Quantum systems allow active state spaces

Chris HeunenUniversity of Oxford

July 2, 2013

1 / 34

Relationship between classical and quantum

2 / 34

Relationship between classical and quantum

2 / 34

Relationship between classical and quantum

2 / 34

Part I

Algebras of observables

3 / 34

Algebras of observables

Observables are primitive, states are derived

C*-algebras⇤-algebra of operators that is closed

AW*-algebrasabstract/algebraic version of W*-algebra

von Neumann algebras / W*-algebras⇤-algebra of operators that is weakly closed

Jordan algebrasJC/JW-algebras: real version of above

4 / 34

Algebras of observables

Observables are primitive, states are derived

C*-algebras⇤-algebra of operators that is closed

AW*-algebrasabstract/algebraic version of W*-algebra

von Neumann algebras / W*-algebras⇤-algebra of operators that is weakly closed

Jordan algebrasJC/JW-algebras: real version of above

4 / 34

Algebras of observables

Observables are primitive, states are derived

C*-algebras⇤-algebra of operators that is closed

AW*-algebrasabstract/algebraic version of W*-algebra

von Neumann algebras / W*-algebras⇤-algebra of operators that is weakly closed

Jordan algebrasJC/JW-algebras: real version of above

4 / 34

Algebras of observables

Observables are primitive, states are derived

C*-algebras⇤-algebra of operators that is closed

AW*-algebrasabstract/algebraic version of W*-algebra

von Neumann algebras / W*-algebras⇤-algebra of operators that is weakly closed

Jordan algebrasJC/JW-algebras: real version of above

4 / 34

Classical mechanics

I If X is a state space,then C (X ) = {f : X ! C} is an operator algebra.

I Theorem: Every commutative operator algebrais of this form.

I Can recover states (as maps C (X ) ! C): “spectrum”Constructions on states transfer to observables:

X + Y 7! C (X )⌦ C (Y )

X ⇥ Y 7! C (X )� C (Y

Equivalence of categories: states determine everything

5 / 34

Classical mechanics

I If X is a state space,then C (X ) = {f : X ! C} is an operator algebra.

I Theorem: Every commutative operator algebrais of this form.

I Can recover states (as maps C (X ) ! C): “spectrum”Constructions on states transfer to observables:

X + Y 7! C (X )⌦ C (Y )

X ⇥ Y 7! C (X )� C (Y

Equivalence of categories: states determine everything

5 / 34

Classical mechanics

I If X is a state space,then C (X ) = {f : X ! C} is an operator algebra.

I Theorem: Every commutative operator algebrais of this form.

I Can recover states (as maps C (X ) ! C): “spectrum”Constructions on states transfer to observables:

X + Y 7! C (X )⌦ C (Y )

X ⇥ Y 7! C (X )� C (Y

Equivalence of categories: states determine everything

5 / 34

Quantum mechanics

I If H is a Hilbert space,then B(H) = {f : H ! H} is an operator algebra.

I Theorem: Every operator algebraembeds into one of this form.

I Recover states?Do states determine everything?“Noncommutative spectrum”?

6 / 34

Quantum mechanics

I If H is a Hilbert space,then B(H) = {f : H ! H} is an operator algebra.

I Theorem: Every operator algebraembeds into one of this form.

I Recover states?Do states determine everything?“Noncommutative spectrum”?

6 / 34

Quantum mechanics

I If H is a Hilbert space,then B(H) = {f : H ! H} is an operator algebra.

I Theorem: Every operator algebraembeds into one of this form.

I Recover states?Do states determine everything?“Noncommutative spectrum”?

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Quantum state spaces?

certain convex sets (states)

sheaves over locales (prime ideals)

quantales (maximal ideals)

orthomodular lattices (projections)

q-spaces (projections of enveloping W*-algebra)

7 / 34

Quantum state spaces?

No!?

commutativeoperator algebras

spectrum//

'� _

✏✏

state spacesoo

G

✏✏

����

operator algebras

F

//_____ quantumstate spaces

I Theorem: If G is continuous,then F degenerates.

I That’s right. (F (Mn) = ; for n � 3.)

I So G better not be continuousSo quantum state spaces must be radically di↵erent ...

8 / 34

Quantum state spaces? No!

?

commutativeoperator algebras

spectrum//

'� _

✏✏

state spacesoo

G

✏✏

����

operator algebrasF

//_____ quantumstate spaces

I Theorem: If G is continuous,then F degenerates.

I That’s right. (F (Mn) = ; for n � 3.)

I So G better not be continuousSo quantum state spaces must be radically di↵erent ...

8 / 34

Quantum state spaces? No!

?

commutativeoperator algebras

spectrum//

'� _

✏✏

state spacesoo

G

✏✏

����

operator algebrasF

//_____ quantumstate spaces

I Theorem: If G is continuous,then F degenerates.

I That’s right. (F (Mn) = ; for n � 3.)

I So G better not be continuousSo quantum state spaces must be radically di↵erent ...

8 / 34

Quantum state spaces? No!?

commutativeoperator algebras

spectrum//

'� _

✏✏

state spacesoo

G

✏✏

����

operator algebrasF

//_____ quantumstate spaces

I Theorem: If G is continuous,then F degenerates.

I That’s right. (F (Mn) = ; for n � 3.)

I So G better not be continuousSo quantum state spaces must be radically di↵erent ...

8 / 34

Part II

Classical viewpoints

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Doctrine of classical concepts

“However far the phenomena transcend thescope of classical physical explanation, the ac-count of all evidence must be expressed in classi-cal terms.... The argument is simply that by theword experiment we refer to a situation wherewe can tell others what we have done and whatwe have learned and that, therefore, the accountof the experimental arrangements and of the re-sults of the observations must be expressed inunambiguous language with suitable applicationof the terminology of classical physics.”

10 / 34

Classical viewpoints

I Invariant that circumvents the obstruction:Given an operator algebra A,consider C(A) = {C ✓ A commutative subalgebra},the collection of classical viewpoints.

ITheorem: Can reconstruct A

as a piecewise algebra.(A ⇠= colim C(A))

11 / 34

Classical viewpoints

I Invariant that circumvents the obstruction:Given an operator algebra A,consider C(A) = {C ✓ A commutative subalgebra},the collection of classical viewpoints.

ITheorem: Can reconstruct A

as a piecewise algebra.(A ⇠= colim C(A))

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Piecewise structures

I A piecewise widget is a widget that forgotoperations between noncommuting elements.

I Theorem: There is no piecewise morphismProj(C3) ! {0, 1}

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Piecewise structures

I A piecewise widget is a widget that forgotoperations between noncommuting elements.

I A piecewise complex *-algebra is a set A with:I a reflexive binary relation � ✓ A2;I (partial) binary operations +, · : � ! A;I (total) functions ⇤ : A ! A and · : C⇥ A ! A;

such that every S ✓ A with S2 ✓ � is contained in a T ✓ Awith T 2 ✓ � where (T ,+, ·, ⇤) is a commutative ⇤-algebra.

I Theorem: There is no piecewise morphismProj(C3) ! {0, 1}

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Piecewise structures

I A piecewise widget is a widget that forgotoperations between noncommuting elements.

I A piecewise Boolean algebra is a set B with:I a reflexive binary relation � ✓ B2;I (partial) binary operations _,^ : � ! B ;I a (total) function ¬ : B ! B ;

such that every S ✓ B with S2 ✓ � is contained in a T ✓ Bwith T 2 ✓ � where (T ,^,_,¬) is a Boolean algebra.

I Theorem: There is no piecewise morphismProj(C3) ! {0, 1}

12 / 34

Piecewise structures

I A piecewise widget is a widget that forgotoperations between noncommuting elements.

I Every projection lattice gives a piecewise Boolean algebra:

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I Theorem: There is no piecewise morphismProj(C3) ! {0, 1}

12 / 34

Piecewise structures

I A piecewise widget is a widget that forgotoperations between noncommuting elements.

I Every projection lattice gives a piecewise Boolean algebra:

•

gggggggggg

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jjjjjjjj

jjjjj

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HHHH

HH

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I Theorem: There is no piecewise morphismProj(C3) ! {0, 1}

12 / 34

Piecewise structures

ITheorem: Can reconstruct A

as a piecewise algebra.(A ⇠= colim C(A))

I What we can say about a quantum system= what we can say about it from classical viewpoints= what we can say about A using just C(A)= what we can say about A as a piecewise algebra

I How much is this? Quite a bit:I Quantum foundations: BohrificationI Quantum logic: BohrificationI Quantum information theory: entropy

13 / 34

Piecewise structures

ITheorem: Can reconstruct A

as a piecewise algebra.(A ⇠= colim C(A))

I What we can say about a quantum system= what we can say about it from classical viewpoints= what we can say about A using just C(A)= what we can say about A as a piecewise algebra

I How much is this? Quite a bit:I Quantum foundations: BohrificationI Quantum logic: BohrificationI Quantum information theory: entropy

13 / 34

Contextual entropy

Define: contextual entropy of state ⇢ of Afunction E⇢ : C(A) ! R,C 7! Shannon entropy H(tr(⇢ �))

Theorem: contextual entropy generalisesvon Neumann entropyS(⇢) = min{E⇢(C ) | C 2 C(A)}

Theorem: E⇢ determines ⇢!(in dim � 3)

14 / 34

Contextual entropy

Define: contextual entropy of state ⇢ of Afunction E⇢ : C(A) ! R,C 7! Shannon entropy H(tr(⇢ �))

Theorem: contextual entropy generalisesvon Neumann entropyS(⇢) = min{E⇢(C ) | C 2 C(A)}

Theorem: E⇢ determines ⇢!(in dim � 3)

14 / 34

Contextual entropy

Define: contextual entropy of state ⇢ of Afunction E⇢ : C(A) ! R,C 7! Shannon entropy H(tr(⇢ �))

Theorem: contextual entropy generalisesvon Neumann entropyS(⇢) = min{E⇢(C ) | C 2 C(A)}

Theorem: E⇢ determines ⇢!(in dim � 3)

14 / 34

Bohrification: history

reformulatewith classicalviewpoints

general topos approach to physics

Bohrification

attempts at dynamics

15 / 34

Bohrification: idea

I Consider “contextual sets”assignment of set S(C ) to each classical viewpoint C 2 C(A)such that C ✓ D implies S(C ) ✓ S(D)

I They form a topos T (A)!category whose objects behave a lot like setsin particular, it has a logic of its own!

I There is one canonical contextual set AA(C ) = C

I Theorem: T (A) believes that A is acommutative operator algebra!

16 / 34

Bohrification: idea

I Consider “contextual sets”assignment of set S(C ) to each classical viewpoint C 2 C(A)such that C ✓ D implies S(C ) ✓ S(D)

I They form a topos T (A)!category whose objects behave a lot like setsin particular, it has a logic of its own!

I There is one canonical contextual set AA(C ) = C

I Theorem: T (A) believes that A is acommutative operator algebra!

16 / 34

Bohrification: idea

I Consider “contextual sets”assignment of set S(C ) to each classical viewpoint C 2 C(A)such that C ✓ D implies S(C ) ✓ S(D)

I They form a topos T (A)!category whose objects behave a lot like setsin particular, it has a logic of its own!

I There is one canonical contextual set AA(C ) = C

I Theorem: T (A) believes that A is acommutative operator algebra!

16 / 34

Bohrification: idea

I Consider “contextual sets”assignment of set S(C ) to each classical viewpoint C 2 C(A)such that C ✓ D implies S(C ) ✓ S(D)

I They form a topos T (A)!category whose objects behave a lot like setsin particular, it has a logic of its own!

I There is one canonical contextual set AA(C ) = C

I Theorem: T (A) believes that A is acommutative operator algebra!

16 / 34

Bohrification: caveats

Change rules to make quantum system classical. Price:

I No proof by contradiction. (P _ ¬P)I No choice. (Si 6= ; =)

Qi Si 6= ;)

I No real numbers. (completions of Q di↵er)

No matter!

Theorem: A determined by state space(within T (A))

Circumvents obstruction ...

17 / 34

Bohrification: caveats

Change rules to make quantum system classical. Price:I No proof by contradiction. (P _ ¬P)I No choice. (Si 6= ; =)

Qi Si 6= ;)

I No real numbers. (completions of Q di↵er)

No matter!

Theorem: A determined by state space(within T (A))

Circumvents obstruction ...

17 / 34

Bohrification: quantum state space?

Change rules to make quantum system classical. Price:

I No proof by contradiction. (P _ ¬P)I No choice. (Si 6= ; =)

Qi Si 6= ;)

I No real numbers. (completions of Q di↵er)

No matter!

Theorem: A determined by state space(within T (A))

Circumvents obstruction ...

17 / 34

Bohrification: quantum state space?

Change rules to make quantum system classical. Price:

I No proof by contradiction. (P _ ¬P)I No choice. (Si 6= ; =)

Qi Si 6= ;)

I No real numbers. (completions of Q di↵er)

No matter!

Theorem: A determined by state space(within T (A))

Circumvents obstruction ...

17 / 34

Piecewise structures: how far can we get?

Theorem: If C(A) ⇠= C(B),then A ⇠= B as Jordan algebras(for W*-algebras without I2 term)

Theorem: If C(A) ⇠= C(B),then A ⇠= B as piecewise Jordan algebras(for all C*-algebras except C2 and M2)

I So need to add more information to C(A) ...

18 / 34

Piecewise structures: how far can we get?

Theorem: If C(A) ⇠= C(B),then A ⇠= B as Jordan algebras(for W*-algebras without I2 term)

Theorem: If C(A) ⇠= C(B),then A ⇠= B as piecewise Jordan algebras(for all C*-algebras except C2 and M2)

I So need to add more information to C(A) ...

18 / 34

Part III

Interaction between classical viewpoints

19 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Five stages of grief

Established psychology:

1. Denial: “These are not groups!”

2. Anger: “Why are you destroying my groups? I hate you!”

3. Bargaining: “At least think in terms of commutative groups?”

4. Depression: “I wasted my life on the wrong groups!”

5. Acceptance: “Noncommutative groups are cool!”

6. Stockholm syndrome: “Commutative groups? Don’t care!”

20 / 34

Active lattices: idea

operatoralgebra

projections

xxqqqqqq

qqqqqq

qqq

classicalviewpoints

lattice activelattice

group

I Replace classical viewpoints C(A)by projection lattice {p 2 A | p⇤ = p = p2}

I Any ⇤-algebra has unitary group {u 2 A | uu⇤ = 1 = u⇤u}

I Unitaries act on projections (u · p = upu⇤)Projections inject into unitaries (p 7! 1� 2p)

So projections act on themselves!

21 / 34

Active lattices: idea

operatoralgebra

projections

xxqqqqqq

qqqqqq

qqq

classicalviewpoints

lattice

activelattice

group

I Replace classical viewpoints C(A)by projection lattice {p 2 A | p⇤ = p = p2}

I Any ⇤-algebra has unitary group {u 2 A | uu⇤ = 1 = u⇤u}

I Unitaries act on projections (u · p = upu⇤)Projections inject into unitaries (p 7! 1� 2p)

So projections act on themselves!

21 / 34

Active lattices: idea

operatoralgebra

projections

xxqqqqqq

qqqqqq

qqq

unitaries

$$

IIII

IIII

IIII

II

classicalviewpoints

lattice

activelattice

group

I Replace classical viewpoints C(A)by projection lattice {p 2 A | p⇤ = p = p2}

I Any ⇤-algebra has unitary group {u 2 A | uu⇤ = 1 = u⇤u}

I Unitaries act on projections (u · p = upu⇤)Projections inject into unitaries (p 7! 1� 2p)

So projections act on themselves!

21 / 34

Active lattices: idea

operatoralgebra

projections

xxqqqqqq

qqqqqq

qqq

unitaries

$$

IIII

IIII

IIII

II

classicalviewpoints

lattice

activelattice

group

I Replace classical viewpoints C(A)by projection lattice {p 2 A | p⇤ = p = p2}

I Any ⇤-algebra has unitary group {u 2 A | uu⇤ = 1 = u⇤u}

I Unitaries act on projections (u · p = upu⇤)Projections inject into unitaries (p 7! 1� 2p)

So projections act on themselves!

21 / 34

Active lattices: idea

operatoralgebra

projections

xxqqqqqq

qqqqqq

qqq

unitaries

$$

IIII

IIII

IIII

II

✏✏

���

classicalviewpoints

lattice activelattice

oo // group

I Replace classical viewpoints C(A)by projection lattice {p 2 A | p⇤ = p = p2}

I Any ⇤-algebra has unitary group {u 2 A | uu⇤ = 1 = u⇤u}

I Unitaries act on projections (u · p = upu⇤)Projections inject into unitaries (p 7! 1� 2p)So projections act on themselves!

21 / 34

Symmetries

I 7�!

p 7�! 1� 2pprojections ⇠= self-adjoint unitaries

=“symmetries”

I Sym(A) is subgroup of unitaries generated by symmetries

I if A type I1, then Sym(A) = { all symmetries }I if A type I2/I3/..., then Sym(A) = { u | det(u)2 = 1 }I if A type I1/II/III, then Sym(A) = { all unitaries }

22 / 34

Symmetries

I 7�!

p 7�! 1� 2pprojections ⇠= self-adjoint unitaries

=“symmetries”

I Sym(A) is subgroup of unitaries generated by symmetries

I if A type I1, then Sym(A) = { all symmetries }I if A type I2/I3/..., then Sym(A) = { u | det(u)2 = 1 }I if A type I1/II/III, then Sym(A) = { all unitaries }

22 / 34

Symmetries

I 7�!

p 7�! 1� 2pprojections ⇠= self-adjoint unitaries

=“symmetries”

I Sym(A) is subgroup of unitaries generated by symmetries

I if A type I1, then Sym(A) = { all symmetries }I if A type I2/I3/..., then Sym(A) = { u | det(u)2 = 1 }I if A type I1/II/III, then Sym(A) = { all unitaries }

22 / 34

Active lattices

I An action of a (piecewise) group G on a (piecewise) lattice Pis a homomorphism G ! Aut(P)

I An active lattice is:

every AW*-algebra A has one:I a complete orthomodular lattice PI a group G generated by PI an action of G on PI an action of G on P

I Theorem: Its active lattice determines A(full and faithful functor)

23 / 34

Active lattices

I An action of a (piecewise) group G on a (piecewise) lattice Pis a homomorphism G ! Aut(P)

I An active lattice is:

every AW*-algebra A has one:

I a piecewise AW*-algebra A

a complete orthomodular lattice P

I a lattice structure P on the projections

a group G generated by P

I a group structure G on the symmetries

an action of G on P

I an action of G on P

I Theorem: Its active lattice determines A(full and faithful functor)

23 / 34

Active lattices

I An action of a (piecewise) group G on a (piecewise) lattice Pis a homomorphism G ! Aut(P)

I An active lattice is:

every AW*-algebra A has one:

I a complete orthomodular lattice PI a group G generated by PI an action of G on P

I an action of G on P

I Theorem: Its active lattice determines A(full and faithful functor)

23 / 34

Active lattices

I An action of a (piecewise) group G on a (piecewise) lattice Pis a homomorphism G ! Aut(P)

I An active lattice is: every AW*-algebra A has one:I a complete orthomodular lattice P Proj(A)I a group G generated by P Sym(A)I an action of G on P u · p = upu⇤

I an action of G on P

I Theorem: Its active lattice determines A(full and faithful functor)

23 / 34

Active lattices

I An action of a (piecewise) group G on a (piecewise) lattice Pis a homomorphism G ! Aut(P)

I An active lattice is: every AW*-algebra A has one:I a complete orthomodular lattice P Proj(A)I a group G generated by P Sym(A)I an action of G on P u · p = upu⇤

I an action of G on P

I Theorem: Its active lattice determines A(full and faithful functor)

23 / 34

Matrix algebras

I If A is an operator algebra, then so is Mn(A)

I “All AW*-algebras are matrix algebras”If A type In, then A ⇠= Mn(C )If A type I1/II1/III, then A ⇠= Mn(A)

ITheorem: Classical viewpoints in

Mn(A) are diagonal.

(8C 2 C(Mn(A)) 9u 2 U(Mn(A)) : uCu⇤ diagonal)

24 / 34

Matrix algebras

I If A is an operator algebra, then so is Mn(A)

I “All AW*-algebras are matrix algebras”If A type In, then A ⇠= Mn(C )If A type I1/II1/III, then A ⇠= Mn(A)

ITheorem: Classical viewpoints in

Mn(A) are diagonal.

(8C 2 C(Mn(A)) 9u 2 U(Mn(A)) : uCu⇤ diagonal)

24 / 34

Matrix algebras

I If A is an operator algebra, then so is Mn(A)

I “All AW*-algebras are matrix algebras”If A type In, then A ⇠= Mn(C )If A type I1/II1/III, then A ⇠= Mn(A)

ITheorem: Classical viewpoints in

Mn(A) are diagonal.

(8C 2 C(Mn(A)) 9u 2 U(Mn(A)) : uCu⇤ diagonal)

24 / 34

Matrix algebras: projections

IEven if A has few projections, Mn(A) has lots!

pij(a) =

✓(1 + aa⇤)�1 (1 + aa⇤)�1aa⇤(1 + aa⇤)�1 a⇤(1 + aa⇤)�1a

◆

a

1

p12(a)

I These vector projections encode algebraic structure of A!pij(a+ b) = polynomial in pij(a), pik(b), pjk(c), . . .pij(ab) = polynomial in pik(a), pkj(b), . . .pij(a⇤) = polynomial in pji (a), . . .

25 / 34

Matrix algebras: projections

IEven if A has few projections, Mn(A) has lots!

pij(a) =

✓(1 + aa⇤)�1 (1 + aa⇤)�1aa⇤(1 + aa⇤)�1 a⇤(1 + aa⇤)�1a

◆

a

1

p12(a)

I These vector projections encode algebraic structure of A!pij(a+ b) = polynomial in pij(a), pik(b), pjk(c), . . .pij(ab) = polynomial in pik(a), pkj(b), . . .pij(a⇤) = polynomial in pji (a), . . .

25 / 34

Matrix algebras: projections

IEven if A has few projections, Mn(A) has lots!

pij(a) =

✓(1 + aa⇤)�1 (1 + aa⇤)�1aa⇤(1 + aa⇤)�1 a⇤(1 + aa⇤)�1a

◆

a

1

p12(a)

I These vector projections encode algebraic structure of A!pij(a+ b) = polynomial in pij(a), pik(b), pjk(c), . . .pij(ab) = polynomial in pik(a), pkj(b), . . .pij(a⇤) = polynomial in pji (a), . . .

25 / 34

Active lattices determine operator algebras

I Lemma: If f : Proj(Mn(A)) ! Proj(Mn(B)) equivariant,then f (pij(a)) = pij('(a)) for some ' : A ! B .

I Lemma: The vector projections generate Proj(Mn(A)).

I Recall: “All AW*-algebras are matrix algebras”

I Theorem: Its active lattice determines A(full and faithful functor)

26 / 34

Active lattices determine operator algebras

I Lemma: If f : Proj(Mn(A)) ! Proj(Mn(B)) equivariant,then f (pij(a)) = pij('(a)) for some ' : A ! B .

I Lemma: The vector projections generate Proj(Mn(A)).

I Recall: “All AW*-algebras are matrix algebras”

I Theorem: Its active lattice determines A(full and faithful functor)

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Active lattices determine operator algebras

I Lemma: If f : Proj(Mn(A)) ! Proj(Mn(B)) equivariant,then f (pij(a)) = pij('(a)) for some ' : A ! B .

I Lemma: The vector projections generate Proj(Mn(A)).

I Recall: “All AW*-algebras are matrix algebras”

I Theorem: Its active lattice determines A(full and faithful functor)

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Active lattices determine operator algebras

I Lemma: If f : Proj(Mn(A)) ! Proj(Mn(B)) equivariant,then f (pij(a)) = pij('(a)) for some ' : A ! B .

I Lemma: The vector projections generate Proj(Mn(A)).

I Recall: “All AW*-algebras are matrix algebras”

I Theorem: Its active lattice determines A(full and faithful functor)

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Conclusion

“Knowing a quantum system =all classical viewpoints + switching between them”

I Physics = dynamics and kinematics in one

I Logic of contextuality

I Operational process categoriesProtocol specification language

I Noncommutative topology

I Game theory, database theory, concurrent programming ...

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Conclusion

“Knowing a quantum system =all classical viewpoints + switching between them”

I Physics = dynamics and kinematics in one

I Logic of contextuality

I Operational process categoriesProtocol specification language

I Noncommutative topology

I Game theory, database theory, concurrent programming ...

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Conclusion

“Knowing a quantum system =all classical viewpoints + switching between them”

I Physics = dynamics and kinematics in one

I Logic of contextuality

I Operational process categoriesProtocol specification language

I Noncommutative topology

I Game theory, database theory, concurrent programming ...

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Conclusion

“Knowing a quantum system =all classical viewpoints + switching between them”

I Physics = dynamics and kinematics in one

I Logic of contextuality

I Operational process categoriesProtocol specification language

I Noncommutative topology

I Game theory, database theory, concurrent programming ...

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Conclusion

“Knowing a quantum system =all classical viewpoints + switching between them”

I Physics = dynamics and kinematics in one

I Logic of contextuality

I Operational process categoriesProtocol specification language

I Noncommutative topology

I Game theory, database theory, concurrent programming ...

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Conclusion

“Knowing a quantum system =all classical viewpoints + switching between them”

I Physics = dynamics and kinematics in one

I Logic of contextuality

I Operational process categoriesProtocol specification language

I Noncommutative topology

I Game theory, database theory, concurrent programming ...

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References (in order of appearance)

“Extending obstructions to

noncommutative functorial spectra”

Journal of Pure and Applied Algebra, 2013

“Noncommutativity as a colimit”

Applied Categorical Structures 20(4):393–414, 2012

“A topos for algebraic quantum theory”

Communications in Mathematical Physics 291:63–110, 2009

“Diagonalizing matrices over AW*-algebras”

Journal of Functional Analysis 264(8):1873–1898, 2013

“Active lattices determine AW*-algebras”

Advances in Mathematics, 2013

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Part IV

Bonus: abstract quantum logic

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Abstract quantum logic

I Topos logic not operational since “set-based”:propositions are “contextual subsets”(In particular, they form a distributive lattice)

I Propositions are closed subspaces(orthomodular projection lattice)

IQuantum logic in abstract categoriesincluding “modal” quantifier 9(“dagger kernel categories” like Hilb or Rel)

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Abstract quantum logic

I Topos logic not operational since “set-based”:propositions are “contextual subsets”(In particular, they form a distributive lattice)

I Propositions are closed subspaces(orthomodular projection lattice)

IQuantum logic in abstract categoriesincluding “modal” quantifier 9(“dagger kernel categories” like Hilb or Rel)

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Abstract quantum logic

I Topos logic not operational since “set-based”:propositions are “contextual subsets”(In particular, they form a distributive lattice)

I Propositions are closed subspaces(orthomodular projection lattice)

IQuantum logic in abstract categoriesincluding “modal” quantifier 9(“dagger kernel categories” like Hilb or Rel)

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Abstract operator algebras

I An abstract operator algebra (Frobenius algebra) in a tensorcategory is a morphism : A⌦ A ! A satisfying

= = = =

I in Hilb: (concrete) operator algebras

(caveats in infinite dimension)

I in Rel: groupoids

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Abstract operator algebras

I An abstract operator algebra (Frobenius algebra) in a tensorcategory is a morphism : A⌦ A ! A satisfying

= = = =

I in Hilb: (concrete) operator algebras

(caveats in infinite dimension)

I in Rel: groupoids

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Abstract operator algebras

I An abstract operator algebra (Frobenius algebra) in a tensorcategory is a morphism : A⌦ A ! A satisfying

= = = =

I in Hilb: (concrete) operator algebras

(caveats in infinite dimension)

I in Rel: groupoids

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Possibilistic quantum logic

I Abstract quantum logic in Relis classical modal logic

I In Rel: projections = subgroupoids

I classical

quantum=

commutative

noncommutative6= distributive

nondistributive

I Can we reconstruct an abstract operator algebrafrom its category of classical viewpoints?

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Possibilistic quantum logic

I Abstract quantum logic in Relis classical modal logic

I In Rel: projections = subgroupoids

I classical

quantum=

commutative

noncommutative6= distributive

nondistributive

I Can we reconstruct an abstract operator algebrafrom its category of classical viewpoints?

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Possibilistic quantum logic

I Abstract quantum logic in Relis classical modal logic

I In Rel: projections = subgroupoids

I classical

quantum=

commutative

noncommutative6= distributive

nondistributive

I Can we reconstruct an abstract operator algebrafrom its category of classical viewpoints?

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Possibilistic quantum logic

I Abstract quantum logic in Relis classical modal logic

I In Rel: projections = subgroupoids

I classical

quantum=

commutative

noncommutative6= distributive

nondistributive

I Can we reconstruct an abstract operator algebrafrom its category of classical viewpoints?

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References (in order of appearance)

“Operational theories and

categorical quantum mechanics”

Logic and algebraic structures in quantum computing and information, 2013

“Quantum logic in dagger kernel categories”

Order 27(2):177–212, 2010

“Complementarity in categorical quantum mechanics”

Foundations of Physics 42(7):856–873, 2012

“A new description of orthogonal bases”

Mathematical Structures in Computer Science 23(3):555–567, 2012

“Categorical formulation of quantum algebras”

Communications in Mathematical Physics 204(3):765–796, 2011

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References (in order of appearance)

“H*-algebras and nonunital Frobenius algebras”

AMS Cli↵ord Lectures 71:1–24, 2010

“Categories of quantum

and classical channels”

Quantum Information Processing, 2013

“Quantum and classical structures

in nondeterministic computation”

Quantum Interaction 143–157, 2009

“Relative Frobenius algebras are groupoids”

Journal of Pure and Applied Algebra 217:114–124, 2012

“Compositional quantum logic”

Computation, Logic, Games, and Quantum Foundations: 21–36, 2013

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