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Contexts, Bi-Heyting Algebras and aNew Logic for Quantum Systems

Oberseminar Theoretische InformatikFAU Erlangen

4. November 2014

Andreas Doring

Theoretische Physik I, FAU Erlangen

[email protected]

Andreas Doring (Erlangen) Bi-Heyting algebras and quantum systems 1 / 49

“Never express yourself more clearly than you are able to think.”

Niels Bohr


References

This talk is based on

AD,“Topos-based Logic for Quantum Systems and Bi-HeytingAlgebras”, to appear in Logic & Algebra in Quantum Computing,Lecture Notes in Logic, Association for Symbolic Logic in conjunctionwith Cambridge University Press (2012); [arXiv:1202.2750]

Some good references on standard quantum logic are:

G. Birkhoff, J. von Neumann, “The Logic of Quantum Mechanics”,Annals of Mathematics 37, No. 4, 823–843 (1936).

Varadarajan, Geometry of Quantum Theory, second ed., Springer(1985).

M. Dalla Chiara, R. Giuntini, “Quantum Logics”, in Handbook ofPhilosophical Logic, Kluwer (2002); [arXiv:quant-ph/0101028v2]


Standard quantum logic





Some basics of standard quantum logic:

Birkhoff and von Neumann suggested to use the completeorthomodular lattice P(H) of projections on a separable, complexHilbert space H as representatives of propositions. (Fine point:modular vs. orthomodular lattices.)

The link between (pre-mathematical) propositions “A ε∆” andprojections is provided by the spectral theorem.

Pure quantum states |ψ〉 (roughly) are models of this propositionallogic, but only specific projections are assigned true or false: ifP |ψ〉 = |ψ〉, then the proposition represented by P is true in thestate |ψ〉; if P |ψ〉 = 0, then the proposition is false in the state |ψ〉.In general, only a probability between 0 and 1 for finding the outcomeof a measurement of A to lie in a Borel set ∆ can be given. Aftermeasurement, the quantum state has changed to the eigenstatecorresponding to the outcome.



Some (problematic) features of quantum logic

Propositions form a non-distributive orthomodular lattice.

Quantum states do not assign true or false to all propositions“A ε∆”; in general, one can only obtain probabilities. Lack oftwo-valued models; instrumentalism.

A disjunction “A ε∆ or B ε Γ” can be true in a state |ψ〉 despite thefact that neither “A ε∆” nor “B ε Γ” are true in the state. This isdue to superposition.

There a many meets in P(H) that are physically meaningless. This isdue to the failure to take contextuality into account.

The implication problem: there is no material implication, and theimplicative rule does not hold. This is due to non-distributivity.

There is no algebra of subsets representing propositions, but an algebra ofsubspaces. Most problematic features result from this (insufficient?)geometric underpinning of quantum logic.


A state space model for quantum systems

A state space model forquantum systems



The topos approach

The topos approach to the formulation of physical theories

was initiated by Chris Isham ’97 and Isham/Butterfield ’98–’02,

addresses certain structural and conceptual issues in the foundationsof physics,

uses topos theory to give new geometric and logical ideas for physics,

aims to provide a new way of formulating physical theories in general,and quantum theory in particular – not based on Hilbert spaces, not‘just another interpretation’,

is motivated by questions on the way to quantum gravity (QG) andquantum cosmology (QC),

most work so far is on standard, non-relativistic quantum theory –natural starting point, testing ground.

Other researchers include: Landsman, Heunen, Spitters, Nakayama,Vickers, Fauser, Flori, ...



Where are we now?

The topos approach is very much work in progress, so think of



Classical physics

In classical physics, a physical quantity A is described by a (measurable)real-valued function fA on the state space (phase space) P:

In a given state s ∈ P, a physical quantity A has the value fA(s) ∈ R.



Classical physics (2)

Propositions such as “A ε∆”, i.e., “the physical quantity A has a value inthe Borel set ∆ ⊆ R”, are represented by subsets of the state space:

The Borel subsets of P form a σ-complete Boolean algebra. Pure statess ∈ P are models of this propositional logic.



The Kochen-Specker theorem and instrumentalism

But can quantum theory be cast in such a form? Obstacle:

Theorem

(Kochen-Specker 1967): If dim(H) ≥ 3, then there is no state spacemodel of quantum theory (under mild and natural conditions). In logicalterms, there is no way to assign true or false to all propositions “A ε∆” atthe same time.

Usual answer: interpret propositions in an instrumentalist way, and useprobabilities, because repeatedly measuring the same physical quantity Ain the same state |ψ〉 can give different outcomes – states do not assigntruth values to all propositions.

This is not what we want to do – we want to provide a quantum statespace in spite of the KS theorem.



C ∗-algebras and von Neumann algebras

As a reminder:

Definition

A C ∗-algebra A is a complex Banach algebra with an involution(−)∗ : A → A such that

∀a ∈ A : |a∗a| = |a|2.

A W ∗-algebra is a C ∗-algebra that is the dual of a Banach space M.

Every C ∗-algebra can be faithfully represented as a norm-closed subalgebraof B(H), the algebra of bounded linear operators on some Hilbert spaceH. Every W ∗-algebra can be represented as a weakly closed subalgebra ofsome B(H). Weakly closed subalgebras of B(H) are called von Neumannalgebras.

We will often use ‘hats’ (as in A, B, ...) when denoting elements ofoperator algebras.



The basic idea

With a quantum system (described by a noncommutative von Neumannalgebra N , e.g. N = B(H)), we associate a topos and define a spectralobject Σ in the topos.

The spectral presheaf Σ is a generalised set, playing the role of a statespace for the quantum system (notwithstanding the Kochen-Speckertheorem!). Structure of state space determines logical structure of atheory:

Classical physics: Borel subsets of state space S representpropositions “A ε∆”, form Boolean algebra.

Quantum physics: clopen subobjects of the state space Σ representpropositions “A ε∆”, form bi-Heyting algebra.



The context category

Let S be a quantum system, described by a noncommutative vonNeumann algebra N ⊆ B(H).

Let V be a non-trivial commutative von Neumann subalgebra of N whichhas the same unit element as N . This gives and is given by a set ofcommuting self-adjoint operators in N . We call V a context.

Each context V is a partial classical perspective on the quantum system.The main idea: take all classical perspectives/contexts together to obtaina complete picture of the quantum system.

Concretely, we consider the set V(N ) of all contexts, partially orderedunder inclusion. This poset is called the context category.



Gelfand spectra

Each commutative von Neumann algebra V has a Gelfand spectrum ΣV ,consisting of the algebra morphisms

λ : V −→ C.

These maps are also pure states of V . The Gelfand topology is the(relative) weak∗-topology on ΣV . With respect to this topology, ΣV is anextremely disconnected compact Hausdorff space.

For a commutative von Neumann algebra V , the projection lattice P(V ) isa complete Boolean algebra. Let λ ∈ ΣV , and let P ∈ P(V ), representingsome proposition about an A ∈ Vsa. Note that

λ(P) = λ(P2) = λ(P)2 ∈ 0, 1 ' false, true,

so the elements of ΣV are the models of the classical logic described byP(V ).



The spectral presheaf

If V ′ ⊂ V is a von Neumann subalgebra, then there is a canonicalrestriction map

rV ,V ′ : ΣV −→ ΣV ′

λ 7−→ λ|V ′ .

This map is continuous, closed, open and surjective.

Definition

Let N be a von Neumann algebra, and let V(N ) be its context category.The spectral presheaf Σ of N over V(N ) is defined

(a) on objects: for all V ∈ V(N ), let ΣV := ΣV ,

(b) on arrows: for all inclusions iV ′V : V ′ → V , let Σ(iV ′V ) = rV ,V ′ .


The bi-Heyting algebra of clopen subobjects

The bi-Heyting algebra ofclopen subobjects



Bi-Heyting algebras – history

Rauszer: bi-Heyting algebras in superintuitionistic logic (’73–’77)

Lawvere: co-Heyting and bi-Heyting algebras in category and topostheory, in particular in connection with continuum physics (’86, ’91)

Reyes/Makkai (’95) and Reyes/Zolfaghari (’96): bi-Heyting algebrasand modal logic

Bezhanishvili et al. (’10): new duality theorems for bi-Heytingalgebras based on bitopological spaces

Majid (’95, ’08): Heyting and co-Heyting algebras within a tentativerepresentation-theoretic approach to the formulation of quantumgravity

As far as I am are aware, nobody has connected quantum systems andtheir logic with bi-Heyting algebras before.



Definition

A bi-Heyting algebra K is a lattice which is a Heyting algebra and aco-Heyting algebra. For each A ∈ K , the functor A ∧ : K → K has aright adjoint A⇒ : K → K , and the functor A ∨ : K → K has a leftadjoint A⇐ : K → K . We write ¬ for the Heyting negation and ∼ forthe co-Heyting negation.

A bi-Heyting algebra K is called complete if it is complete as a Heytingalgebra and complete as a co-Heyting algebra.

Canonical example: Boolean algebra B (where ¬ = ∼).



Projections and clopen subsets

Let V be a commutative von Neumann algebra, and let Cl(ΣV ) denotethe clopen subsets of the Gelfand spectrum of V . There is an isomorphismof complete Boolean algebras

αV : P(V ) −→ Cl(ΣV )

P 7−→ λ ∈ ΣV | λ(P) = 1.

Hence, within each context, i.e., each commutative subalgebra V ⊂ N , wecan freely switch between clopen subsets of ΣV and projections in V .

We will write SP := αV (P) and PS := α−1V (S). Note that for acommutative von Neumann algebra V , the Gelfand spectrum ΣV ishomeomorphic to the Stone space of the complete Boolean algebra P(V ).



Clopen subobjects of the spectral presheaf

A subobject of the spectral presheaf Σ is simply a subfunctor S . It isdetermined by specifying a subset SV of ΣV for each context V such that

∀V ,V ′ ∈ V(N ) : V ′ ⊂ V implies Σ(iV ′V )(SV ) ⊆ SV ′ .

A subobject S of Σ is called clopen if SV ⊆ ΣV is a clopen subset for allV ∈ V(N ).

Equivalently, we can consider the family (PSV)V∈V(N ) of corresponding

projections. The subobject condition becomes

∀V ,V ′ ∈ V(N ) : V ′ ⊂ V implies PSV≤ PSV ′ .

The set of clopen subobjects of Σ is denoted by Subcl(Σ).



Contextuality and coarse-graining

The concept of contextuality is implemented by this construction: Σ is apresheaf over the context category V(N ).

Within each context, we have classical Boolean logic; local propositionsabout the value of some physical quantity in V are represented byelements of the complete Boolean algebra P(V ).

Moreover, the concept of coarse-graining is implemented by the fact thatwe use subobjects of Σ: if V ′ ⊂ V , then PSV ′ ≥ PSV

(since S is asubobject), so SV ′ represents a local proposition at the smaller contextV ′ ⊂ V that is coarser than (i.e., it is weaker than, a consequence of) thelocal proposition represented by SV .

Clopen subobjects S ∈ Subcl(Σ) hence are interpreted as contextualisedfamilies of local propositions, compatible w.r.t. coarse-graining.



Subcl(Σ) as a lattice

It is obvious that Subcl(Σ) is a partially ordered set if we set

∀S ,T ∈ Subcl(Σ) : S ≤ T iff (∀V ∈ V(N ) : SV ⊆ TV ).

Meets and joins with respect to this order are defined as follows: for allfamilies (S i )i∈I ⊆ Subcl(Σ) and all V ∈ V(N ),

(∧i∈I

S i )V := int⋂i∈I

S i ;V ,

(∨i∈I

S i )V := cl⋃i∈I

S i ;V .

Since the lattice operations are defined locally, i.e., at each stageV ∈ V(N ) separately, we obtain a distributive lattice by the fact that forall V ∈ V(N ),

Cl(ΣV ) ' P(V )

is distributive.Andreas Doring (Erlangen) Bi-Heyting algebras and quantum systems 24 / 49


Subcl(Σ) as a complete Heyting algebra

In fact, we can say more: each Cl(ΣV ) is a complete Boolean algebra, sofor each S ∈ Subcl(Σ) the functor

∧ S : Subcl(Σ) −→ Subcl(Σ)

R 7−→ R ∧ S

preserves all joins (note that meets and joins are defined stagewise) andhence has a right adjoint

S ⇒ : Subcl(Σ) −→ Subcl(Σ).

This map, the Heyting implication from S , makes Subcl(Σ) into acomplete Heyting algebra. The Heyting implication is given by theadjunction

R ∧ S ≤ T if and only if R ≤ (S ⇒ T ).

Note that this is the implicative rule, and ⇒ is a material implication.Andreas Doring (Erlangen) Bi-Heyting algebras and quantum systems 25 / 49


Subcl(Σ) as a complete co-Heyting algebra

There is more structure: again since each Cl(ΣV ) is a complete Booleanalgebra, for each S ∈ Subcl(Σ) the functor

S ∨ : Subcl(Σ) −→ Subcl(Σ)

preserves all meets, so it has a left adjoint

S ⇐ : Subcl(Σ) −→ Subcl(Σ)

which we call co-Heyting implication from S . This map makes Subcl(Σ)into a complete co-Heyting algebra. It is characterised by the adjunction

(S ⇐ T ) ≤ R iff S ≤ T ∨ R,

which means that

(S ⇐ T ) =∧R ∈ Subcl(Σ) | S ≤ T ∨ R.



Subcl(Σ) as a complete bi-Heyting algebra

The Heyting implication ⇒ induces the Heyting negation

¬ : Subcl(Σ) −→ Subcl(Σ)op

S 7−→ ¬S := (S ⇒ ∅) =∨T ∈ Subcl(Σ) | S ∧ T = ∅.

Analogously, the co-Heyting implication ⇐ induces the co-Heytingnegation

∼: Subcl(Σ) −→ Subcl(Σ)op

S 7−→∼ S := (Σ⇐ S) =∧T ∈ Subcl(Σ) | S ∨ T = Σ.

Summing up, we have shown

Proposition

(Subcl(Σ),∧,∨, ∅,Σ,⇒,¬,⇐,∼) is a complete bi-Heyting algebra.

It is easy to see that Subcl(Σ) is not a Boolean algebra.Andreas Doring (Erlangen) Bi-Heyting algebras and quantum systems 27 / 49


Two small results on negation

Lemma

For all S ∈ Subcl(Σ), we have ¬S ≤ ∼ S.

Proof.

For all V ∈ V(N ), it holds that (¬S)V ⊆ ΣV \SV , since(¬S ∧ S)V = (¬S)V ∩ SV = ∅, while (∼ S)V ⊇ ΣV \SV since(∼ S ∨ S)V = (∼ S)V ∪ SV = ΣV .

The above lemma and the fact that ¬S is the largest subobject such that¬S ∧ S = ∅ imply

Corollary

In general, ∼ S ∧ S ≥ ∅.

This means that the co-Heyting negation does not give a system in whicha central axiom of most logical systems, viz. freedom from contradiction,holds. We have a paraconsistent logic for quantum systems.


Daseinisation

Daseinisation


Daseinisation

Daseinisation

Standard quantum theory: proposition “A ε∆” represented by projectionP = E [A ε∆].

There is a way of ‘translating’ from standard quantum logic to the newlogic based on clopen subobjects of the quantum state space Σ: firstconsider a single commutative subalgebra V ⊂ N . There is an inclusion

P(V ) → P(N )

that is a morphism of complete orthomodular lattices, so it preserves allmeets in particular. Hence, it has a left adjoint

δoN ,V : P(N ) −→ P(V )

P 7−→ δoN ,V (P) =∧Q ∈ P(V ) | Q ≥ P.

Note that commutativity of V plays no role here.


Daseinisation

Daseinisation (2)

Then consider this map for all contexts V ∈ V(N ) at once:

Definition

Let N be a von Neumann algebra, and let P(N ) be its lattice ofprojections. The map

δo : P(N ) −→ Subcl(Σ)

P 7−→ δo(P) := (αV (δoN ,V (P)))V∈V(N )

is called outer daseinisation of projections.


Daseinisation

Daseinisation (3)

Some of the main properties of daseinisation are:

δo : P(N )→ Subcl(Σ) is monotone.

δo(0) = ∅ and δo(1) = Σ.

δo is injective, but not surjective.

δo preserves all joins (disjunctions). This means that the part ofstandard quantum logic that comes from superposition is preserved,despite the fact that Σ is not a vector space.

For meets (conjunctions), we have

∀P, Q ∈ P(N ) : δo(P ∧ Q) ≤ δo(P) ∧ δo(Q).

All conjunctions exist, but coarse-graining and contextualityguarantee that they all have a good physical interpretation.


Negations and regular elements

Negations andregular elements



The Heyting negation explicitly

Recall that ¬S is the largest element of Subcl(Σ) such that

S ∧ ¬S = 0.

The stagewise expression for ¬S is

(¬S)V = λ ∈ ΣV | ∀V ′ ⊆ V : λ|V ′ /∈ SV ′.

One can show:

Proposition

Let S ∈ Subcl(Σ), and let V ∈ V(N ). Then

P(¬S)V = 1−∨

V ′∈mV

PSV ′ ,

where mV = V ′ ⊆ V | V ′ minimal.



Heyting-regular elements

With this, it is easy to check that double negation is given (stagewise) by

P(¬¬S)V =∧

V ′∈mV

PSV ′ ≥ PSV,

so ¬¬S ≥ S as expected. We obtain:

Proposition

An element S of Subcl(Σ) is Heyting-regular, i.e., ¬¬S = S, if and onlyif for all V ∈ V(N ), it holds that

PSV=

∧V ′∈mV

PSV ′ ,

where mV = V ′ ⊆ V | V ′ minimal.



The co-Heyting negation explicitly

Dually, ∼ S is the smallest element of Subcl(Σ) such that

S∨ ∼ S = Σ.

One can show:

Proposition

Let S ∈ Subcl(Σ), and let V ∈ V(N ). Then

P(∼S)V =∨

V∈MV

(δoV ,V

(1− PS V)),

where MV = V ⊇ V | V maximal.



Co-Heyting regular elements

One checks that double co-Heyting negation is given (stagewise) by

P(∼∼S)V =∨

V∈MV

δoV ,V

(PS V),

which implies ∼∼ S ≤ S . We obtain:

Proposition

An element S of Subcl(Σ) is co-Heyting-regular, i.e., ∼∼ S = S, if andonly if for all V ∈ V(N ) it holds that

PSV=

∨V∈MV

δoV ,V

(PS V),

where MV = V ⊇ V | V maximal.



Daseinisation and regularity

Definition

A clopen subobject S ∈ Subcl(Σ) is called tight if

Σ(iV ′V )(SV ) = SV ′

for all V ′,V ∈ V(N ) such that V ′ ⊆ V .

Proposition

Tight subobjects are both Heyting-regular and co-Heyting regular.

Theorem

A subobject δo(P) obtained from daseinisation is tight and hence bothregular and co-Heyting regular.


States and truth values




Pure states

In classical physics, a (pure) state is given by an element s of the statespace S. A proposition “A ε∆” is represented by a Borel subsetf −1A (∆) ⊂ S.

The truth value of the proposition in the given state is

v(B; s) = (s ∈ B),

which is a Boolean formula that is either false or true.

For quantum theory, we need an analogue of the pure state s ∈ S. But:

Theorem

(Isham, Butterfield ’00, D ’05): The spectral presheaf Σ of a vonNeumann algebra N has no global elements if N has no summand of typeI2. This is equivalent to the Kochen-Specker theorem.



Pseudo-states

In standard quantum theory (for N = B(H)), one uses vector states: letψ ∈ H be a unit vector, then

wψ : B(H) −→ C

A 7−→ 〈ψ, A(ψ)〉.

We simply ‘daseinise’ such a vector state: let Pψ the projection onto theline Cψ, then the pseudo-state wψ is given as

wψ := δo(Pψ).

This is a ‘small subobject’, (one of) the smallest non-empty subobjectsone can obtain from daseinisation.



Truth values

Let wψ be a pseudo-state, and let S ∈ Subcl(Σ) be a proposition (e.g.,S = δo(E [A ε∆])). We can interpret the expression

v(S ;wψ) := (wψ ∈ S)

in the Mitchell-Benabou language of the topos SetV(N )op , which gives atruth value in the multi-valued, intuitionistic logic of the topos.

Concretely, such a truth value is a global element of the subobjectclassifier of the topos, which is the presheaf of sieves Ω.

Since the base category V(N ) of our topos is a poset, this becomesparticularly simple: the global elements of Ω correspond bijectively to thelower sets in V(N ),

Γ(Ω) = L(V(N )).


Generalisation to OMLs

Spectral presheaves of OMLs

What about generalisations to orthomodular lattices (OMLs)?

Given an OML L, one can consider the poset B(L) of its Booleansubalgebras. Each Boolean subalgebra B has a Stone space Ω(B), so onecan define a spectral presheaf Ω(L) of L.

Moreover, there is a suitable category in which these spectral presheavesare objects. Morphisms φ : L1 → L2 of OMLs induce morphismsΦ : Ω(L2)→ Ω(L1).

Theorem

(Sarah Cannon, AD ’13; unpublished) Let L1, L2 be two OMLs. There isan isomorphism Φ : Ω(L2)→ Ω(L1) if and only if there is an isomorphismφ : L1 → L2.

Hence, the spectral presheaf of an OML L is a complete invariant. Theclopen subobjects form a bi-Heyting algebra Subcl(Ω(L)).


Summary

Summary


Summary

Some features of topos-based logic for quantum systems

Propositions form a distributive, complete bi-Heyting algebraSubcl(Σ).

Each pseudo-state wψ assigns a truth-value to every proposition“A ε∆”, represented by δo(E [A ε∆]). Multi-valued, contextual logic.

‘Translation’ map δo : P(N )→ Subcl(Σ) preserves all joins –‘superposition without linearity’.

All meets in Subcl(Σ) can be interpreted due to coarse-graining andcontextuality.

There is a material implication, the Heyting implication, and theimplicative rule holds.

Additional paraconsistent logic.

All this relates to the fact that this new form of logic for quantum systemshas a geometric underpinning in the form of generalised sets (i.e., objectsin a topos).


Summary

Two ways of generalising classical logic

Birkhoff and von Neumann explicitly emphasised in their 1936 article thatthey find it most plausible to give up distributivity, but to keep negationintact when generalising from classical Boolean logic.

This can be seen as a claim that intuitionistic logic, which was muchdebated at the time, is not appropriate for quantum systems.

In our new form of topos-based logic for quantum systems, we depart fromclassical logic in a different way, by keeping distributivity, but ‘splitting’negation into two concepts.

This leads to a much better-behaved logic for quantum systems.Bi-Heyting algebras are a comparatively mild generalisation of Booleanalgebras.

Moreover, many other features of the spectral presheaf Σ have goodphysical meaning in the topos approach to quantum theory.


Summary

Some open questions

Dynamics: Time evolution? (See [arXiv:1212.4882].) State change bymeasurements?

(More) physical interpretation of intuitionistic and paraconsistentparts of this logic?

Universal property of outer daseinisation δo?

Higher-order aspects, making more use of the internal logic of thetopos?

Behaviour of logic under morphisms between topoi? → current workwith Rui Soares Barbosa, Jonathon Funk, Pedro Resende

Treatment of composite systems?

...


Summary

References

Selection of references on the new topos-based form of quantum logic:

AD, C.J. Isham, “Topos Theory in the Foundations of Physics 1–4”,JMP 49, Issue 5, 053515–18 (2008). [arXiv:quant-ph/0703060,62,64and 66]

AD, “Topos theory and ‘neo-realist’ quantum theory”, in QuantumField Theory, Competitive Models, eds. B. Fauser et al., Birkhauser(2009). [arXiv:0712.4003]

AD, “Topos-Based Logic for Quantum Systems and Bi-HeytingAlgebras”, to appear in Logic & Algebra in Quantum Computing,Lecture Notes in Logic, published by the ASL/CUP. [arxiv:1202.2750]

(There’s more stuff in the ArXiv.)


Summary

Thanks for listening!


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