= = = f f BOB BOB meaning vectors of words not does like - - PowerPoint PPT Presentation

The logic of quantum mechanics - take II

Bob Coecke — arXiv:1204.3458

=

f f

=

f f f

=

f

=

like

Bob Alice does Alice

like

not Bob

• An alternative Gospel of structure:
• rder, composition, processes

arXiv:1307.4038

• The logic of quantum mechanics - Take II

arXiv:1204.3458

• A universe of processes and some of its guises

arXiv:1009.3786

• Quantum Picturalism

arXiv:0908.1787

• Introducing categories to the practicing physicist

arXiv:0808.1032

• Kindergarten Quantum Mechanics

arXiv:quant-ph/0510032

— CROCL ’99 —

— genesis —

— genesis — [von Neumann 1932] Formalized quantum mechanics in “Mathematische Grundlagen der Quantenmechanik”

— genesis — [von Neumann 1932] Formalized quantum mechanics in “Mathematische Grundlagen der Quantenmechanik” [von Neumann to Birkhoff 1935] “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)

— genesis — [von Neumann 1932] Formalized quantum mechanics in “Mathematische Grundlagen der Quantenmechanik” [von Neumann to Birkhoff 1935] “I would like to make a confession which may seem immoral: I do not believe 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 mechanics in “Mathematische Grundlagen der Quantenmechanik” [von Neumann to Birkhoff 1935] “I would like to make a confession which may seem immoral: I do not believe 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 mechanics in “Mathematische Grundlagen der Quantenmechanik” [von Neumann to Birkhoff 1935] “I would like to make a confession which may seem immoral: I do not believe 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. Schr¨

• dinger (1935): the stuff which is the true soul of

quantum 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. Schr¨

• dinger (1935): the stuff which is the true soul of

quantum theory is how quantum systems compose. Last 20 year discoveries: Schr¨

• dinger 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 A

f

✲ B

A

f ′

✲ B

A

f ′′

✲ 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 A

f

✲ B

A

f ′

✲ B

A

f ′′

✲ B

be boiling, frying, baking. States are processes I := unspecified

ψ ✲ A.

• 3. Let

A

g ◦ f✲C

be the composite process of first boiling A

f

✲ B and

then salting B

g ✲C.

• 3. Let

A

g ◦ f✲C

be the composite process of first boiling A

f

✲ B and

then salting B

g ✲C. Let

X

1X ✲ 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 ⊗ D

f⊗h✲ B ⊗ E

be boiling potato while frying carrot.

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

A ⊗ D

f⊗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 ⊗ D

f⊗h

✲ B ⊗ E

g⊗k

✲ C ⊗ F

x ✲ M= A ⊗ D x◦(g⊗k)◦( f⊗h)

✲ M.

• 5. Total process:

A ⊗ D

f⊗h

✲ B ⊗ E

g⊗k

✲ C ⊗ F

x ✲ M= A ⊗ D x◦(g⊗k)◦( f⊗h)

✲ M.

• 6. Recipe = composition structure on processes.

• 5. Total process:

A ⊗ D

f⊗h

✲ B ⊗ E

g⊗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 ⊗ D

f⊗h

✲ B ⊗ E

g⊗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 ⊗ D

f⊗h

✲ B ⊗ E

g⊗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. Andr´ e 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 f g

— 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 — quantum classical =

= =

— asserting (pure) entanglement — quantum classical =

= =

⇒ introduce ‘parallel wire’ between systems: subject to: only topology matters!

— quantum-like — E.g.

=

Transpose:

f f

=

Conjugate:

f f

=

classical data flow? f

=

f f f

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 is derivable 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 only if it is derivable in the dagger compact category of fi- nite dimensional Hilbert spaces, linear maps, tensor product 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 Kissinger Picturing Quantum Processes Cambridge 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, or negation (¬) : not (cf. nothing = not something) entailment (⇒) : then quantifiers (∀, ∃) : every(thing), some(thing) constants (a, b) : thing variable (x) : Alice, Bob predicates (P(x), R(x, y)) : eating, getting sick truth 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, or negation (¬) : not (cf. nothing = not something) entailment (⇒) : then quantifiers (∀, ∃) : every(thing), some(thing) constants (a, b) : thing variable (x) : Alice, Bob predicates (P(x), R(x, y)) : eating, getting sick truth 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 LANGUAGE FOR NATURAL LANGUAGE MEANING

BC, Mehrnoosh Sadrzadeh & Stephen Clark (2010) Mathematical foundations for 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

• bject
• bject

subject subject

— the from-words-to-a-sentence process — Information flow within a verb:

verb

• bject
• bject

subject 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

• r 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

• r 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 A A 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 on categorical 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 n s n 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:

verb n n

flow flow 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 :: →       

• 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 :: →       

• 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

Bob Alice

— − − − − → Alice ⊗ − − − → does ⊗ − − → not ⊗ − − → like ⊗ − − → Bob —

Alice

like

Bob

meaning vectors of words grammar not

=

not

like

Bob Alice

=

not

like

Bob Alice

Using:

=

like like

=

like like

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 group

for 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 via

their Euler angle decomposition yield all unitaries: ΛZ(γ) ◦ ΛX(β) ◦ ΛZ(α) = γ α β .                1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0                = .

— completeness II —

• Thm. (Miriam Backens, about a year ago) The above

described graphical calculus with phases restricted to

π 2-multiples, is complete for qubit stabiliser fragment

• f quantum computing, provided that we add the Euler

angle 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