Categorical quantum mechanics

Transcription

1 Categorical quantum mechanics Chris Heunen 1 / 76

2 Categorical Quantum Mechanics? Study of compositional nature of (physical) systems Primitive notion: forming compound systems 2 / 76

3 Categorical Quantum Mechanics? Study of compositional nature of (physical) systems Primitive notion: forming compound systems Operational yet algebraic Why non-unit state vectors? Why non-hermitean operators? Why complex numbers? 2 / 76

4 Categorical Quantum Mechanics? Study of compositional nature of (physical) systems Primitive notion: forming compound systems Operational yet algebraic Why non-unit state vectors? Why non-hermitean operators? Why complex numbers? Powerful graphical calculus 2 / 76

5 Categorical Quantum Mechanics? Study of compositional nature of (physical) systems Primitive notion: forming compound systems Operational yet algebraic Why non-unit state vectors? Why non-hermitean operators? Why complex numbers? Powerful graphical calculus llows different interpretation in many different fields Physics: quantum theory, quantum information theory Computer science: logic, topology Mathematics: representation theory, quantum algebra 2 / 76

6 3 / 76

7 4 / 76

8 Outline Lecture 1: monoidal categories: graphical calculus dual objects: entanglement (co)monoids: no-cloning Lecture 2: Frobenius structures: observables bialgebras: complementarity complete positivity: mixed states 5 / 76

9 Part I Monoidal categories 6 / 76

10 Category = systems and processes: physical systems, and physical processes governing them; data types, and algorithms manipulating them; algebraic structures, and structure-preserving functions; logical propositions, and implications between them. 7 / 76

11 Category = systems and processes: physical systems, and physical processes governing them; data types, and algorithms manipulating them; algebraic structures, and structure-preserving functions; logical propositions, and implications between them. Monoidal category = category + parallelism: independent physical systems evolve simultaneously; running computer algorithms in parallel; products or sums of algebraic or geometric structures; using proofs of P and Q to prove conjunction (P and Q). 7 / 76

12 category C is monoidal when equipped with: a tensor product functor : C C C a unit object I Ob(C) an associator natural isomorphism ( B) C α,b,c (B C) a left unitor natural isomorphism I λ a right unitor natural isomorphism I ρ 8 / 76

13 This data must satisfy the triangle and pentagon equations: α,i,b ( I) B (I B) ρ id B B id λ B α,b C,D ( (B C) ) D ( (B C) D ) α,b,c id D ( ) ( B) C D ( id α B,C,D B (C D) ) α B,C,D ( B) (C D) α,b,c D

14 This data must satisfy the triangle and pentagon equations: α,i,b ( I) B (I B) ρ id B B id λ B α,b C,D ( (B C) ) D ( (B C) D ) α,b,c id D ( ) ( B) C D ( id α B,C,D B (C D) ) α B,C,D ( B) (C D) α,b,c D Theorem (coherence for monoidal categories): If the pentagon and triangle equations hold, then so does any well-typed equation built from α, λ, ρ and their inverses using,, and id. 9 / 76

15 Example: Hilbert spaces Hilbert spaces and bounded linear maps form a monoidal category: tensor product is the tensor product of Hilbert spaces unit object is one-dimensional Hilbert space C left unitors C H λ H right unitors H C ρ H H are unique linear maps with 1 u u H are unique linear maps with u 1 u associators (H J) K αh,j,k H (J K) are unique linear maps with (u v) w u (v w) 10 / 76

16 Example: sets and functions Sets and functions form a monoidal category Set: tensor product is Cartesian product of sets unit object is a chosen singleton set { } left unitors I λ right unitors I ρ are (, a) a are (a, ) a associators ( B) C α,b,c (B C) are functions ( (a, b), c ) ( a, (b, c) ) (Other tensor products exist.) 11 / 76

17 Example: sets and relations relation R B between sets is a subset R B R B 12 / 76

18 Example: sets and relations relation R B between sets is a subset R B R B B S C S R C ; = Different notion of process: nondeterministic evolution of states 12 / 76

19 Example: sets and relations relation R B between sets is a subset R B R B Composition is matrix multiplication, with OR and ND for + and. 12 / 76

20 Example: sets and relations Sets and relations form a monoidal category Rel: tensor product is Cartesian product of sets, acting on relations as: (a, c)(r S)(b, d) iff arb and csd unit object is a chosen singleton set = { } associators ( B) C α,b,c (B C) are catrelations ( (a, b), c ) ( a, (b, c) ) left unitors I λ right unitors I ρ are given by (, a) a are given by (a, ) a 13 / 76

21 Example: sets and relations Sets and relations form a monoidal category Rel: tensor product is Cartesian product of sets, acting on relations as: (a, c)(r S)(b, d) iff arb and csd unit object is a chosen singleton set = { } associators ( B) C α,b,c (B C) are catrelations ( (a, b), c ) ( a, (b, c) ) left unitors I λ right unitors I ρ are given by (, a) a are given by (a, ) a Cartesian product is not a categorical product in Rel: If Set is classical, and Hilb is quantum, Rel is in the middle 13 / 76

22 Graphical calculus For f B and B g C, draw their composition g f B as C g B f 14 / 76

23 Graphical calculus For f B and B g C, draw their composition g f B as C g B f For f B and C g D, draw their tensor product C f g B D as B f D C g Time runs upwards, space runs sideways 14 / 76

24 Graphical calculus The tensor unit is drawn as the empty diagram: 15 / 76

25 Graphical calculus The tensor unit is drawn as the empty diagram: Unitors are also not drawn: B C λ ρ α,b,c Coherence is essential: as there can only be a single morphism built from associators and unitors of given type, it doesn t matter that their depiction encodes no information 15 / 76

26 Graphical calculus For example, interchange law: (g f) (j h) = (g j) (f h) 16 / 76

27 Graphical calculus For example, interchange law: (g f) (j h) = (g j) (f h) f g h j C B F E D = f g h j C B F E D 16 / 76

28 Graphical calculus: sound and complete Think of diagram as within rectangular region of R n, with wires terminating at upper and lower boundaries only, morphisms as points. Two diagrams are isotopic when one can be deformed continuously into the other, keeping the boundaries fixed. 17 / 76

29 Graphical calculus: sound and complete Think of diagram as within rectangular region of R n, with wires terminating at upper and lower boundaries only, morphisms as points. Two diagrams are isotopic when one can be deformed continuously into the other, keeping the boundaries fixed. Theorem (correctness): well-formed equation between morphisms in a monoidal category follows from the axioms iff it holds in the graphical calculus up to planar isotopy. 17 / 76

30 Graphical calculus: sound and complete Think of diagram as within rectangular region of R n, with wires terminating at upper and lower boundaries only, morphisms as points. Two diagrams are isotopic when one can be deformed continuously into the other, keeping the boundaries fixed. Theorem (correctness): well-formed equation between morphisms in a monoidal category follows from the axioms iff it holds in the graphical calculus up to planar isotopy. Soundness: algebraic equality graphical isotopy Completeness: algebraic equality graphical isotopy 17 / 76

31 States state of an object in a monoidal category is a morphism I. a Tensor unit is trivial system; state is way to bring into existence 18 / 76

32 States state of an object in a monoidal category is a morphism I. a Tensor unit is trivial system; state is way to bring into existence in Hilb: linear functions C H, correspond to elements of H in Set: functions { }, correspond to elements of in Rel: relations { } R, correspond to subsets of 18 / 76

33 Joint states morphism I c B is a joint state. B c 19 / 76

34 Joint states morphism I c B is a joint state. It is a product state when: B B c = a b 19 / 76

35 Joint states morphism I c B is a joint state. It is a product state when: B B c = a b It is entangled when it is not a product state 19 / 76

36 Joint states morphism I c B is a joint state. It is a product state when: B B = c a b It is entangled when it is not a product state entangled states in Hilb: vectors of H K with Schmidt rank > 1 entangled states in Set: don t exist entangled states in Rel: non-square subsets of B 19 / 76

37 Braiding monoidal category is braided when equipped with natural iso B σ,b B satisfying the hexagon equations (B C) σ,b C (B C) σ B,C ( B) C C ( B) α 1,B,C α 1 B,C, α,b,c α C,,B ( B) C B (C ) (B C) (C ) B σ,b id C id B σ,c (B ) C B ( C) α B,,C id σ B,C (C B) α 1,C,B σ,c id B ( C) B 20 / 76

38 Braiding Draw σ as and its inverse as 21 / 76

39 Braiding Invertibility: = = Naturality: f g = g f f g = g f Hexagon: = = 21 / 76

40 Braiding Invertibility: = = Naturality: f g = g f f g = g f Hexagon: = = Theorem (correctness): well-formed equation between morphisms in a braided monoidal category follows from the axioms iff it holds in the graphical language up to 3-dimensional isotopy. 21 / 76

41 Symmetry braided monoidal category is symmetric when σ B, σ,b = id B Graphically: no knots = 22 / 76

42 Symmetry braided monoidal category is symmetric when σ B, σ,b = id B Graphically: no knots = Theorem (correctness): well-formed equation between morphisms in a symmetric monoidal category follows from the axioms iff it holds in graphical language up to 4-dimensional isotopy. in Hilb: linear extension of a b b a in Set: function (a, b) (b, a) in Rel: relation (a, b) (b, a) 22 / 76

43 Scalars scalar in a monoidal category is a morphism I a I a 23 / 76

44 Scalars scalar in a monoidal category is a morphism I a I a Lemma: in a monoidal category, scalar composition is commutative Proof: either algebraically: λ 1 I I ρ 1 I b I a a λ 1 I I ρ 1 I b I I I id I b λ I ρ I I I I I a id I id I b a id I λ I ρ I I I 23 / 76

45 Scalars scalar in a monoidal category is a morphism I a I Lemma: in a monoidal category, scalar composition is commutative Proof: or graphically: b a a = a b 23 / 76

46 Scalar multiplication Scalar multiplication a f B of scalar I a I and morphism f B s f satisfies many familiar properties in any monoidal category: id I f = f a b = a b a (b f) = (a b) f (b g) (a f) = (b a) (g f) 24 / 76

47 Scalar multiplication Scalar multiplication a f B of scalar I a I and morphism f B s f satisfies many familiar properties in any monoidal category: id I f = f a b = a b a (b f) = (a b) f (b g) (a f) = (b a) (g f) In our examples: in Hilb: a f is the morphism x af(x) in Set: id 1 f = f is trivial in Rel: true R = R and false R = 24 / 76

48 Dagger dagger on a category C is a contravariant functor : C satisfying = on objects and f = f on morphisms. C Hilb is a dagger category using adjoints Rel is a dagger category using converse: br a iff arb Set is not a dagger category 25 / 76

49 Dagger dagger on a category C is a contravariant functor : C satisfying = on objects and f = f on morphisms. C Hilb is a dagger category using adjoints Rel is a dagger category using converse: br a iff arb Set is not a dagger category Graphically: flip about horizontal axis B f f B 25 / 76

50 Dagger dagger on a category C is a contravariant functor : C satisfying = on objects and f = f on morphisms. C Hilb is a dagger category using adjoints Rel is a dagger category using converse: br a iff arb Set is not a dagger category Graphically: flip about horizontal axis B f f B 25 / 76

51 Dagger morphism f in a dagger category is: self-adjoint when f = f unitary when f f = id and f f = id positive when f = g g for some g 26 / 76

52 Dagger morphism f in a dagger category is: self-adjoint when f = f unitary when f f = id and f f = id positive when f = g g for some g In a monoidal dagger category: (f g) = f g the associators and unitors are unitary 26 / 76

53 Dagger morphism f in a dagger category is: self-adjoint when f = f unitary when f f = id and f f = id positive when f = g g for some g In a monoidal dagger category: (f g) = f g the associators and unitors are unitary In a braided/symmetric monoidal dagger category, the braiding is additionally unitary 26 / 76

54 Part II Dual objects 27 / 76

55 Dual objects n object L is left-dual to an object R, and R is right-dual to L, written L R, when there are morphisms I η R L and L R ε I with: id L ρ 1 L id L η L L I L (R L) α 1 L,R,L L I L (L R) L λ L ε id L λ 1 R η id R R I R (R L) R id R α R,L,R R R I R (L R) ρ R id R ε 28 / 76

56 Dual objects n object L is left-dual to an object R, and R is right-dual to L, written L R, when there are morphisms I η R L and L R ε I with: = = where we draw η and ε as: R L L R 28 / 76

57 Dual objects: examples Every finite-dimensional Hilbert space H is both right dual and left dual to its dual Hilbert space H : cap H H C is evaluation: φ ψ ψ φ cup C H H is maximally entangled state: 1 i i i for any orthonormal basis { i } Infinite-dimensional Hilbert spaces do not have duals 29 / 76

58 Dual objects: examples Every finite-dimensional Hilbert space H is both right dual and left dual to its dual Hilbert space H : cap H H C is evaluation: φ ψ ψ φ cup C H H is maximally entangled state: 1 i i i for any orthonormal basis { i } Infinite-dimensional Hilbert spaces do not have duals In Rel, every object is self-dual: cap cup 1 1 is (a, a) for all a is (a, a) 29 / 76

59 Map-state duality The category Set only has duals for singletons. The name I f B and coname B f I of a morphism f B, given dual objects and B B, are B f f B 30 / 76

60 Map-state duality The category Set only has duals for singletons. The name I f B and coname B f I of a morphism f B, given dual objects and B B, are B f f B Conversely, B B f = f Proof: There is only one function 1, so all conames B 1 are equal, so all functions B are equal. 30 / 76

61 Dual objects: properties Robustly defined: Suppose L R. Then L R iff R R. If (L, R, η, ε) and (L, R, η, ε ) are dualities, then ε = ε. 31 / 76

62 Dual objects: properties Robustly defined: Suppose L R. Then L R iff R R. If (L, R, η, ε) and (L, R, η, ε ) are dualities, then ε = ε. Monoidal: lways I I. If L R and L R, then L L R R. R R = = L L L L L L 31 / 76

63 Dual objects: properties Robustly defined: Suppose L R. Then L R iff R R. If (L, R, η, ε) and (L, R, η, ε ) are dualities, then ε = ε. Monoidal: lways I I. If L R and L R, then L L R R. Symmetric: If L R in a braided monoidal category, then also R L. = = 31 / 76

64 Duals functor For f B and, B B, the right dual B f is defined as: f := f B B 32 / 76

65 Duals functor For f B and, B B, the right dual B f is defined as: f := f =: f B B B 32 / 76

66 Duals functor For f B and, B B, the right dual B f is defined as: f := f =: f B B B Examples: in FHilb: usual dual f : K in Rel: R = R H given by f (e) = e f 32 / 76

67 Duals functor: properties Lemma: (g f) = f g, and f = f f = f 33 / 76

68 Duals functor: properties Lemma: (g f) = f g, and f = f f = f Lemma: B ( B) ( B) ε B η ( B) B 33 / 76

69 Compact categories symmetric monoidal category is compact when every object has a (simultaneously left and right) dual. Theorem (correctness): well-formed equation between morphisms in a compact category follows from the axioms if and only if it holds in the graphical language up to spatial oriented isotopy. f 34 / 76

70 Compact categories symmetric monoidal category is compact when every object has a (simultaneously left and right) dual. Theorem (correctness): well-formed equation between morphisms in a compact category follows from the axioms if and only if it holds in the graphical language up to spatial oriented isotopy. f = f 34 / 76

71 Compact categories symmetric monoidal category is compact when every object has a (simultaneously left and right) dual. Theorem (correctness): well-formed equation between morphisms in a compact category follows from the axioms if and only if it holds in the graphical language up to spatial oriented isotopy. f = f = f 34 / 76

72 Compact categories symmetric monoidal category is compact when every object has a (simultaneously left and right) dual. Theorem (correctness): well-formed equation between morphisms in a compact category follows from the axioms if and only if it holds in the graphical language up to spatial oriented isotopy. f = f = f = f 34 / 76

73 Dagger dual objects Lemma: In a monoidal dagger category, L R R L. 35 / 76

74 Dagger dual objects Lemma: In a monoidal dagger category, L R R L. In a symmetric monoidal dagger category, a dagger dual has: η = ε 35 / 76

75 Dagger dual objects Lemma: In a monoidal dagger category, L R R L. In a symmetric monoidal dagger category, a dagger dual has: η = ε Lemma: Dagger dualities correspond to maximally entangled states η η = ε η = ε η = 35 / 76

76 Dagger dual objects Lemma: In a monoidal dagger category, L R R L. In a symmetric monoidal dagger category, a dagger dual has: η = ε Lemma: Dagger dualities correspond to maximally entangled states η η = ε η = ε η = Dagger duals, and hence maximally entangled states, are unique up to unique unitary 35 / 76

77 Compact dagger categories compact dagger category is both compact and dagger, and duals are dagger duals. ( ) = ( ) = 36 / 76

78 Compact dagger categories compact dagger category is both compact and dagger, and duals are dagger duals. ( ) = ( ) = Lemma: Duals and daggers commute (f ) = f = f = f = (f ) Conjugation is the functor ( ) := ( ) = ( ) 36 / 76

79 Conjugation f f f : B f : B conjugation f f f : B f : B dual dagger 37 / 76

80 Traces The trace of f is the scalar f 38 / 76

81 Traces The trace of f is the scalar f Examples: in FHilb, it is the ordinary trace in Rel, it detects fixed points 38 / 76

82 Traces The trace of f is the scalar f Examples: in FHilb, it is the ordinary trace in Rel, it detects fixed points Lemma: Trace is cyclic, Tr(f g) = Tr(f) Tr(g), and Tr(f ) = Tr(f) g f = f g = f g 38 / 76

83 Dimension The dimension of is the scalar dim() := Tr(id ) Lemma: dim(i) = id I dim( B) = dim() dim(b) if B then dim() = dim(b) 39 / 76

84 Dimension The dimension of is the scalar dim() := Tr(id ) Lemma: dim(i) = id I dim( B) = dim() dim(b) if B then dim() = dim(b) infinite-dimensional Hilbert spaces do not have duals 39 / 76

85 Teleportation In FHilb: begin with a single system L prepare a joint system R L in a maximally entangled state perform a joint measurement on the first two systems perform a unitary operation on the remaining system L U i U i L 40 / 76

86 Teleportation In FHilb: begin with a single system L prepare a joint system R L in a maximally entangled state perform a joint measurement on the first two systems perform a unitary operation on the remaining system L L U i U i U i = U i L L 40 / 76

87 Teleportation In FHilb: begin with a single system L prepare a joint system R L in a maximally entangled state perform a joint measurement on the first two systems perform a unitary operation on the remaining system L L L U i U i U i = U i = L L L 40 / 76

88 Teleportation In FHilb: begin with a single system L prepare a joint system R L in a maximally entangled state perform a joint measurement on the first two systems perform a unitary operation on the remaining system L L L L U i U i U i = U i = = L L L L 40 / 76

89 Teleportation In FHilb: begin with a single system L prepare a joint system R L in a maximally entangled state perform a joint measurement on the first two systems perform a unitary operation on the remaining system L L L L U i U i U i = U i = = L L L L In Rel: encrypted communication using one-time pad 40 / 76

90 Part III (Co)monoids 41 / 76

91 Comonoids comonoid in a monoidal category is an object with comultiplication d and counit e I satisfying d d = d d d e = = e d 42 / 76

92 Comonoids comonoid in a monoidal category is an object with comultiplication and counit I satisfying = = = Examples: in Set, any object has unique cocommutative comonoid with comultiplication a (a, a) and counit a in Rel, any group forms a comonoid with comultiplication g (h, h 1 g) and counit 1 in FHilb, any choice of basis {e i } gives cocommutative comonoid with comultiplication e i e i e i and counit e i 1 42 / 76

93 Monoids monoid in a monoidal category consists of maps I satisfying associativity and unitality. Lemma: In braided monoidal category, two (co)monoids combine Lemma: In monoidal dagger category, monoid gives comonoid 43 / 76

94 Pair of pants Map-state duality: composition g f becomes I g f. g f = g f 44 / 76

95 Pair of pants Map-state duality: composition g f becomes I g f. Lemma: If, then is pair of pants monoid 44 / 76

96 Pair of pants Map-state duality: composition g f becomes I g f. Lemma: If, then is pair of pants monoid Example: Pair of pants on C n in FHilb is n-by-n matrices M n Proof: define (C n ) C n M n by j i e ij 44 / 76

97 Pair of pants: one size fits all Map-state duality: composition g f becomes I g f. Lemma: If, then is pair of pants monoid Example: Pair of pants on C n in FHilb is n-by-n matrices M n Proof: define (C n ) C n M n by j i e ij Proposition: ny monoid (,, ) embeds into (,, ) R = 44 / 76

98 Cloning braided monoidal category has cloning if there is natural d with cocommutativity, coassociativity, d I = ρ I, and B B B B d d B = d B B B 45 / 76

99 Cloning braided monoidal category has cloning if there is natural d with cocommutativity, coassociativity, d I = ρ I, and B B B B d d B = d B B B Set has cloning, but compact categories like Rel or FHilb cannot Lemma: If compact category has cloning, then = 45 / 76

100 Cloning braided monoidal category has cloning if there is natural d with cocommutativity, coassociativity, d I = ρ I, and B B B B d d B = d B B B Set has cloning, but compact categories like Rel or FHilb cannot Lemma: If compact category has cloning, then... Proof: First, consider the following equality ( ). = d I = d = d d 45 / 76

101 Cloning braided monoidal category has cloning if there is natural d with cocommutativity, coassociativity, d I = ρ I, and B B B B d d B = d B B B Set has cloning, but compact categories like Rel or FHilb cannot Lemma: If compact category has cloning, then = d d = = d d 45 / 76

102 No-cloning Theorem: If braided monoidal category with duals has cloning, then f = Tr(f) id for any f 46 / 76

103 No-cloning Theorem: If braided monoidal category with duals has cloning, then f = Tr(f) id for any f Proof: First, consider equation ( ): = = 46 / 76

104 No-cloning Theorem: If braided monoidal category with duals has cloning, then f = Tr(f) id for any f Proof: First, consider equation ( ). Then: f = f = f = f 46 / 76

105 No-cloning Theorem: If braided monoidal category with duals has cloning, then f = Tr(f) id for any f Proof: First, consider equation ( ). Then: f f = f = = f 46 / 76

106 Summary Monoidal categories scalars, sound and complete graphical calculus Dual objects entanglement, teleportation, encrypted communication Monoids no cloning 47 / 76

107 Part IV Frobenius structures 48 / 76

108 Frobenius structure Frobenius structure in a monoidal category is pair of comonoid (,, ) and monoid (,, ) satisfying Frobenius law: = If =, called dagger Frobenius structure. 49 / 76

109 Frobenius structure Frobenius structure in a monoidal category is pair of comonoid (,, ) and monoid (,, ) satisfying Frobenius law: = If =, called dagger Frobenius structure. Example in FHilb: copying an orthogonal basis δ ij e i e j e i δ ij e j e j = e i e i e j e i e j 49 / 76

110 Frobenius structure Frobenius structure in a monoidal category is pair of comonoid (,, ) and monoid (,, ) satisfying Frobenius law: = If =, called dagger Frobenius structure. Example in FHilb: matrix algebra δ jk m e im e ml m e im δ jk e ml e km = e mj e ij e kl e ij e kl 49 / 76

111 Frobenius structure Frobenius structure in a monoidal category is pair of comonoid (,, ) and monoid (,, ) satisfying Frobenius law: = If =, called dagger Frobenius structure. Example in FHilb: group algebra of finite group k gk 1 kh k gk k 1 h k 1 = k g h g h 49 / 76

112 Frobenius structure Frobenius structure in a monoidal category is pair of comonoid (,, ) and monoid (,, ) satisfying Frobenius law: = If =, called dagger Frobenius structure. Example in Rel: set of morphisms of groupoid k l k l x h l 1 = g 1 k x g h g h 49 / 76

113 Frobenius structure Frobenius structure in a monoidal category is pair of comonoid (,, ) and monoid (,, ) satisfying Frobenius law: = If =, called dagger Frobenius structure. Example in any compact dagger category: pair of pants = 49 / 76

114 Frobenius law Lemma: ny Frobenius structure satisfies: = = Proof: Suffices to prove one of the equalities = = = = = 50 / 76

115 Classical structures classical structure is a special and commutative Frobenius structure = = 51 / 76

116 Classical structures classical structure is a special and commutative Frobenius structure = = Examples: in FHilb: copying orthonormal basis is classical structure in FHilb: matrix algebra only special when trivial in FHilb: group algebra only special when trivial in Rel: groupoid always special in general: pair of pants only special when trivial 51 / 76

117 Symmetry Frobenius structure in monoidal category is symmetric when: = 52 / 76

118 Symmetry Frobenius structure in monoidal category is symmetric when: = In braided monoidal category, this is equivalent to: = 52 / 76

119 Symmetry Frobenius structure in monoidal category is symmetric when: = In braided monoidal category, this is equivalent to: = Examples: in FHilb: copying orthonormal basis is symmetric in FHilb: matrix algebra symmetric as Tr(ab) = Tr(ba) in FHilb: group algebra symmetric as inverses are two-sided in Rel: groupoid symmetric as inverses are two-sided in general: pair of pants symmetric 52 / 76

120 Self-duality Proposition: If (,,,, ) Frobenius structure in monoidal category, then is self-dual = = 53 / 76

121 Self-duality Proposition: If (,,,, ) Frobenius structure in monoidal category, then is self-dual = = Proof: Snake equation: = = = 53 / 76

122 Self-duality Proposition: If (,,,, ) Frobenius structure in monoidal category, then is self-dual = = Conversely, monoid (,, ) forms Frobenius structure with some comonoid (,, ) iff allows nondegenerate form: map : I with part of self-duality. 53 / 76

123 Normal forms Two ways to think about graphical calculus diagram: representing morphism; shorthand for e.g. linear map entity in its own right; can be manipulated by replacing parts 54 / 76

124 }{{} m 54 / 76 Normal forms Two ways to think about graphical calculus diagram: representing morphism; shorthand for e.g. linear map entity in its own right; can be manipulated by replacing parts Theorem: if (,,,, ) is Frobenius structure, any connected morphism m n built out of finitely many pieces,,, and id, using and equals normal form n {}}{

125 Normal forms Two ways to think about graphical calculus diagram: representing morphism; shorthand for e.g. linear map entity in its own right; can be manipulated by replacing parts Theorem: if (,,,, ) is special Frobenius structure, any connected morphism m n built out of finitely many pieces,,, and id, using and equals normal form n {}}{ } {{ } m 54 / 76

126 Normal forms Two ways to think about graphical calculus diagram: representing morphism; shorthand for e.g. linear map entity in its own right; can be manipulated by replacing parts Theorem: if (,,,, ) is special commutative Frobenius structure, any connected morphism m n built out of finitely many pieces,,, and id and, using and equals normal form n {}}{ } {{ } m 54 / 76

127 Involutive monoids Map-state duality: f f is involution on pair of pants = H H 55 / 76

128 Involutive monoids Map-state duality: f f is involution on pair of pants = H H anti-linear, so ; but = (H H) H H morphism to opposite monoid: (g f) = f g 55 / 76

129 Involutive monoids Map-state duality: f f is involution on pair of pants = H H anti-linear, so ; but = (H H) H H morphism to opposite monoid: (g f) = f g If (, m, u) monoid, dagger dual, then (, m, u ) monoid too 55 / 76

130 Involutive monoids Map-state duality: f f is involution on pair of pants = H H anti-linear, so ; but = (H H) H H morphism to opposite monoid: (g f) = f g If (, m, u) monoid, dagger dual, then (, m, u ) monoid too Monoid on object with dagger dual is involutive monoid when equipped with monoid morphism i satisfying i i = id i i = 55 / 76

131 Involutive monoids Map-state duality: f f is involution on pair of pants = H H anti-linear, so ; but = (H H) H H morphism to opposite monoid: (g f) = f g If (, m, u) monoid, dagger dual, then (, m, u ) monoid too Monoid on object with dagger dual is involutive monoid when equipped with monoid morphism i satisfying i i = id B B i i = i B = f f i Morphisms f B of involutive monoids satisfy i B f = f i 55 / 76

132 Frobenius law and dagger set of maps closed under composition = submonoid of pair of pants. 56 / 76

133 Frobenius law and dagger set of maps closed under composition and dagger = involutive submonoid of pair of pants. 56 / 76

134 Frobenius law and dagger set of maps closed under composition and dagger = involutive submonoid of pair of pants. Theorem: Let be duals. Monoid (,, ) is dagger Frobenius structure iff Cayley embedding is involutive monoid morphism with i = Frobenius law = coherence law between dagger and closure 56 / 76

135 Frobenius law and dagger set of maps closed under composition and dagger = involutive submonoid of pair of pants. Theorem: Let be duals. Monoid (,, ) is dagger Frobenius structure iff Cayley embedding is involutive monoid morphism with i = Frobenius law = coherence law between dagger and closure matrix algebra in FHilb: involution M n M n is f f groupoid in Rel: involution G G is g g 1 pair of pants in general: involution invisible ( ) = = 56 / 76

136 Classification Corollary: Dagger Frobenius structures in FHilb are C*-algebras Proof: Correspond to M n closed under addition, scalar multiplication, matrix multiplication, adjoint, and contain identity 57 / 76

137 Classification Corollary: Dagger Frobenius structures in FHilb are C*-algebras Proof: Correspond to M n closed under addition, scalar multiplication, matrix multiplication, adjoint, and contain identity Corollary: Classical structures in FHilb are orthonormal bases Proof: If M k1 M kn commutive must have k 1 = = k n = 1 57 / 76

138 Classification Corollary: Dagger Frobenius structures in FHilb are C*-algebras Proof: Correspond to M n closed under addition, scalar multiplication, matrix multiplication, adjoint, and contain identity Corollary: Classical structures in FHilb are orthonormal bases Proof: If M k1 M kn commutive must have k 1 = = k n = 1 Theorem: Special dagger Frobenius structures in Rel are groupoids id dom(f) f id dom(f) f g = f f id cod(f) f id cod(f) 57 / 76

139 Classification Corollary: Dagger Frobenius structures in FHilb are C*-algebras Proof: Correspond to M n closed under addition, scalar multiplication, matrix multiplication, adjoint, and contain identity Corollary: Classical structures in FHilb are orthonormal bases Proof: If M k1 M kn commutive must have k 1 = = k n = 1 Theorem: Special dagger Frobenius structures in Rel are groupoids id dom(f) f id dom(f) f g = f f id cod(f) f id cod(f) Corollary: Classical structures in Rel are abelian groupoids 57 / 76

140 Phases state I a of a Frobenius structure is a phase when a a = = a a 58 / 76

141 Phases state I a of a Frobenius structure is a phase when a a = = a a Proposition: phases of dagger Frobenius structure in (braided) monoidal dagger category form (abelian) phase group 58 / 76

142 Phases state I a of a Frobenius structure is a phase when a a = = a a Proposition: phases of dagger Frobenius structure in (braided) monoidal dagger category form (abelian) phase group phase group of C*-algebra is its unitary group phase group of orthonormal basis are powers of circle group phase group of a group is group itself phase group of pair or pants are unitary endomorphisms = f f = f f 58 / 76

143 Part V Complementarity 59 / 76

144 Complementarity Symmetric dagger Frobenius structures and on the same object in a braided monoidal dagger category are complementary when = = 60 / 76

145 Complementarity Symmetric dagger Frobenius structures and on the same object in a braided monoidal dagger category are complementary when = = Black and white not obviously interchangeable. But by symmetry = = 60 / 76

146 Complementarity: examples Proposition: classical structures in FHilb are complementary iff they copy mutually unbiased orthonormal bases d i e j 2 = 1 dim(h) 61 / 76

147 Complementarity: examples Proposition: classical structures in FHilb are complementary iff they copy mutually unbiased orthonormal bases d i e j 2 = 1 dim(h) Lemma: if dagger self-dual in braided monoidal dagger category, then pair of pants and twisted knickers on are complementary Symmetric Frobenius structure gives complementary pair on 61 / 76

148 Complementarity in Rel Example: Let G and H be nontrivial groups, = G H, G totally disconnected groupoid: objects G and G(g, g) = H; H totally disconnected groupoid: objects H and H(h, h) = G. Then G and H give rise to complementary Frobenius structures. 62 / 76

149 Complementarity in Rel Example: Let G and H be nontrivial groups, = G H, G totally disconnected groupoid: objects G and G(g, g) = H; H totally disconnected groupoid: objects H and H(h, h) = G. Then G and H give rise to complementary Frobenius structures. Proposition: equivalent for groupoids G, H on set of morphisms: give complementary Frobenius structures map Ob(G) Ob(H), a ( dom G (a), dom H (a) ) is bijective 62 / 76

150 Complementarity and dagger Proposition: symmetric dagger Frobenius structures in braided category complementary iff the following morphism is unitary 63 / 76

151 Complementarity and dagger Proposition: symmetric dagger Frobenius structures in braided category complementary iff the following morphism is unitary Proof: = = 63 / 76

152 Oracles n oracle is a morphism f B together with Frobenius structures on and on B such that the following morphism is unitary: B f B 64 / 76

153 Oracles n oracle is a morphism f B together with Frobenius structures on and on B such that the following morphism is unitary: B f B Example: Extension of function between mutually unbiased bases 64 / 76

154 Oracles n oracle is a morphism f B together with Frobenius structures on and on B such that the following morphism is unitary: B f B Example: Extension of function between mutually unbiased bases Proposition: Let (, ), (B, ) and (B, ) be symmetric dagger Frobenius structures. Self-conjugate comonoid morphism (, ) f (B, ) is oracle (, ) (B, ) iff complements 64 / 76

155 Deutsch Josza algorithm Let function {1,..., n} f {0, 1} be promised balanced or constant. Extend to oracle C) n C 2 ; latter with computational and X bases. Write b =. ( 1/ 2 1/ 2 1/ n C 2 Measure the first system f pply a unitary map 1/ n b Prepare initial states 65 / 76

156 Deutsch Josza algorithm Let function {1,..., n} f {0, 1} be promised balanced or constant. Extend to oracle C) n C 2 ; latter with computational and X bases. Write b =. ( 1/ 2 1/ 2 1/ n C 2 Measure the first system f pply a unitary map 1/ n b Prepare initial states History is certain when f constant, impossible when f balanced. 65 / 76

157 Bialgebras Complementarity related to Hopf algebras = = = = 66 / 76

158 Bialgebras Complementarity related to Hopf algebras = = = = Strong complementarity = complementarity + bialgebra (in FHilb and Rel: complementary only if commutative bialgebra) 66 / 76

159 Bialgebras Complementarity related to Hopf algebras = = = = Strong complementarity = complementarity + bialgebra (in FHilb and Rel: complementary only if commutative bialgebra) Theorem: the strongly complementary classical structures in FHilb are the group algebras 66 / 76

160 Part VI Complete positivity 67 / 76

161 Morphisms of Frobenius structures Lemma: If a morphism between Frobenius structures preserves (co)multiplication and (co)unit, then it is an isomorphism. Proof: B B B B f f = = = f B B B B 68 / 76

162 Mixed states State I m of dagger Frobenius structure is mixed when m = m X m 69 / 76

163 Mixed states State I m of dagger Frobenius structure is mixed when m = m X m Examples: In FHilb: mixed state of C*-algebra is positive element a = b b In Rel: mixed state of groupoid is inverse-closed set of arrows In general: mixed state of pair of pants is name of positive map 69 / 76

164 Completely positive maps If (,, ) and (B,, ) are dagger Frobenius structures, a morphism f B is completely positive when I (f id) m B E is mixed state for mixed state I m E and any dagger Frobenius structure (E,, ). Examples: Unitary evolution u u for unitary u 70 / 76

165 Completely positive maps If (,, ) and (B,, ) are dagger Frobenius structures, a morphism f B is completely positive when I (f id) m B E is mixed state for mixed state I m E and any dagger Frobenius structure (E,, ). Examples: Unitary evolution u u for unitary u Preparation of mixed state: completely positive map I 70 / 76

166 Completely positive maps If (,, ) and (B,, ) are dagger Frobenius structures, a morphism f B is completely positive when I (f id) m B E is mixed state for mixed state I m E and any dagger Frobenius structure (E,, ). Examples: Unitary evolution u u for unitary u Preparation of mixed state: completely positive map I Measurement: completely positive map C n in FHilb is precisely positive-operator valued measure 70 / 76

167 Completely positive maps If (,, ) and (B,, ) are dagger Frobenius structures, a morphism f B is completely positive when I (f id) m B E is mixed state for mixed state I m E and any dagger Frobenius structure (E,, ). Examples: Unitary evolution u u for unitary u Preparation of mixed state: completely positive map I Measurement: completely positive map C n in FHilb is precisely positive-operator valued measure Completely positive maps G H in Rel respect inverses: g h implies g 1 h 1 and id dom(g) id dom(h) 70 / 76

168 The CP condition Lemma: ssume f id E 0 = f 0. If f B is completely positive, then CP condition holds B B f f = X f B B 71 / 76

169 The CP condition Lemma: ssume f id E 0 = f 0. If f B is completely positive, then CP condition holds B B f f = X f B B Proof: Let E = be pair of pants, define I m E as: 71 / 76

170 The CP condition Lemma: ssume f id E 0 = f 0. If f B is completely positive, then CP condition holds B B f f = X f B B Proof: Let E = be pair of pants, define I m E Then m is a mixed state E m = = = E 71 / 76

171 The CP condition Lemma: ssume f id E 0 = f 0. If f B is completely positive, then CP condition holds B B f f = X f B B Proof: Let E = be pair of pants, define I m E Then m is a mixed state, as is (f id E ) m. B B f h = Y h B B 71 / 76

172 The CP condition Lemma: ssume f id E 0 = f 0. If f B is completely positive, then CP condition holds B B f f = X f B B Proof: Let E = be pair of pants, define I m E Then m is a mixed state, as is (f id E ) m. Hence: B B B f = f h = Y h B B B 71 / 76

173 The CP condition Lemma: ssume f id E 0 = f 0. If f B is completely positive, then CP condition holds B B f f = X f B B Theorem (Stinespring): If f then it is completely positive. B satisfies CP condition, 71 / 76

174 The CP construction Theorem: If C is a monoidal dagger category, there is a new category CP[C]: objects in CP[C] are special dagger Frobenius structures in C morphisms in CP[C] are morphisms in C satisfying CP condition C C C g f g g f = f = X Y f g C C C 72 / 76

175 The CP construction Theorem: If C is a braided monoidal dagger category, there is a new monoidal category CP[C]: objects in CP[C] are special dagger Frobenius structures in C morphisms in CP[C] are morphisms in C satisfying CP condition tensor product in CP[C] is as in C B C D B C D f g = f f g g B C D B C D 72 / 76

176 The CP construction Theorem: If C is a symmetric monoidal dagger category, there is a new symmetric monoidal category CP[C]: objects in CP[C] are special dagger Frobenius structures in C morphisms in CP[C] are morphisms in C satisfying CP condition tensor product in CP[C] is as in C B B B B = B B B B 72 / 76

177 The CP construction Theorem: If C is a symmetric monoidal dagger category, there is a new symmetric monoidal dagger category CP[C]: objects in CP[C] are special dagger Frobenius structures in C morphisms in CP[C] are morphisms in C satisfying CP condition tensor product in CP[C] is as in C dagger in CP[C] is as in C B B B B B f = f = f = f = f f B B B B B (duality between Schrödinger and Heisenberg pictures) 72 / 76

178 The CP construction Theorem: If C is a symmetric monoidal dagger category, there is a new compact dagger category CP[C]: objects in CP[C] are special dagger Frobenius structures in C morphisms in CP[C] are morphisms in C satisfying CP condition tensor product in CP[C] is as in C dagger in CP[C] is as in C dual in CP[C] is (,, ) := (,, ) with : I = = = 72 / 76

179 The CP construction Theorem: If C is a symmetric monoidal dagger category, there is a new dagger category CP[C]: objects in CP[C] are special dagger Frobenius structures in C morphisms in CP[C] are morphisms in C satisfying CP condition tensor product in CP[C] is as in C dagger in CP[C] is as in C dual in CP[C] is (,, ) := (,, ) with : I Examples: CP[FHilb] = fin-dim C*-algebras and completely positive maps CP[Rel] = groupoids and inverse-respecting relations 72 / 76

180 Classical and quantum structures Consider full subcategory CP c [C] of classical structures. Lemma: CP c [FHilb] stochastic matrices (rows sum to 1) 73 / 76

181 Classical and quantum structures Consider full subcategory CP c [C] of classical structures. Lemma: CP c [FHilb] stochastic matrices (rows sum to 1) Consider full subcategory CP q [C] of pairs of pants. (Normalizable Frobenius structures are isomorphic to special ones.) 73 / 76

182 Classical and quantum structures Consider full subcategory CP c [C] of classical structures. Lemma: CP c [FHilb] stochastic matrices (rows sum to 1) Consider full subcategory CP q [C] of pairs of pants. (Normalizable Frobenius structures are isomorphic to special ones.) Example: CP q [FHilb] Hilbert spaces and completely positive maps 73 / 76

183 Classical and quantum structures Consider full subcategory CP c [C] of classical structures. Lemma: CP c [FHilb] stochastic matrices (rows sum to 1) Consider full subcategory CP q [C] of pairs of pants. (Normalizable Frobenius structures are isomorphic to special ones.) Example: CP q [FHilb] Hilbert spaces and completely positive maps Proposition: ssuming all objects have positive dimension There is functor P: C CP q [C] given by P() = (,, ) and P(f) = f f that preserves tensor products and daggers 73 / 76

184 Classical and quantum structures Consider full subcategory CP c [C] of classical structures. Lemma: CP c [FHilb] stochastic matrices (rows sum to 1) Consider full subcategory CP q [C] of pairs of pants. (Normalizable Frobenius structures are isomorphic to special ones.) Example: CP q [FHilb] Hilbert spaces and completely positive maps Proposition: ssuming all objects have positive dimension There is functor P: C CP q [C] given by P() = (,, ) and P(f) = f f that preserves tensor products and daggers Lemma ( P is faithful up to phase ): if P(f) = P(g), then s f = t g and s s = t t for some I s,t if s f = t g and s s = id I = t t, then P(f) = P(g) I 73 / 76

185 Environment n environment structure for compact dagger category C pure is a compact dagger category C of which C pure is subcategory for each object in C pure, a discarding map : I in C 74 / 76

186 Environment n environment structure for compact dagger category C pure is a compact dagger category C of which C pure is subcategory for each object in C pure, a discarding map : I in C with: = = = f f = g g in C pure f = g in C 74 / 76

187 Environment n environment structure with purification for category C pure is a compact dagger category C of which C pure is subcategory for each object in C pure, a discarding map : I in C with: = = = f f = g g in C pure f = g in C every map in C is of the form f for f in C pure (Hence C and C pure must have same objects) 74 / 76

188 Environment n environment structure with purification for category C pure is a compact dagger category C of which C pure is subcategory for each object in C pure, a discarding map : I in C with... Example: C pure has environment structure in CP q [C pure ]. 74 / 76

189 Environment n environment structure with purification for category C pure is a compact dagger category C of which C pure is subcategory for each object in C pure, a discarding map : I in C with... Example: C pure has environment structure in CP q [C pure ]. Theorem: If C pure has environment structure with purification, there is isomorphism F : CP q [C pure ] C with F() = on objects, F(f g) = F(f) F(g) on morphisms, that preserves daggers 74 / 76

190 Decoherence decoherence structure for C pure is an environment structure C with discarding maps : I in C for each special dagger Frobenius structure (,, ) in C pure, an object in C and a measuring map : in C, with: 75 / 76

191 Decoherence decoherence structure for C pure is an environment structure C with discarding maps : I in C for each special dagger Frobenius structure (,, ) in C pure, an object in C and a measuring map : in C, with: I I = ( B) B = = = 75 / 76

192 Decoherence decoherence structure with purification for C pure is an environment structure C with discarding maps : I in C for each special dagger Frobenius structure (,, ) in C pure, an object in C and a measuring map : in C, with: I I = ( B) B = = = B every map in C is of the form f for f in C pure (Hence each object in C is of form ) 75 / 76

193 Decoherence decoherence structure with purification for C pure is an environment structure C with discarding maps : I in C for each special dagger Frobenius structure (,, ) in C pure, an object in C and a measuring map : in C, with... Example: C pure has decoherence structure in CP[C pure ] with (,, ) = (,, ) and measuring map 75 / 76

194 Decoherence decoherence structure with purification for C pure is an environment structure C with discarding maps : I in C for each special dagger Frobenius structure (,, ) in C pure, an object in C and a measuring map : in C, with... Example: C pure has decoherence structure in CP[C pure ] with (,, ) = (,, ) and measuring map Theorem: If C pure has decoherence structure with purification, there is isomorphism F : CP[C pure ] C that preserves daggers and satisfies F(f g) = F(f) F(g) 75 / 76

195 Teleportation Theorem: If (,, ) and (,, ) complementary dagger Frobenius in braided monoidal dagger category C, and commutative, then in CP[C]: classical communication output lice correction = measurement Bob input preparation 76 / 76

196 Teleportation Theorem: If (,, ) and (,, ) complementary dagger Frobenius in braided monoidal dagger category C, and commutative, then in CP[C]: mixed states arbitrary systems classical communication only in sense of copied by Frobenius structures, one of which noncommutative two bits of classical communication: two channels used, maybe more than two copyable states tensor product and composition only 76 / 76

197 Summary Monoidal categories scalars, sound and complete graphical calculus Dual objects entanglement, teleportation, encrypted communication Monoids no cloning Frobenius structures normal form, classical structures, observables, classification Complementarity Deutsch-Josza Completely positive maps mixed states, axiomatization, teleportation 77 / 76