The Logic of Quantum Mechanics - take II Bob Coecke Oxford Univ. Computing Lab. Quantum Group

Transcription

1 The Logic of Quantum Mechanics - take II Bob Coecke Oxford Univ. Computing Lab. Quantum Group ALICE f f BOB = f f = f f = ALICE BOB does not Alice not like BC (2010) Quantum picturalism. Contemporary physics 51, arxiv: BC & Ross Duncan (2011) Interacting quantum observables. New Journal of Physics 13, arxiv: BC, Mehrnoosh Sadrzadeh & Stephen Clark (2011) Mathematical foundations for a compositional distributional model of meaning. Linguistic Analysis - Lambek Festschrift. arxiv: BC & Aleks Kissinger (2010) The compositional structure of multipartite quantum entanglement. ICALP 10. arxiv: Lucas Dixon, R. Duncan, A. Kissinger & Alex Merry. quantomatic. Bob = Alice like not Bob

2 INTRODUCTION

3 genesis [von Neumann 1932] Formalized quantum mechanics in Mathematische Grundlagen der Quantenmechanik

4 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)

5 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 Quantum Mechanics in Annals of Mathematics.

6 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 Quantum Mechanics in Annals of Mathematics. [ ] many followed them,... and FAILED.

7 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 Quantum Mechanics in Annals of Mathematics. [ ] many followed them,... and FAILED.

8 the mathematics of it

9 the mathematics of it Hilber space stuff: continuum, field structure of complex numbers, vector space over it, inner-product, etc.

10 the mathematics of it Hilber space stuff: continuum, field structure of complex numbers, vector space over it, inner-product, etc. WHY?

11 the mathematics of it Hilber space stuff: continuum, field structure of complex numbers, vector space over it, inner-product, etc. WHY? von Neumann: only used it since it was available.

12 the mathematics of it Hilber space stuff: continuum, field structure of complex 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.

13 the mathematics of it Hilber space stuff: continuum, field structure of complex 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. it comes with little conceptual content

14 the mathematics of it Hilber space stuff: continuum, field structure of complex 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. it comes with little conceptual content SO WHAT ELSE SHOULD ONE USE THEN?

15 the physics of it

16 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition.

17 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition. Schrödinger (1935): the stuff which is the true soul of quantum theory is how quantum systems compose.

18 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition. Schrödinger (1935): the stuff which is the true soul of quantum theory is how quantum systems compose. Quantum Computer Scientists: Schrödinger is right!

19 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition. Schrödinger (1935): the stuff which is the true soul of quantum theory is how quantum systems compose. Quantum Computer Scientists: Schrödinger is right! 75 years later Birkhoff-von Neumann quantum logicians still fail to elegantly capture composition of quantum systems,... nor much else neiter,...

20 the game plan

21 the game plan Task 0. Solve: tensor product structure the other stuff =???

22 the game plan Task 0. Solve: tensor product structure the other stuff =??? i.e. axiomatize without reference to spaces.

23 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 structure) on is needed to deduce physical phenomena?

24 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 structure) on is needed to deduce physical phenomena? Task 2. Investigate wether such an interaction structure appear elsewhere in our classical reality

25 Outcome 1a: Sheer ratio of results to assumptions

26 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 Quantum World through Mathematical Innovation, Cambridge University Press.

27 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 Quantum World through Mathematical Innovation, Cambridge University Press.

28 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 ICALP 10) E.g.: Ross Duncan & Simon Perdrix (2010) Rewriting measurement-based quantum computations with generalised flow. ICALP 10.

29 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 ICALP 10) Outcome 1c: Framework is a simple intuitive (but rigorous) diagrammatic language, meanwhile adopted by others e.g. Lucien Hardy in arxiv: :... we join the quantum picturalism revolution [1] [1] BC (2010) Quantum picturalism. Contemporary Physics 51,

30 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 ICALP 10) Outcome 1c: Framework is a simple intuitive (but rigorous) diagrammatic language, meanwhile adopted by others e.g. Lucien Hardy in arxiv: :... we join the quantum picturalism revolution [1] [1] BC (2010) Quantum picturalism. Contemporary Physics 51,

31 Outcome 2a:

32 Outcome 2a: Behaviors of matter (Abramsky-C; LiCS 04, quant-ph/ ) : ALICE f f BOB = f f = f f = ALICE BOB Meaning in language (Clark-C-Sadrzadeh; Linguistic Analysis, arxiv: ) : meaning vectors of words does not Alice not like Bob pregroup grammar = like Alice not Bob Knowledge updating (C-Spekkens; Synthese, arxiv: ) : P(C AB) = P(AB C) P(A C) P(B C) P(C A) P(C B) A = conditional independence A = B A A B A = C C -1 C -1 P(C B) P(C A) B A (BA) -1

33 the logic of it

34 the logic of it WHAT IS LOGIC?

35 the logic of it WHAT IS LOGIC? Definitely not Birkhoff-von Neumann quantum logic, which, by definition, is logic without deduction, nor is it

36 the logic of it WHAT IS LOGIC? Definitely not Birkhoff-von Neumann quantum logic, which, by definition, is logic without deduction, nor is it some metaphysical jargon (cf. Putnam, Omnés,...) aiming to solve the measurement problem.

37 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language.

38 the logic of it WHAT IS LOGIC? Pragmatic option 1: 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)) Sick(a), Sick(b)

39 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason.

40 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason. Cf. the soft incarnation of AI in robotics, automated theorem proving, automated theory exploration,...

41 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason. Our framework appeals to both senses of logic, and moreover induces important new applications:

42 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason. Our framework appeals to both senses of logic, and moreover induces important new applications: From truth to meaning in natural language processing: (December 2010)

43 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason. Our framework appeals to both senses of logic, and moreover induces important new applications: From truth to meaning in natural language processing: (December 2010) Automated theorem generation for graphical theories:

44 BASIC TENSOR LOGIC Roger Penrose (1971) Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications, Academic Press. André Joyal and Ross Street (1991) The Geometry of tensor calculus I. Advances in Mathematics 88,

45 the data of tensor logic Box : = f output wire(s) input wire(s)

46 the data of tensor logic Box : = f output wire(s) input wire(s) Interpretation: wire := system ; box := process

47 the data of tensor logic Box : = f output wire(s) input wire(s) Interpretation: wire := system ; box := process one system: n subsystems: no system:... }{{} 1 } {{ } n }{{} 0

48 the connectives of tensor logic

49 the connectives of tensor logic sequential or causal or connected composition: g f g f

50 the connectives of tensor logic sequential or causal or connected composition: g f g f parallel or acausal or disconnected composition: f g f fg

51 merely a new notation?

52 merely a new notation? (g f) (k h) = (g k) (f h)

53 merely a new notation? (g f) (k h) = (f h) (g k) g f k h

54 merely a new notation? (f g) (h k) = (g k) (f h) g f k h

55 merely a new notation? (g f) (k h) = (g k) (f h) g k g k f h = f h

56 merely a new notation? (g f) (k h) = (g k) (f h) g k g k f h = f h 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

57 topological connectedness as a paradigm g f f fg connected disconnected

58 topological connectedness as a paradigm g f f fg connected This is a digital paradigm: disconnected

59 topological connectedness as a paradigm g f f fg connected disconnected This is a digital paradigm: 0 no information-flow 1 information-flow

60 = = = topological connectedness as a paradigm dot = ground = mixedness decoherence eigenstate complementarity = GHZ-state vs W-state

61 A MINIMAL LANGUAGE FOR QUANTUM REASONING Samson Abramsky & BC (2004) A categorical semantics for quantum protocols. In: IEEE-LiCS 04. quant-ph/ BC (2005) Kindergarten quantum mechanics. quant-ph/

62 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

63 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

64 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

65 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

66 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

67 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

68 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

69 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

70 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

71 graphical notation for states and effects ψ : I A π : A I π ψ : I I ψ π π ψ

72 adjoint f : A B B f A

73 adjoint f : B A A f B

74 = = asserting (pure) entanglement quantum classical =

75 = = asserting (pure) entanglement quantum classical = introduce parallel connection between systems :

76 quantum-like =

77 quantum-like =

78 sliding f f = f = f

79 classical data flow? f f f f =

80 classical data flow? f f =

81 classical data flow? f f =

82 classical data flow? ALICE f ALICE f = BOB BOB quantum teleportation

83 symbolically: dagger compact categories

84 symbolically: dagger compact categories Thm. [Kelly-Laplaza 80; Selinger 05] An equational statement between expressions in dagger compact categorical language holds if and only if it is derivable in the graphical notation via homotopy.

85 symbolically: dagger compact categories Thm. [Kelly-Laplaza 80; Selinger 05] An equational statement between expressions in dagger compact categorical language holds if and only if it is derivable in the graphical notation via homotopy. there are many models, i.e. concrete dagger compact categories, e.g. linear maps, relations,...

86 symbolically: dagger compact categories Thm. [Kelly-Laplaza 80; Selinger 05] An equational statement between expressions in dagger compact 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 dagger compact category of finite dimensional Hilbert spaces, linear maps, tensor product and adjoints.

87 symbolically: dagger compact categories 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 if it can be derived in the graphical language via homotopy.

88 MEANING IN NATURAL LANGUAGE BC, Mehrnoosh Sadrzadeh & Stephen Clark (2010) Mathematical foundations for a compositional distributional model of meaning. arxiv:

89 the from-words-to-a-sentence process Consider meanings of words (for Google these are vectors): word 1 word 2 word n...

90 the from-words-to-a-sentence process How do we/machines compute meaning of sentences?? word 1 word 2 word n...

91 the from-words-to-a-sentence process How do we/machines compute meaning of sentences? grammar word 1 word 2 word n...

92 the from-words-to-a-sentence process Information flow within a verb: object subject verb

93 = the from-words-to-a-sentence process Information flow within a verb: object subject verb Again we have:

94 grammar as pregroups Lambek 99 I ηl A A l A l A ɛl I Al A A A l Ar A A A A I ηr A r A A A r ɛ r I A A A r A A = = = A A l A l A A r A r = A l A l A r A r

95 grammar as pregroups Lambek 99 For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n = s

96 grammar as pregroups Lambek 99 For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n = s Diagrammatic typing: n n s -1 n-1 n

97 from gammer to meaning For noun type n and verb type 1 n s n 1 : n 1 n s n 1 n = s Diagrammatic meaning: flow flow n verb n

98 Alice does not like Bob

99 Alice does not like Bob grammar Alice does not not like Bob meaning vectors of words

100 Alice does not like Bob grammar Alice like Bob not meaning vectors of words

101 Alice does not like Bob grammar Alice like Bob not meaning vectors of words

102 Alice does not like Bob grammar Alice like Bob not meaning vectors of words = Alice not like Bob

103 experiment: word disambiguation E.g. what is saw in: Alice saw Bob with a saw. Edward Grefenstette & Mehrnoosh Sadrzadeh (2011) Experimental support for a categorical compositional distributional model of meaning. Accepted for: Empirical Methods in Natural Language Processing (EMNLP 11).

104 grammar Alice hates Bob meaning vectors of words f measurements f states

105 analogy: non-local info-flows English (& French): Hindi: Persian: Arabic (and Hebrew): Mehrnoosh Sadrzadeh (2008) Pregroup analysis of Persian sentences.

106 analogy: quantizing grammar! Topological quantum field theory: F : ncob FVect C :: V Grammatical quantum field theory: F : P regroup FVect R+ :: V Louis Crane was the first one to notice this analogy.

107 THE EXTENDED LANGUAGE: BASES AND COMPLEMENTARITY Coecke, Pavlovic & Vicary (2008) A new description of orthogonal bases. Mathematical Structures in Computer Science, TA. arxiv: BC & Ross Duncan (IS HERE) (2008) Interacting quantum observables. ICALP 08 & New Journal of Physics 13, arxiv:

108 commutative Frobenius algebras

109 commutative Frobenius algebras A commutative monoid is a set X with a binary map : X X X which is commutative, associative and unital i.e (a b) c = a (b c) a b = b a a 1 = a

110 commutative Frobenius algebras A commutative monoid is a set X with a binary map µ(, ) : X X X which is commutative, associative and unital i.e µ(µ(a, b), c) = µ(a, µ(b, c)) µ(a, b) = µ(b, a) µ(a, 1) = a

111 commutative Frobenius algebras A commutative monoid is a set X with a binary map µ : X X X which is commutative, associative and unital i.e µ (µ 1 X ) = µ (1 X µ) µ = µ σ µ (1 X e) = 1 X with: σ : X X X X :: (a, b) (b, a) e : { } X :: 1

112 commutative Frobenius algebras A commutative monoid is a set X with a binary map µ : X X X which is commutative, associative and unital i.e µ (µ 1 X ) = µ (1 X µ) µ = µ σ µ (1 X e) = 1 X A cocomutative comonoid is a set X with a binary map δ : X X X which is cocommutative, coassociative and counital i.e (δ 1 X ) δ = (1 X δ) δ δ = σ δ (1 X e ) δ = 1 X

113 commutative monoid cocommutative comonoid + Z2 :: B :: { (0, 0), (1, 1) 0 (0, 1), (1, 0) 1 { (0, 0), (0, 1), (1, 0) 0 (1, 1) 1... :: { 0 (0, 0) 1 (1, 1)

114 commutative Frobenius algebras A commutative monoid is vect. space V & lin. map µ : V V V which is commutative, associative and unital i.e µ (µ 1 V ) = µ (1 V µ) µ = µ σ µ (1 V e) = 1 V A cocomutative comonoid is vect. space V & lin. map δ : V V V which is cocommutative, coassociative and counital i.e (δ 1 V ) δ = (1 V δ) δ δ = σ δ (1 V e ) δ = 1 V

115 commutative monoid cocommutative comonoid ˆ+ Z2 :: { 00, , 10 1 ˆ+ Z 2 :: { 0 00, , 10 ˆ B :: ˆ B :: δ Z :: { δ X :: { δ Z :: δ X :: { { δ Y :: δ Y ::

116 commutative Frobenius algebras A commutative monoid is: such that = = = A cocomutative comonoid is: such that = = =

117 commutative Frobenius algebras A commutative Frobenius algebra is a commutative monoid and a commutative comonoid: which moreover satisfy: =

118 commutative monoid cocommutative comonoid ˆ+ Z2 :: { 00, , 10 1 ˆ+ Z 2 :: { 0 00, , 10 δ Z :: { δ X :: { δ Z :: δ X :: { { δ Y :: δ Y ::

119 bases in tensor logic The notion of a(n) (orthonormal) basis can be captured purely in the language of tensor logic: Thm. Dagger special commutative Frobenius algebras: ( ) ( ) = = = are the same thing as orthonormal bases; the basis vectors are the vectors copied by the comultiplication. BC, Dusko Pavlovic & Jamie Vicary (2008) A new description of orthogonal bases. Mathematical Structures in Computer Science. arxiv:

120 ... CFA normal form depends on inputs, outputs, loops:......

121 such that, for k > 0: spiders m {}}... { spiders =... }{{} m+m k {}}{ } {{ } n+n k = n Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/ ,

122 such that, for k > 0: spiders m {}}... { (co-)mult. =... }{{} m+m k {}}{ } {{ } n+n k = n......

123 such that, for k > 0: spiders m {}}... { (co-)mult. =... }{{} m+m k {}}{ } {{ } n+n k = n......

124 such that, for k > 0: spiders m {}}... { cups/caps =... }{{} m+m k {}}{ } {{ } n+n k = n Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/ ,

125 complementary bases

126 Thm. complementary bases BC & Ross Duncan (IS HERE) (2008) Interacting quantum observables. ICALP 08 & New Journal of Physics 13, arxiv:

127 quantum gates Z-spin: δ Z : i ii X-spin: δ X : ± ± ±

128 quantum gates i.e. (δ Z 1) (1 δ X) = = CNOT

129 quantum gates =?

130 quantum gates

131 quantum gates

132 quantum gates

133 quantum gates

134 classical communication and control

135 classical communication and control

136 classical communication and control

137 classical communication and control

138 classical communication and control

139 classical communication and control

140 classical communication and control

141 qubit 1 the leftmost, and qubit 4 is the rightmost. By virtue of the soundness of R and Proposition 10, if D P can be rewritten to a circuit-like diagram without any conditional operations, then the rewrite sequence constitutes a proof that the pattern computes the same operation as applications to QC models the derived circuit. Example 19. Returning to the CNOT pattern of Example 18, there is a rewrite Translating measurement based quantum computations sequence, the key steps of which are shown below, which reduces the D P to to thecircuits unconditional without circuit-like additional pattern for CNOT resources: introduced in Example 7. This proves two things: firstly that P indeed computes the CNOT unitary, and that the pattern P is deterministic. H π, {2} H π, {3} H π, {2} π, {3} π, {2} H H H π, {2} π, {3} π, {2} π, {2} π, {3} π, {2} H H H π, {2} π, {2} π, {3} π, {3} π, {2} π, {2} π, {2} π, {2} π, {2} π, {2} π, {2} One can clearly see in this example how the non-determinism introduced by measurements is corrected by conditional operations later in the pattern. The Ross possibility Duncan of performing (IS HERE) such & Simon corrections Perdrix depends (2010) onrewriting the geometry measurementbasetern, the quantum entanglement computations graph with implicitly generalised definedflow. by the ICALP 10. of the pat- pattern. Definition 20. Let P be a pattern; the geometry of P is an open graph γ(p) = (G, I, O) whose vertices are the qubits of P and where i j iff E ij occurs in the command sequence of P. Definition 21. Given a geometry Γ = ((V, E), I, O) we can define a diagram

142 applications to quantum foundations Spekkens toy qubit qubit theory and stabilizer quantum theory within one framework. (Non-)locality can be reduced to the phase group : Spekkens qubit QM stabilizer qubit QM = Z 2 Z 2 Z 4 = local non-local Z 4 Z 2 Z 2 BC, Bill Edwards (COMES HERE) & Robert W. Spekkens (2010) Phase groups and the origin of non-locality for qubits. arxiv:

143 automated theory exploration Lucas Dixon (Google), Ross Duncan (Brussels, IS HERE), Aleks Kissinger (Oxford, IS HERE AND HAS A POSTER ON THIS) & Alex Merry (Oxford); website:

144 THE EXTENDED LANGUAGE II: GHZ vs W ENTANGLEMENT BC & Aleks Kissinger (IS HERE) (2010) The compositional structure of multipartite quantum entanglement. ICALP 10. arxiv: BC, A. Kissinger, Alex Merry & Shibdas Roy (2011) The GHZ/W-calculus contains rational arithmetic. arxiv:

145 Defn. Special CFA (SCFA): =

146 = Defn. Special CFA (SCFA): Lem. Loop invertible special up to phase.

147 = = Defn. Special CFA (SCFA): Lem. Loop invertible special up to phase. Defn. Anti-special CFA (ACFA):

148 = = Defn. Special CFA (SCFA): Lem. Loop invertible special up to phase. Defn. Anti-special CFA (ACFA): Lem. [Herrmann] Loop disconnected antispecial.

149 C(F)As tripartite states A comultiplication & unit yield a tripartite state: =

150 = Theorem. If a CFA on C 2 is special, that is, it induces a GHZ-class Frobenius state, and vice versa.

151 = = Theorem. If a CFA on C 2 is special, that is, it induces a GHZ-class Frobenius state, and vice versa. Theorem. If a CFA on C 2 is anti-special, that is, it induces a W-class Frobenius state, and vice versa.

152 GHZ algebra ( ) ( 1 ) = = 1 = = ( 1 1 ) = ( ) =

153 W algebra ( ) ( 1 ) = = 0 = = ( 0 1 ) = ( ) =

154 Data: Rules: GHZ spiders m {}}{ n, m N }{{} n m+m k( 1) {}}{ } {{ } n+n k( 1) = m+m k( 1) {}}{ }{{} n+n k( 1)

155 Data: Rules: W exploding spiders { m }}... { } {{...,, } n m+m 1 {}}{ n, m N m+m 1 {}}{ } {{ } n+n 1 =... } {{ } n+n 1

156 Data: Rules: ACFA exploding spiders { m }}... { } {{...,, } n m+m k( 2) {}}{ } {{ } n+n k( 2) = n, m N m+m k( 2) {}}{ }{{} n+n k( 2)

157 GHZ-W interaction

158 Informatics information flow vs no information flow GHZ-W interaction Topology connected vs disconnected Algebra vs Entanglement GHZ vs W = = vs

159 Informatics information flow vs no information flow GHZ-W interaction Topology connected vs disconnected Calculus vs + Algebra vs Entanglement GHZ vs W = = vs

160 ( ) }{{} GHZ-mult. (( 1 z ) ( 1 z )) = ( 1 z z )

161 ( ) }{{} GHZ-mult. ( ) }{{} W-mult. (( 1 z (( 1 z ) ) ( 1 z ( 1 z )) )) = = ( 1 z z ( 1 z + z ) )

162 ( ) }{{} GHZ-mult. ( ) }{{} W-mult. Encoding states (( 1 z (( 1 z ( 1 z ) ) ( 1 z ( 1 z )) )) = = ( 1 z z ( 1 z + z ) ) ) as z GHZ =, W = +

163 ( ) }{{} GHZ-mult. ( ) }{{} W-mult. Encoding states (( 1 z (( 1 z ( 1 z ) ) ) ( 1 z ( 1 z )) )) = = ( 1 z z ( 1 z + z ) ) as z GHZ =, W = + GHZ and W interact via distributivity

164 challenge What can we do with stuff of this kind: Given that: GHZ/W-calculus encodes rational arithmetic We have automation support

165 Posters on this line of work: Aleks Kissinger: Synthesising physical theories. Raymond Lal: Causal structure in categorical quantum mechanics. Johan Paulsson: Towards a diagrammatic representation of classical data. Also from the Oxford gang: Shane Mansfield: Hardy s non-locality paradox and other possibilistic non-locality conditions. Want to learn this kind of category theory? Try: BC & E. O. Paquette (2010) Categories for the practicing physicist. In: New Structures for Physics. BC (Ed.), Springer. arxiv: