bob coecke- quantum picturalism
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Quantum Picturalism
(or, The Logic of Quantum Mechanics take 2)
PSA Montreal November 2010
=
f
f=
f f
f
ALICE
BOB
=
ALICE
BOB
f
=
not
like
BobAlice
does
Alice not like
not
Bob
Bob CoeckeOxford University Computing Laboratory
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The Hilbert space quantum formalism
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The Hilbert space quantum formalism
[von Neumann 1932] Formalized quantum mechanics
in Mathematische Grundlagen der Quantenmechanik
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The Hilbert space quantum formalism
[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)
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The Hilbert space quantum formalism
[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, Annals of Mathematics.
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The Hilbert space quantum formalism
[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, Annals of Mathematics.
[1936 2000] many followed them, ... and FAILED.
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The Hilbert space quantum formalism
[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, Annals of Mathematics.
[1936 2000] many followed them, ... and FAILED.
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The Hilbert space quantum formalism
Hilber space stuff: continuum, field structure of com-
plex numbers, vector space over it, inner-product, etc.
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The Hilbert space quantum formalism
Hilber space stuff: continuum, field structure of com-
plex numbers, vector space over it, inner-product, etc.
WHY?
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The Hilbert space quantum formalism
Hilber 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.
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The Hilbert space quantum formalism
Hilber 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.
Model theory: one can do almost anything with it.
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The Hilbert space quantum formalism
Hilber 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.
Model theory: one can do almost anything with it.
Schrodinger (1935): the stuff which is the true soul of
quantum theory is how quantum systems compose.
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tensor product structurethe other stuff
= ?
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tensor product structurethe other stuff
= ?
Conceptually: not about properties of the individual,
but about relationships among the individuals
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tensor product structurethe other stuff
= ?
Conceptually: not about properties of the individual,
but about relationships among the individuals
Mathematically: axiomatize an abstract tensor prod-
uct without reference to underlying spaces
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tensor product structurethe other stuff
= ?
Conceptually: not about properties of the individual,
but about relationships among the individuals
Mathematically: axiomatize an abstract tensor prod-
uct without reference to underlying spaces
1. Game plan: Which assumptions (i.e. which struc-
ture) on is needed to deduce physical phenomena?
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tensor product structurethe other stuff
= ?
Conceptually: not about properties of the individual,
but about relationships among the individuals
Mathematically: axiomatize an abstract tensor prod-
uct without reference to underlying spaces
1. Game plan: Which assumptions (i.e. which struc-
ture) on is needed to deduce physical phenomena?
2. Additional question: Does such an interaction struc-
ture appear elsewhere in our classical reality?
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Outcome 1a: Sheer ratio of results to assumptions
Hans Halvorson (2010) Editorial to: Deep Beauty: Understanding the Quan-
tum World through Mathematical Innovation, Cambridge University Press.
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Outcome 1a: Sheer ratio of results to assumptions
confirms that we are probing something very essential.
Hans Halvorson (2010) Editorial to: Deep Beauty: Understanding the Quan-
tum World through Mathematical Innovation, Cambridge University Press.
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Outcome 1a: Sheer ratio of results to assumptions
confirms that we are probing something very essential.
Outcome 1b: Exposing this structure has already helped
to solve open problems elsewhere. (e.g. 2 ICALP10)
EG: Ross Duncan & Simon Perdrix (2010)Rewriting measurement-based quan-
tum computations with generalised flow. ICALP10.
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Outcome 1a: Sheer ratio of results to assumptions
confirms that we are probing something very essential.
Outcome 1b: Exposing this structure has already helped
to solve open problems elsewhere. (e.g. 2 ICALP10)
Outcome 1c: Simple intuitive (but rigorous) dia-grammatic language, meanwhile adopted by others:
... we join the quantum picturalism revolution [1]
Lucien Hardy (2010) A formalism-local framework for general
probabilistic theories including quantum theory. arXiv:1005.5164
[1] Coecke (2010) Quantum picturalism. Contemporary Physics 51, 5983.
arXiv:0908.1787 (survey)
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Outcome 1a: Sheer ratio of results to assumptions
confirms that we are probing something very essential.
Outcome 1b: Exposing this structure has already helped
to solve open problems elsewhere. (e.g. 2 ICALP10)
Outcome 1c: Simple intuitive (but rigorous) dia-grammatic language, meanwhile adopted by others:
... we join the quantum picturalism revolution [1]
Lucien Hardy (2010) A formalism-local framework for general
probabilistic theories including quantum theory. arXiv:1005.5164
[1] Coecke (2010) Quantum picturalism. Contemporary Physics 51, 5983.
arXiv:0908.1787 (survey)
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Outcome 2a:Behaviors of matter:
=
f
f =
f f
f
ALICE
BOB
=
ALICE
BOB
f
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Outcome 2a:Behaviors of matter:
=
f
f =
f f
f
ALICE
BOB
=
ALICE
BOB
f
Meaning in language:
=
not
like
BobAlice
does
Alice not like
not
Bob
meaning vectors of words
pregroup grammar
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Outcome 2a:Behaviors of matter:
=
f
f =
f f
f
ALICE
BOB
=
ALICE
BOB
f
Meaning in language:
=
not
like
BobAlice
does
Alice not like
not
Bob
meaning vectors of words
pregroup grammar
Knowledge updating :
conditionalindependence
=P(C|AB)
A A
=
A
=
A
B
A
B
=
B
(BA) 1-A
C 1- C 1-
C
P(AB|C) P(A|C) P(B|C) P(C|A) P(C|B)
P(C|B) P(C|A)
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Outcome 2b: The structure is a true (quantum) logic:
- I can give you a demo backstage -
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A MINIMAL LANGUAGE
FOR QUANTUM REASONING
Abramsky & Coecke (2004) A categorical semantics for quantum protocols.arXiv:quant-ph/0402130
Coecke (2005) Kindergarten quantum mechanics.
arXiv:quant-ph/0510032
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(physical) data in the language
Systems:
A B C
Processes:
Af-A A
g-B B
h-C
Compound systems:
A B I A Cfg
-B D
Temporal composition:
Ahg
-C := Ag-B
h-C A
1A-A
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graphical notation
g f
g
f
f g f fg
Roger Penrose (1971) Applications of negative dimensional tensors.
In: Combinatorial Mathematics and its Applications. Academic Press.
Andre Joyal & Ross Street (1991) The geometry of tensor calculus I.
Advances in Mathematics 88, 55112.
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merely a new notation?
(g f) (k h) = (g k) (f h)
=
f h
g k
f h
g k
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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
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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graphical notation
: I A : A I : I I
A
A
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adjoint
f : A B
f
A
B
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adjoint
f : B A
f
B
A
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asserting (pure) entanglement
quantum
classical=
=
=
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quantum-like
A A
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quantum-like
A A
=
A
A
A
A
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quantum-like
A A
=
A
A
A
A
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quantum-like
A A
=
A
A
A
A
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quantum-like
A A
=
A
A
A
A
lik
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quantum-like
A A
=
A
A
A
A
lik
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quantum-like
ff=
lidi
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sliding
=f f
=f
f
lidi
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sliding
=f f
=f
f
In QM: cups = Bell-states, caps =Bell-effects, -rotations = transpose
classical data flow?
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classical data flow?
f=
f
ff
classical data flow?
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classical data flow?
f
=
f
classical data flow?
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classical data flow?
f
=
f
classical data flow?
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classical data flow?
f
ALICE
BOB
=
ALICE
BOB
f
quantum teleportation
Applying decorated normalization 3
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Applying decorated normalization 3
=
f
f
f f
Entanglement swapping
classical data flow?
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classical data flow?
f
=
f
g
g
gate teleportation computation
dagger compact categories
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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.
dagger compact categories
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gg p g
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. [Selinger 08] An equational statement between
expressions in dagger compact categorical language
holds if and only if it is derivable in the category of
finite dimensional Hilbert spaces, linear maps, tensorproduct, andadjoints.
dagger compact categories
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gg p g
In words: Any equation involving:
states, operations, effects
unitarity, adjoints (e.g. self-adjoint), projections
Bell-states/effects, transpose, conjugation
inner-product, trace, Hilbert-Schmidt norm
positivity, completely positive maps, ...
holds in quantum theory if and only ifit can be derived
in the graphical language via homotopy.
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A SLIGHTLY DIFFERENT LANGUAGE
FOR NATURAL LANGUAGE MEANING
Coecke, Sadrzadeh & Clark (2010) Mathematical Foundations for a Compo-
sitional Distributional Model of Meaning.
arXiv:1003.4394
the from-words-to-a-sentence process
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Consider meanings of words (e.g. vectors as in Google):
word 1 word 2 word n
...
?
How do we/machines compute meaning of sentences?
the from-words-to-a-sentence process
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Consider meanings of words (e.g. vectors as in Google):
word 1 word 2 word n
...
grammar
How do we/machines compute meaning of sentences?
the from-words-to-a-sentence process
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Information flow within a verb:
verb
object subject
the from-words-to-a-sentence process
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Information flow within a verb:
verb
object subject
Again we have:
=
going non-symmetric
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I
l
A Al
Al
Al
I I
r
Ar
A A Ar
r
I
A Al A A
A Al
r
A Ar
=
A
A
A
A
=A
A A
A
r
r
=
A
A
A
A
=A
A A
All
ll
r
r
Alice
does
not
like
Bob
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Alice
does
not
like
Bob
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Alice not like Bob
meaning vectors of words
not
grammar
does
Alice
does
not
like
Bob
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Alice like Bob
meaning vectors of words
grammar
not
Alice
does
not
like
Bob
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Alice like Bob
meaning vectors of words
grammar
not
Alice
does
not
like
Bob
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Alice like Bob
meaning vectors of words
grammar
not
= not
like
BobAlice
grammar
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Alice hates Bob
meaning vectors of words
g
f
f
states
measurements
analogy: non-local info-flows English (& French):
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English (& French):
Hindi:
Persian:
Arabic (and Hebrew):
Mehrnoosh Sadrzadeh (2008) Pregroup analysis of Persian sentences.
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THE EXTENDED LANGUAGE:
COMPLEMENTARITY & CLASSICALITY
observables
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observables m
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spiders =
m
....
....
n
such that, for k > 0:
m+mk
........
....
....
....
n+nk
=
....
....
Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812
observables
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Theorem 1. In any dagger symmetric monoidal cate-
gory families of spiders and dagger special commuta-
tive Frobenius algebra are in bijective correspondence.
observables m
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spiders =
m
....
....
n
such that, for k > 0:
m+mk
........
....
....
....
n+nk
=
....
....
observables
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Theorem 1. In any dagger symmetric monoidal cate-
gory families of spiders and dagger special commuta-
tive Frobenius algebra are in bijective correspondence.
Theorem 2. (Coecke-Pavlovic-Vicary) InFdHilb dag-
ger special commutative Frobenius algebra are exactlyorthonormal bases, namely those of copyable elts.
Coecke & Pavlovic (2007) Quantum measurement without sums. In: Mathe-
matics of Quantum Computing and Technology. quant-ph/0608035
Coecke, Pavlovic & Vicary (2008) A new description of orthogonal bases.
Mathematical Structures in Computer Science. 0810.0812
observables m
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spiders =
m
....
....
n
such that, for k > 0:
m+mk
........
....
....
....
n+nk
=
....
....
Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812
complementarity
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complementarity
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Coecke & Duncan (2008)Interacting quantum observables. arXiv:0906.4725
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environment
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ground =
f=
g f
=
g
f g
Thm. mixed states, CP maps, class. probabilities.
Coecke & Perdrix (2010)Environment and class. chan. ... arXiv:1004.1598
environment
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ground =
f=
g f
=
g
f g
Thm. mixed states, CP maps, class. probs in Hilb.
Coecke & Perdrix (2010)Environment and class. chan. ... arXiv:1004.1598
environment
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ground =
f=
f
=f
Thm. mixed states, CP maps, class. probabilities.
Coecke & Perdrix (2010)Environment and class. chan. ... arXiv:1004.1598
Prop 1:
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=
Prop 2:
=
Destructive measurement:
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=
Non-destructive measurement:
= =
Indeed measurement:
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=
Indeed controlled unitary:
=
key distribution
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=
=
key distribution
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=
=
phase groups and universality for QC
Translation to circuits and determinism for so-called
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measurement based quantum computations:
H
H
H
, {3}
, {2}
, {2}
, {3}, {2}
-
H
H
H
, {3}
, {2}
,
{2}, {2} , {3}
-
H
H
H
, {3}
, {3}
, {2}
, {2}
, {2}
-
, {2}
, {2}
, {2}
-
, {2}, {2}
, {2} , {2}
-
Ross Duncan & Simon Perdrix (2010) Rewriting measurement-based quantum
computations with generalised flow. ICALP10.
phase groups and quantum non-locality
Toy qubits vs. true quantum theory in one language:
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Spekkens qubit QM
stabilizer qubit QM=
Z2 Z2Z4
=local
non-local
Bob Coecke, Bill Edwards & Rob Spekkens (2010) Phase groups and the ori-
gin of non-locality for qubits. arXiv:1003.5005
entanglement classification
Tripartite SLOCC-classes as comm. Frobenius algs:
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GHZ = |000 + |111W = |001 + |010 + |100
=special CFAs
anti-special CFAs
=
=
=
Coecke & Aleks Kissinger (2010) The compositional structure of multipartite
quantum entanglement. ICALP10. arXiv:1002.2540
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PUNCHLINE ON LOGIC
Taking (de)composition as primitive (vs. an individual
entities properties) we get very far with very little!
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Taking (de)composition as primitive (vs. an individual
entities properties) we get very far with very little!
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An interaction logic is also present in natural lan-guage (= also the source of static orthodox logic).
Taking (de)composition as primitive (vs. an individual
entities properties) we get very far with very little!
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An interaction logic is also present in natural lan-guage (= also the source of static orthodox logic).
Overall stance: Quantum richness is not going to be
understood by weakening standard logical tools, but
by considering radically different (classical?) ones.
Taking (de)composition as primitive (vs. an individual
entities properties) we get very far with very little!
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An interaction logic is also present in natural lan-guage (= also the source of static orthodox logic).
Overall stance: Quantum richness is not going to be
understood by weakening standard logical tools, but
by considering radically different (classical?) ones.
Logic indeed (vs. quantum (non-)logic): Automa-
tion is demonstrated via the quantomatic software.
Dixon, Duncan & Kissinger. http://dream.inf.ed.ac.uk/projects/quantomatic/