The logic of quantum mechanics - take IIBob Coecke — arXiv:1204.3458

=

f

f =

f f

f

ALICE

BOB

=

ALICE

BOB

f

=

not

likeBobAlice

does

Alice not like

not

Bob

meaning vectors of words

pregroup grammar

• An alternative Gospel of structure:order, composition, processesarXiv:1307.4038

• The logic of quantum mechanics - Take IIarXiv:1204.3458

• A universe of processes and some of its guisesarXiv:1009.3786

• Quantum PicturalismarXiv:0908.1787

• Introducing categories to the practicing physicistarXiv:0808.1032

• Kindergarten Quantum MechanicsarXiv:quant-ph/0510032

— CROCL ’99 —

— genesis —

— genesis —

[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”

— genesis —

[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”

[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)

— genesis —

[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”

[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)

[Birkhoff and von Neumann 1936] The Logic of Quan-tum Mechanics in Annals of Mathematics.

— genesis —

[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”

[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)

[Birkhoff and von Neumann 1936] The Logic of Quan-tum Mechanics in Annals of Mathematics.

[1936 – 2000] many followed them, ... and FAILED.

— genesis —

[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”

[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)

[Birkhoff and von Neumann 1936] The Logic of Quan-tum Mechanics in Annals of Mathematics.

[1936 – 2000] many followed them, ... and FAILED.

— the mathematics of it —

— the mathematics of it —

Hilbert space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.

— the mathematics of it —

Hilbert space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.

WHY?

— the mathematics of it —

Hilbert space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.

WHY?

von Neumann: only used it since it was ‘available’.

— the physics of it —

— the physics of it —

von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.

— the physics of it —

von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.

Schrodinger (1935): the stuff which is the true soul ofquantum theory is how quantum systems compose.

— the physics of it —

von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.

Schrodinger (1935): the stuff which is the true soul ofquantum theory is how quantum systems compose.

Last 20 year discoveries: Schrodinger was right!

— the game plan —

— the game plan —

Task 0. Solve:tensor product structure

the other stuff= ???

— the game plan —

Task 0. Solve:tensor product structure

the other stuff= ???

i.e. axiomatize “⊗” without reference to spaces.

— the game plan —

Task 0. Solve:tensor product structure

the other stuff= ???

i.e. axiomatize “⊗” without reference to spaces.

Task 1. Investigate which assumptions (i.e. which struc-ture) on ⊗ is needed to deduce physical phenomena.

— the game plan —

Task 0. Solve:tensor product structure

the other stuff= ???

i.e. axiomatize “⊗” without reference to spaces.

Task 1. Investigate which assumptions (i.e. which struc-ture) on ⊗ is needed to deduce physical phenomena.

Task 2. Investigate wether such an “interaction struc-ture” appear elsewhere in “our classical reality”.

COMPOSITION WITHOUT SPACES

1. Let A be a raw potato.

1. Let A be a raw potato.A admits many states e.g. dirty, clean, skinned, ...

1. Let A be a raw potato.A admits many states e.g. dirty, clean, skinned, ...

2. We want to process A into cooked potato B.B admits many states e.g. boiled, fried, deep fried,baked with skin, baked without skin, ...

1. Let A be a raw potato.A admits many states e.g. dirty, clean, skinned, ...

2. We want to process A into cooked potato B.B admits many states e.g. boiled, fried, deep fried,baked with skin, baked without skin, ... Let

Af

- B Af ′

- B Af ′′

- B

be boiling, frying, baking.

1. Let A be a raw potato.A admits many states e.g. dirty, clean, skinned, ...

2. We want to process A into cooked potato B.B admits many states e.g. boiled, fried, deep fried,baked with skin, baked without skin, ... Let

Af

- B Af ′

- B Af ′′

- B

be boiling, frying, baking. States are processes

I := unspecifiedψ

- A.

3. LetA

g ◦ f- C

be the composite process of first boiling Af

- B andthen salting B

g- C.

3. LetA

g ◦ f- C

be the composite process of first boiling Af

- B andthen salting B

g- C. Let

X1X - X

be doing nothing. We have 1Y ◦ ξ = ξ ◦ 1X = ξ.

4. Let A ⊗ D be potato A and carrot D and let

4. Let A ⊗ D be potato A and carrot D and let

A ⊗ Df⊗h

- B ⊗ E

be boiling potato while frying carrot.

4. Let A ⊗ D be potato A and carrot D and let

A ⊗ Df⊗h

- B ⊗ E

be boiling potato while frying carrot. Let

C ⊗ F x- M

be mashing spice-cook-potato and spice-cook-carrot.

5. Total process:

A ⊗ Df⊗h

- B ⊗ Eg⊗k

- C ⊗ F x- M= A ⊗ D

x◦(g⊗k)◦( f⊗h)- M.

5. Total process:

A ⊗ Df⊗h

- B ⊗ Eg⊗k

- C ⊗ F x- M= A ⊗ D

x◦(g⊗k)◦( f⊗h)- M.

6. Recipe = composition structure on processes.

5. Total process:

A ⊗ Df⊗h

- B ⊗ Eg⊗k

- C ⊗ F x- M= A ⊗ D

x◦(g⊗k)◦( f⊗h)- M.

6. Recipe = composition structure on processes.

7. Laws governing recipes:

(1B ⊗ g) ◦ ( f ⊗ 1C) = ( f ⊗ 1D) ◦ (1A ⊗ g)

5. Total process:

A ⊗ Df⊗h

- B ⊗ Eg⊗k

- C ⊗ F x- M= A ⊗ D

x◦(g⊗k)◦( f⊗h)- M.

6. Recipe = composition structure on processes.

7. Laws governing recipes:

(1B ⊗ g) ◦ ( f ⊗ 1C) = ( f ⊗ 1D) ◦ (1A ⊗ g)i.e.

boil potato then fry carrot = fry carrot then boil potato

5. Total process:

A ⊗ Df⊗h

- B ⊗ Eg⊗k

- C ⊗ F x- M= A ⊗ D

x◦(g⊗k)◦( f⊗h)- M.

6. Recipe = composition structure on processes.

7. Laws governing recipes:

(1B ⊗ g) ◦ ( f ⊗ 1C) = ( f ⊗ 1D) ◦ (1A ⊗ g)i.e.

boil potato then fry carrot = fry carrot then boil potato

⇒Monoidal Category

BOXES AND WIRES

Roger Penrose (1971) Applications of negative dimensional tensors. In: Com-binatorial Mathematics and its Applications, 221–244. Academic Press.

Andre Joyal and Ross Street (1991) The Geometry of tensor calculus I. Ad-vances in Mathematics 88, 55–112.

— wire and box language —

f

Interpretation: wire := system ; box := process

— composing boxes —

— composing boxes —

sequential composition:

g ◦ f ≡g

f

— composing boxes —

sequential composition:

g ◦ f ≡g

f

parallel composition:

f ⊗ g ≡ f fg

— merely a new notation? —

— merely a new notation? —

(g ◦ f ) ⊗ (k ◦ h) = (g ⊗ k) ◦ ( f ⊗ h)

=f h

g k

f h

g k

— merely a new notation? —

(g ◦ f ) ⊗ (k ◦ h) = (g ⊗ k) ◦ ( f ⊗ h)

=f h

g k

f h

g k

peel potato and then fry it,while,

clean carrot and then boil it=

peel potato while clean carrot,and then,

fry potato while boil carrot

QUANTUM PROCESSES

Samson Abramsky & BC (2004) A categorical semantics for quantum proto-cols. In: IEEE-LiCS’04. quant-ph/0402130

BC (2005) Kindergarten quantum mechanics. quant-ph/0510032

— quantitative metric —

f : A→ B

f

A

B

— quantitative metric —

f † : B→ A

f

B

A

— asserting (pure) entanglement —

quantumclassical

=

==

— asserting (pure) entanglement —

quantumclassical

=

==

⇒ introduce ‘parallel wire’ between systems:

subject to: only topology matters!

— quantum-like —

E.g.

=

Transpose:

ff

=Conjugate:

ff

=

classical data flow?

f

=

fff

classical data flow?

f

=

f

classical data flow?

f

=

f

classical data flow?

f

ALICE

BOB

=

ALICE

BOB

f

⇒ quantum teleportation

— symbolically: dagger compact categories —

Thm. [Kelly-Laplaza ’80; Selinger ’05] An equa-tional statement between expressions in dagger com-pact categorical language holds if and only if it isderivable in the graphical notation via homotopy.

Thm. [Hasegawa-Hofmann-Plotkin; Selinger ’08]An equational statement between expressions in dag-ger compact categorical language holds if and onlyif it is derivable in the dagger compact category of fi-nite dimensional Hilbert spaces, linear maps, tensorproduct and adjoints.

— expressivity —

Full-blown QM/QC course at Oxford:

Quantum Computer Science

– quantum circuits, MBQC,– quantum algorithms,– quantum cryptography,– quantum non-locality, ...

— expressivity —

Full-blown QM/QC course at Oxford:

Quantum Computer Science

– quantum circuits, MBQC,– quantum algorithms,– quantum cryptography,– quantum non-locality, ...

Lecture notes forthcoming as book (approx. 400 pp):

Bob Coecke & Aleks KissingerPicturing Quantum ProcessesCambridge University Press

— kindergarten quantum mechanics: the experiment —

Contest in problem solving between:

• Children using quantum picturalism

• Physics teachers using ordinary QM

The children will win!

BC (2010) Quantum picturalism. Contemporary Physics 51, 59–83.

— what is logic? —

— what is logic? —

Pragmatic option: Logic is structure in language.

— what is logic? —

Pragmatic option: Logic is structure in language.

“Alice and Bob ate everything or nothing, then got sick.”

connectives (∧,∨) : and, ornegation (¬) : not (cf. nothing = not something)entailment (⇒) : thenquantifiers (∀,∃) : every(thing), some(thing)constants (a, b) : thingvariable (x) : Alice, Bobpredicates (P(x),R(x, y)) : eating, getting sicktruth valuation (0, 1) : true, false

(∀z : Eat(a, z) ∧ Eat(b, z)) ∧ ¬(∃z : Eat(a, z) ∧ Eat(b, z))⇒ S ick(a), S ick(b)

— what is logic? —

Pragmatic option: Logic is structure in language.

“Alice and Bob ate everything or nothing, then got sick.”

connectives (∧,∨) : and, ornegation (¬) : not (cf. nothing = not something)entailment (⇒) : thenquantifiers (∀,∃) : every(thing), some(thing)constants (a, b) : thingvariable (x) : Alice, Bobpredicates (P(x),R(x, y)) : eating, getting sicktruth valuation (0, 1) : true, false

(∀z : Eat(a, z) ∧ Eat(b, z)) ∧ ¬(∃z : Eat(a, z) ∧ Eat(b, z))⇒ S ick(a), S ick(b)

This is about truth. What about genuine meaning?

A SLIGHTLY DIFFERENT LANGUAGEFOR NATURAL LANGUAGE MEANING

BC, Mehrnoosh Sadrzadeh & Stephen Clark (2010) Mathematical foundationsfor a compositional distributional model of meaning. arXiv:1003.4394

— the from-words-to-a-sentence process —

— the from-words-to-a-sentence process —

Consider meanings of words, e.g. as vectors (cf. Google):

word 1 word 2 word n...

— the from-words-to-a-sentence process —

What is the meaning the sentence made up of these?

word 1 word 2 word n...

— the from-words-to-a-sentence process —

I.e. how do we/machines produce meanings of sentences?

word 1 word 2 word n...?

— the from-words-to-a-sentence process —

I.e. how do we/machines produce meanings of sentences?

word 1 word 2 word n...grammar

— the from-words-to-a-sentence process —

Information flow within a verb:

verb

object subject

— the from-words-to-a-sentence process —

Information flow within a verb:

verb

object subject

Again we have:

=

— pregoup grammar —

Lambek’s residuated monoids (1950’s):b ≤ a( c⇔ a · b ≤ c⇔ a ≤ c � b

— pregoup grammar —

Lambek’s residuated monoids (1950’s):b ≤ a( c⇔ a · b ≤ c⇔ a ≤ c � b

or equivalently,

a · (a( c) ≤ c ≤ a( (a · c)

(c � b) · b ≤ c ≤ (c · b) � b

— pregoup grammar —

Lambek’s residuated monoids (1950’s):b ≤ a( c⇔ a · b ≤ c⇔ a ≤ c � b

or equivalently,

a · (a( c) ≤ c ≤ a( (a · c)

(c � b) · b ≤ c ≤ (c · b) � b

Lambek’s pregroups (2000’s):a · −1a ≤ 1 ≤ −1a · a

b−1 · b ≤ 1 ≤ b · b−1

— Lambek’s pregoup grammar —

A A A AA A A A

-1

-1

-1

-1

=

A

A

A

A

=A

A A

A=A

A

A

A

=A

A A

A-1

-1

-1

-1 -1

-1 -1

-1

Jim Lambek mentioned this connection first to (to me) during a seminar oncategorical quantum mechanics at McGill here in Montreal in 2004.

— Lambek’s pregoup grammar —

For noun type n, verb type is −1n · s · n−1, so:

— Lambek’s pregoup grammar —

For noun type n, verb type is −1n · s · n−1, so:

n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s

— Lambek’s pregoup grammar —

For noun type n, verb type is −1n · s · n−1, so:

n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s

— Lambek’s pregoup grammar —

For noun type n, verb type is −1n · s · n−1, so:

n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s

— Lambek’s pregoup grammar —

For noun type n, verb type is −1n · s · n−1, so:

n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s

Diagrammatic type reduction:

n nsn n-1 -1

— Lambek’s pregoup grammar —

For noun type n, verb type is −1n · s · n−1, so:

n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s

Diagrammatic meaning:

verbn n

flow flow

— algorithm for meaning of sentences —

— algorithm for meaning of sentences —

1. Perform type reduction:

(word type 1) . . . (word type n) { sentence type

— algorithm for meaning of sentences —

1. Perform type reduction:

(word type 1) . . . (word type n) { sentence type

2. Interpret diagrammatic type reduction as linear map:

f :: 7→

∑i

〈ii|

⊗ id ⊗

∑i

〈ii|

— algorithm for meaning of sentences —

1. Perform type reduction:

(word type 1) . . . (word type n) { sentence type

2. Interpret diagrammatic type reduction as linear map:

f :: 7→

∑i

〈ii|

⊗ id ⊗

∑i

〈ii|

3. Apply this map to tensor of word meaning vectors:

f(−→v 1 ⊗ . . . ⊗

−→v n

)

—−−−−→Alice ⊗

−−−→does ⊗ −−→not ⊗

−−→like ⊗

−−→Bob —

—−−−−→Alice ⊗

−−−→does ⊗ −−→not ⊗

−−→like ⊗

−−→Bob —

Alice not like Bob

meaning vectors of words

not

grammar

does

—−−−−→Alice ⊗

−−−→does ⊗ −−→not ⊗

−−→like ⊗

−−→Bob —

Alice like Bob

meaning vectors of words

grammar

not

—−−−−→Alice ⊗

−−−→does ⊗ −−→not ⊗

−−→like ⊗

−−→Bob —

Alice like Bob

meaning vectors of words

grammar

not

—−−−−→Alice ⊗

−−−→does ⊗ −−→not ⊗

−−→like ⊗

−−→Bob —

Alice like Bob

meaning vectors of words

grammar

not= not

like

BobAlice

—−−−−→Alice ⊗

−−−→does ⊗ −−→not ⊗

−−→like ⊗

−−→Bob —

Alice like Bob

meaning vectors of words

grammar

not= not

like

BobAlice = not

likeBobAlice

Using: =

likelike

=

likelike

CLASSICAL DATA i.e. BASIS

WIRES?

SPIDERS!

B.C. and D. Pavlovic – arXiv:quant-ph/0608035

B.C., D. Pavlovic and J. Vicary – arXiv:0810.0812

B.C., E.O. Paquette and D. Pavlovic – arXiv:0904.1997

B.C. and S. Perdrix – arXiv:1004.1598

— spiders —

‘spiders’ =

m︷ ︸︸ ︷....

....

︸ ︷︷ ︸n

such that, for k > 0:

m+m′−k︷ ︸︸ ︷........

....

....

....

︸ ︷︷ ︸n+n′−k

=

....

....

Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812

— spiders —

‘(co-)mult.’ =

m︷ ︸︸ ︷....

....

︸ ︷︷ ︸n

such that, for k > 0:

m+m′−k︷ ︸︸ ︷........

....

....

....

︸ ︷︷ ︸n+n′−k

=

....

....

— spiders —

‘(co-)mult.’ =

m︷ ︸︸ ︷....

....

︸ ︷︷ ︸n

such that, for k > 0:

m+m′−k︷ ︸︸ ︷........

....

....

....

︸ ︷︷ ︸n+n′−k

=

....

....

— spiders —

‘cups/caps’ =

m︷ ︸︸ ︷....

....

︸ ︷︷ ︸n

such that, for k > 0:

m+m′−k︷ ︸︸ ︷........

....

....

....

︸ ︷︷ ︸n+n′−k

=

....

....

Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812

MATING AND FIGHTING SPIDERS

B.C. and R. Duncan – arXiv:0906.4725

— decorated spiders —

— decorated spiders —

Thm. ‘Unbiased points’ always form an Abelian groupfor spider multiplication with conjugate as inverse.

m︷ ︸︸ ︷....

....α︸ ︷︷ ︸

n

∣∣∣∣∣∣ n,m ∈ N0, α ∈ G

m︷ ︸︸ ︷

........

....

....

....α

β︸ ︷︷ ︸n

=

m︷ ︸︸ ︷....

....

α+β

︸ ︷︷ ︸n

— complementary spiders —

— complementary spiders —

Thm. Complementarity⇒

BC & Ross Duncan (2008) Interacting quantum observables. ICALP’08 &New Journal of Physics 13, 043016. arXiv:0906.4725

— complementary spiders —

Def. Strong complementarity :=

BC & Ross Duncan (2008) Interacting quantum observables. ICALP’08 &New Journal of Physics 13, 043016. arXiv:0906.4725

— universality —

Thm. (BC & Ross Duncan) The above described graph-ical language, is universal for computing with qubits.

Proof. CNOT-gate + arbitrary one-qubit unitaries viatheir Euler angle decomposition yield all unitaries:

ΛZ(γ) ◦ ΛX(β) ◦ ΛZ(α) =γ

αβ .

1 0 0 00 1 0 00 0 0 10 0 1 0

= .

— completeness II —

Thm. (Miriam Backens, about a year ago) The abovedescribed graphical calculus with phases restricted toπ2-multiples, is complete for qubit stabiliser fragmentof quantum computing, provided that we add the Eulerangle decomposition of the Hadamard gate.

— spiders —

Bob = (the) man whom Alice hates = Bob

— spiders —

Bob = (the) man whom Alice hates = Bob

— spiders —

Bob = (the) man whom Alice hates = Bob