Future Work

On a Connection between Piron Lattices andKripke Frames

Shengyang Zhong([email protected])

Institute for Logic, Language and Computation, University of Amsterdam

November 30th, 2013

Whither Quantum Structures in the XXIth Century?Brussels, Belgium

Shengyang Zhong ([email protected]) On a Connection between Piron Lattices and Kripke Frames



Outline

1 Introduction

2 Quantum Kripke FramesDefinition and Main ResultRelations with Other StructuresProbabilistic Quantum Kripke Frames

3 Directions for Future Work




Orthogonality Relation in Quantum Theory

Consider an isolated quantum system described by someHilbert 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 onone-dimensional subspaces of H and thus on states of thequantum system, which is also called orthogonality relation.

By studying this relation, we get many representationtheorems for lattices emerging from quantum logic via Kripkeframes.




Ortholattices and Orthogonality Spaces [Goldblatt, 1974]

‘That the ⊥-closed subsets of an orthogonality space form anortholattice under the partial ordering of set inclusion is aresult of long standing (cf. Birkhoff,[1]§V.7).’‘Every ortholattice is, within isomorphism, a subortholattice ofthe lattice of ⊥-closed subsets of some orthogonality space.’Ortholattice: an orthocomplemented latticeOrthogonality space: a Kripke frame in which the binaryrelation is irreflexive and symmetric.




Property Lattices and State Spaces [Moore, 1995]

Property lattice: a complete atomistic orthocomplementedlattice.

State space: a Kripke frame (Σ,⊥) in which the binaryrelation ⊥ is irreflexive, symmetric and separated in thefollowing sense:

there is w ∈ Σ such that w ⊥ s and w 6⊥ t, for anys, t ∈ Σ such that s 6= t.

The main result in this paper is a duality between

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




What about Piron Lattices?

Compared to lattices emerging from quantum theory, i.e.lattices of closed linear subspaces of Hilbert spaces, bothortholattices and property lattices are too general.

Piron lattice: an irreducible, complete, atomistic,orthocomplemented lattices satisfying weak modularity andthe Covering Law.

They are also called irreducible propositional systems.




Piron Lattices and Hilbert Spaces

Piron’s Theorem (1964)

The lattice of bi-orthogonally closed subspaces of a generalizedHilbert space is always a Piron lattice; and every Piron lattice ofrank 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 complexnumbers and the quaternions are Hilbert spaces over these ∗-fields,in such a way that bi-orthogonally closed subspaces are exactlyclosed linear subspaces.




Piron Lattices and Quantum Dynamic Frames

In [Baltag and Smets, 2005], the authors give a representationtheorem for Piron lattices (satisfying Mayet’s condition) usingquantum dynamic frames.

A quantum dynamic frame is a tuple (Σ, {P?→}P∈L), where Σis 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.




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 alternativeway of defining quantum dynamic frame;

continues the logical study of the orthogonality relationextending Moore’s result and thus Goldblatt’s result.




Definition and Main ResultRelations with Other StructuresProbabilistic Quantum Kripke Frames

Outline

1 Introduction







Some Terminologies of Kripke Frames

Kripke Frame

A Kripke frame F is a tuple (Σ,→), where Σ is a non-empty setand → ⊆ Σ× Σ.

Write s 6→ t for (s, t) 6∈ →.Given P ⊆ Σ, the orthocomplement of P (w.r.t. →) is definedas follows:

∼P def= {s ∈ Σ | s 6→ t, for every t ∈ P}

P is bi-orthogonally closed, if P = ∼∼P.





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 6∈ ∼P and ∼∼P = P, then there is t ∈ P such thats → u if and only if t → u for each u ∈ P;

(iii) (Separation) if s 6= t, then there is w ∈ Σ such that w → sand w 6→ t;

(iv) (Superposition) for any s, t ∈ Σ, there is w ∈ Σ such thatw → s and w → t.





Good Approximations are the Best

Consider s ∈ Σ and P ⊆ Σ such that good approximation of sin 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 thisproperty is unique.

This t will be called the best approxiamtion of s in P.

Given a bi-orthogonally closed P ⊆ Σ, define a partialfunction P?(·) : Σ 99K Σ as follows:

P?(s)def=

{the best approxiamtion t of s in P, if s 6∈ ∼Pundefined, otherwise





Main Results

Theorem 1

For any quantum Kripke frame F = (Σ,→), (LF,⊆,∼(·)) is aPiron lattice, where LF = {P ⊆ Σ | ∼∼P = P} and ∼(·) is theorthocomplement operation (w.r.t. →).

Theorem 2

Every Piron lattice L is isomorphic to (LF,⊆,∼(·)) for somequantum Kripke frame F.





Outline

1 Introduction







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 onstate spaces, despite the fact that quantum Kripke frames take asprimitive the non-orthogonality relation instead of theorthogonality relation.





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.





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× Σsuch that:

sP?→ t ⇐⇒ s 6∈ ∼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) forsome quantum Kripke frame F = (Σ,→).





Hilbert Geometries

Hilbert Geometry

A Hilbert geometry is a tuple (G , l ,⊥), where (G , l) is a projectivegeometry, 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 6= 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 6= b;(O2) if a ⊥ b, then b ⊥ a;(O3) if a 6= b, a ⊥ p, b ⊥ p and l(c , a, b), then c ⊥ p;(O4) if a 6= 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].





Quantum Kripke Frames and Irreducible Hilbert Geometries

Proposition

Let F = (Σ,→) be a quantum Kripke frame. Define a relationlF ⊆ Σ× Σ× Σ such that for any u, v ,w ∈ Σ, (u, v ,w) ∈ lF, ifand 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, 6→) is an irreducible Hilbert geometry.

Proposition

Every irreducible Hilbert geometry (G , l ,⊥) is isomorphic to(Σ, lF, 6→) for some quantum Kripke frame F = (Σ,→).





Quantum Kripke Frames and Classical Frames

Definition (Classical Frame)

A classical frame F is a Kripke frame (Σ,→) in which → is theidentity relation, i.e. → = {(s, t) ∈ Σ× Σ | s = t}.

Bi-orthogonally closed subsets of a classical frame form a Booleanlattice.

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 andtransitivity of → can not hold simultaneously.





Outline

1 Introduction







Minimal Requirement of Probabilistic QKFs

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

µs(P) =

{0 if s ∈ ∼P,ρ(s,P?(s)) otherwise.

Minimal Requirement of Probabilistic Quantum Kripke Frames

For every s ∈ Σ, µs defined in the above way is a quantumprobability measure on the Piron lattice (LF,⊆,∼(·)).





Quantum Probability Measure

Quantum Probability Measure

A quantum probability measure is a function p from a Piron latticeL = (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 andbi ≤ b′j when i 6= 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].





Probabilistic Quantum Kripke Frames

Probabilistic Quantum Kripke Frame

A probabilistic quantum Kripke frame FP is a tuple (F, ρ), whereF = (Σ,→) is a quantum Kripke frame and ρ is a function fromΣ× 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) 6∈ →;3 if {ti | i ∈ I} ⊆ Σ satisfies that I is at most countable and

ti ⊥ tj whenever i 6= j , then∑

i∈I ρ(s, ti ) ≤ 1;and equality holds if and only if s ∈ ∼∼

⋃{ti | i ∈ I};

4 if P ∈ LF, s 6∈ ∼P and t ∈ P, thenρ(s, t) = ρ(s,P?(s)) · ρ(P?(s), t).

Proposition

This definition satisfies the minimal requirement.




Outline

1 Introduction






Simplifying the Definition of Quantum Kripke Frames

Condition (ii), i.e. existence of good approximation, lookscomplicated, because it involves quantification over subsets ofΣ.

Theorem in [Goldblatt, 1984]

There is no first-order formula ϕ in the language with one binaryrelation symbol such that, for any pre-Hilbert space P, thefollowing are equivalent:

(P,⊥) |= ϕ;

orthomodularity holds in the lattice of orthoclosed subspaces of P.

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




Axiomatizing Quantum Kripke Frames

Kripke frames with various properties are often described bythe modal propositional language with one unary modality �.

Try to find a proof system in this language, which is soundand 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) involvesaying that a state can not access another state, which is acharacteristic feature of undefinable properties of modallanguage.




More Work on Probabilistic Quantum Kripke Frames

Characterize quantum Kripke frames that are induced byHilbert spaces with some conditions involving probability.

Capture the notions of quantum probability measure andmixed states in this framework.




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.

Solèr, 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 QuantumStructures: Quantum Structures, pages 477–523. Elsevier B.V.




Thank you very much!


IntroductionQuantum Kripke FramesDefinition and Main ResultRelations with Other StructuresProbabilistic Quantum Kripke Frames


On a Connection between Piron Lattices and Kripke Frames · 2013. 12. 18. · Introduction Quantum...

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