the logic of quantum mechanics - take ii logic of quantum mechanics - take ii ... quantum classical...
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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