Categories and Quantum Informatics

Transcription

1 Categories and Quantum Informatics Week 6: Frobenius structures Chris Heunen 1 / 41

2 Overview Frobenius structure: interacting co/monoid, self-duality Normal forms: coherence theorem Frobenius law: coherence between dagger and closure Classification: in FHilb and Rel Phases: unitary operators 2 / 41

3 Idea Orthonormal basis { } for H in FHilb gives comonoid :. Its adjoint is comparison: and e j 0 if i j. 3 / 41

4 Idea Orthonormal basis { } for H in FHilb gives comonoid :. Its adjoint is comparison: and e j 0 if i j. These cooperate: e j e j if i j 0 if i j e j 3 / 41

5 Idea Orthonormal basis { } for H in FHilb gives comonoid :. Its adjoint is comparison: and e j 0 if i j. These cooperate: e j e j if i j 0 if i j e j This monoid/comonoid interaction is called the Frobenius law. 3 / 41

6 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: 4 / 41

7 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. 4 / 41

8 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. Examples of dagger Frobenius structures: In FHilb: a Hilbert space equipped with an orthogonal basis 4 / 41

9 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. Examples of dagger Frobenius structures: In FHilb: a Hilbert space equipped with an orthogonal basis In FHilb: let G be finite group, spanning Hilbert space A. Define group algebra : g h gh, and : z z 1 G. 4 / 41

10 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. Examples of dagger Frobenius structures: In FHilb: a Hilbert space equipped with an orthogonal basis In FHilb: let G be finite group, spanning Hilbert space A. Define group algebra : g h gh, and : z z 1 G. Adjoint: : h G gh 1 h, and : 1 G g and 1 G g 0. 4 / 41

11 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. Examples of dagger Frobenius structures: In FHilb: a Hilbert space equipped with an orthogonal basis In FHilb: let G be finite group, spanning Hilbert space A. Define group algebra : g h gh, and : z z 1 G. Adjoint: : h G gh 1 h, and : 1 G g and 1 G g 0. Frobenius law: LHS(g h) k G gk 1 kh RHS(g h). 4 / 41

12 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. Examples of dagger Frobenius structures: In FHilb: a Hilbert space equipped with an orthogonal basis In FHilb: let G be finite group, spanning Hilbert space A. Define group algebra : g h gh, and : z z 1 G. Adjoint: : h G gh 1 h, and : 1 G g and 1 G g 0. Frobenius law: LHS(g h) k G gk 1 kh RHS(g h). In Rel: let G be groupoid. Monoid in Rel: : (g, h) g h, and : id X. 4 / 41

13 Frobenius structures In a monoidal category, a Frobenius structurs a comonoid (A,, ) and monoid (A,, ) satisfying the Frobenius law: If, called dagger Frobenius structure. Examples of dagger Frobenius structures: In FHilb: a Hilbert space equipped with an orthogonal basis In FHilb: let G be finite group, spanning Hilbert space A. Define group algebra : g h gh, and : z z 1 G. Adjoint: : h G gh 1 h, and : 1 G g and 1 G g 0. Frobenius law: LHS(g h) k G gk 1 kh RHS(g h). In Rel: let G be groupoid. Monoid in Rel: : (g, h) g h, and : id X. Frobenius law: (g, h) (a, b h) for g a b, t(h) s(b). 4 / 41

14 Pair of pants In a dagger monoidal category, if A A, the pair of pants monoid A A carries a dagger Frobenius structure. 5 / 41

15 Pair of pants In a dagger monoidal category, if A A, the pair of pants monoid A A carries a dagger Frobenius structure. Proof. 5 / 41

16 Extended Frobenius law Any Frobenius structure satisfies: 6 / 41

17 Extended Frobenius law Any Frobenius structure satisfies: Proof. 6 / 41

18 Speciality If copies orthogonal basis { }, can find (squared) norm of : and 7 / 41

19 Speciality If copies orthogonal basis { }, can find (squared) norm of : and So can characterize orthonormality via Frobenius structure. 7 / 41

20 Speciality If copies orthogonal basis { }, can find (squared) norm of : and So can characterize orthonormality via Frobenius structure. A Frobenius structurs special if: 7 / 41

21 Speciality If copies orthogonal basis { }, can find (squared) norm of : and So can characterize orthonormality via Frobenius structure. A Frobenius structurs special if: Examples: 7 / 41

22 Speciality If copies orthogonal basis { }, can find (squared) norm of : and So can characterize orthonormality via Frobenius structure. A Frobenius structurs special if: Examples: Group algebra in FHilb is only special for trivial group 7 / 41

23 Speciality If copies orthogonal basis { }, can find (squared) norm of : and So can characterize orthonormality via Frobenius structure. A Frobenius structurs special if: Examples: Group algebra in FHilb is only special for trivial group Orthogonal basis in FHilb is special just when basis is orthonormal 7 / 41

24 Speciality If copies orthogonal basis { }, can find (squared) norm of : and So can characterize orthonormality via Frobenius structure. A Frobenius structurs special if: Examples: Group algebra in FHilb is only special for trivial group Orthogonal basis in FHilb is special just when basis is orthonormal Groupoid Frobenius structurn Rel is always special 7 / 41

25 Classical structures In a braided monoidal dagger category, a classical structurs a special commutative dagger Frobenius structure. 8 / 41

26 Classical structures In a braided monoidal dagger category, a classical structurs a special commutative dagger Frobenius structure. Examples: In FHilb: an orthonormal basis In Rel: abelian group 8 / 41

27 Classical structures In a braided monoidal dagger category, a classical structurs a special commutative dagger Frobenius structure. Examples: In FHilb: an orthonormal basis In Rel: abelian group Definition of classical structure redundant: (Co)commutativity implies half of (co)unitality Speciality and Frobenius law imply (co)associativity Dual object and Frobenius law imply (co)unitality 8 / 41

28 Classical structures In a braided monoidal dagger category, a classical structurs a special commutative dagger Frobenius structure. Examples: In FHilb: an orthonormal basis In Rel: abelian group Definition of classical structure redundant: (Co)commutativity implies half of (co)unitality Speciality and Frobenius law imply (co)associativity Dual object and Frobenius law imply (co)unitality To check that (A,, ) is classical structure, only need: 8 / 41

29 Symmetry Pair of pants hardly ever commutative. However: A Frobenius structurs symmetric when: 9 / 41

30 Symmetry Pair of pants hardly ever commutative. However: A Frobenius structurs symmetric when: In a compact category, this is equivalent to the following: 9 / 41

31 Symmetry Pair of pants hardly ever commutative. However: A Frobenius structurs symmetric when: In a compact category, this is equivalent to the following: Examples: Pair of pants: in FHilb this says Tr(ab) Tr(ba) Group algebras: inverses in groups are two-sided inverses Groupoid Frobenius structure: inverses are two-sided 9 / 41

32 Self-duality If (A,,,, ) Frobenius structurn monoidal category, then A A is self-dual with: A A A A A A A A 10 / 41

33 Self-duality If (A,,,, ) Frobenius structurn monoidal category, then A A is self-dual with: A A A A A A A A Proof. 10 / 41

34 Nondegenerate forms Monoid (A,, ) forms Frobenius structure with comonoid (A,, ) iff allows nondegenerate form: map : A I with part of self-duality A A. 11 / 41

35 Nondegenerate forms Monoid (A,, ) forms Frobenius structure with comonoid (A,, ) iff allows nondegenerate form: map : A I with part of self-duality A A. Proof. One direction is the previous theorem. 11 / 41

36 Nondegenerate forms Monoid (A,, ) forms Frobenius structure with comonoid (A,, ) iff allows nondegenerate form: map : A I with part of self-duality A A. Proof. One direction is the previous theorem. Conversely, suppose I η A A satisfies: η η 11 / 41

37 Nondegenerate forms Monoid (A,, ) forms Frobenius structure with comonoid (A,, ) iff allows nondegenerate form: map : A I with part of self-duality A A. Proof. One direction is the previous theorem. Conversely, suppose I η A A satisfies: η η Define comultiplication as: η 11 / 41

38 Nondegenerate forms Proof (continued.) Could have defined the comultiplication with η left or right: η η η η η η 12 / 41

39 Nondegenerate forms Proof (continued.) Could have defined the comultiplication with η left or right: η η η η η η Counitality: η η η 12 / 41

40 Nondegenerate forms Proof (continued.) Coassociativity: η η η η η η 13 / 41

41 Nondegenerate forms Proof (continued.) Coassociativity: η η η η η η Frobenius law: η η 13 / 41

42 Homomorphisms A homomorphism of Frobenius structures is morphism which is both monoid and comonoid homomorphism. 14 / 41

43 Homomorphisms A homomorphism of Frobenius structures is morphism which is both monoid and comonoid homomorphism. They arsomorphisms. Proof. Given homomorphism A f B, construct inverse as: f 1 f 14 / 41

44 Homomorphisms A homomorphism of Frobenius structures is morphism which is both monoid and comonoid homomorphism. They arsomorphisms. Proof. Given homomorphism A f B, construct inverse as: f 1 f Indeed: f f f 14 / 41

45 Normal forms Two ways to think about graphical calculus: diagram represents morphism: merely shorthand to write down e.g. linear map; diagram is entity in its own right: can be manipulated by replacing equal parts. 15 / 41

46 Normal forms Two ways to think about graphical calculus: diagram represents morphism: merely shorthand to write down e.g. linear map; diagram is entity in its own right: can be manipulated by replacing equal parts. First viewpoint: ok if different diagrams represent same morphism. Second viewpoint: combinatorial/graph theoretic flavour. 15 / 41

47 Normal forms Two ways to think about graphical calculus: diagram represents morphism: merely shorthand to write down e.g. linear map; diagram is entity in its own right: can be manipulated by replacing equal parts. First viewpoint: ok if different diagrams represent same morphism. Second viewpoint: combinatorial/graph theoretic flavour. A normal form theorem connects the two: proving that all diagrams representing fixed morphism can be rewritten into canonical diagram (like coherence theorem) 15 / 41

48 Normal forms Two ways to think about graphical calculus: diagram represents morphism: merely shorthand to write down e.g. linear map; diagram is entity in its own right: can be manipulated by replacing equal parts. First viewpoint: ok if different diagrams represent same morphism. Second viewpoint: combinatorial/graph theoretic flavour. A normal form theorem connects the two: proving that all diagrams representing fixed morphism can be rewritten into canonical diagram (like coherence theorem) Unique way to copy ( ), discard ( ), fuse ( ), create ( ) data! 15 / 41

49 Spider theorem Let (A,,,, ) be a special Frobenius structure. Any connected morphism A m A n built out of finitely many pieces,,,, and id, using and, equals: n m 16 / 41

50 Spider theorem Let (A,,,, ) be a special Frobenius structure. Any connected morphism A m A n built out of finitely many pieces,,,, and id, using and, equals: n m Proof. Induction on the number of dots. 16 / 41

51 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. 17 / 41

52 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. 17 / 41

53 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. Use naturality to write diagram in form with topmost dot. 17 / 41

54 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. Use naturality to write diagram in form with topmost dot. Topmost dot is : use counitality to eliminatt. 17 / 41

55 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. Use naturality to write diagram in form with topmost dot. Topmost dot is : use counitality to eliminatt. Topmost dot is : use coassociativity to reach normal form. 17 / 41

56 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. Use naturality to write diagram in form with topmost dot. Topmost dot is : use counitality to eliminatt. Topmost dot is : use coassociativity to reach normal form. Topmost dot is : impossible by connectedness. 17 / 41

57 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. Use naturality to write diagram in form with topmost dot. Topmost dot is : use counitality to eliminatt. Topmost dot is : use coassociativity to reach normal form. Topmost dot is : impossible by connectedness. Topmost dot is : the most interesting case. 17 / 41

58 Spider theorem Proof. (continued.) Base case. Trivial, as the diagram must be one of,,,. Induction step. Assume all diagrams with at most n dots can be brought in normal form, and consider a diagram with n + 1 dots. Use naturality to write diagram in form with topmost dot. Topmost dot is : use counitality to eliminatt. Topmost dot is : use coassociativity to reach normal form. Topmost dot is : impossible by connectedness. Topmost dot is : the most interesting case. Is the diagram underneath the connected? If so, use coassociativity and speciality. 17 / 41

59 Spider theorem Proof. (continued.) Supposnstead the rest of the diagram is disconnected: 18 / 41

60 More spider theorems In a monoidal category, let (A,,,, ) be a Frobenius structure. Any connected morphism A m A n built out of finitely many pieces,,,, and id, using and, equals ( ). ( ) 19 / 41

61 More spider theorems In a monoidal category, let (A,,,, ) be a Frobenius structure. Any connected morphism A m A n built out of finitely many pieces,,,, and id, using and, equals ( ). ( ) In a symmetric monoidal category, let (A,,,, ) be a commutative Frobenius structure. Any connected morphism A m A n built out of finitely many pieces,,,, id,, using and, equals ( ). 19 / 41

62 No braided spider theorem In a braided non-symmetric monoidal category, thers no normal form for commutative Frobenius algebras. 20 / 41

63 No braided spider theorem In a braided non-symmetric monoidal category, thers no normal form for commutative Frobenius algebras. Proof. Regard the following diagram as a piece of string on which an overhand knot is tied: 20 / 41

64 No braided spider theorem In a braided non-symmetric monoidal category, thers no normal form for commutative Frobenius algebras. Proof. Regard the following diagram as a piece of string on which an overhand knot is tied: The Frobenius algebra axioms induce homotopy equivalences ( deformations ) of the corresponding graph. Such moves cannot untie the knot. 20 / 41

65 Involutive monoids If (A, m, u) is monoid, so is (A, m, u ). 21 / 41

66 Involutive monoids If (A, m, u) is monoid, so is (A, m, u ). An involution for a monoid (A,, ) is a monoid homomorphism A i A satisfying i i id A. A A i i A A 21 / 41

67 Involutive monoids If (A, m, u) is monoid, so is (A, m, u ). An involution for a monoid (A,, ) is a monoid homomorphism A i A satisfying i i id A. A A B B i i i B f f i A A A A A A morphism of involutive monoids is monoid homomorphism A f satisfying i B f f i A. B 21 / 41

68 Examplnvolutive monoids Matrix algebra. M n is an involutive monoid in FHilb. Opposite monoid M n: multiplication ab in M n is ba in M n. Canonical involution M n M n given by f f. 22 / 41

69 Examplnvolutive monoids Matrix algebra. M n is an involutive monoid in FHilb. Opposite monoid M n: multiplication ab in M n is ba in M n. Canonical involution M n M n given by f f. Pair of pants. A A involutivn a dagger pivotal category. Identity map as involution, because of conventions: ( ) 22 / 41

70 Examplnvolutive monoids Matrix algebra. M n is an involutive monoid in FHilb. Opposite monoid M n: multiplication ab in M n is ba in M n. Canonical involution M n M n given by f f. Pair of pants. A A involutivn a dagger pivotal category. Identity map as involution, because of conventions: ( ) Groupoids. G in Rel is involutive. Opposite monoid: induced by opposite groupoid G op Canonical involution G G given by g g / 41

71 Frobenius law from way of the dagger Monoid (A,, ) is dagger Frobenius if and only if i is involution: i 23 / 41

72 Frobenius law from way of the dagger Monoid (A,, ) is dagger Frobenius if and only if i is involution: i Proof. Assume dagger Frobenius. 23 / 41

73 Frobenius law from way of the dagger Monoid (A,, ) is dagger Frobenius if and only if i is involution: i Proof. Assume dagger Frobenius. i preserves multiplication: i i i 23 / 41

74 Frobenius law from way of the dagger Monoid (A,, ) is dagger Frobenius if and only if i is involution: i Proof. Assume dagger Frobenius. i preserves multiplication: i i i i preserves units: easy. 23 / 41

75 Frobenius law from way of the dagger Monoid (A,, ) is dagger Frobenius if and only if i is involution: i Proof. Assume dagger Frobenius. i preserves multiplication: i i i i preserves units: easy. i is involution: i i 23 / 41

76 Frobenius law from way of the dagger Proof. (continued.) Conversely, suppos i id. Then: and by applying, 24 / 41

77 Frobenius law from way of the dagger Proof. (continued.) Conversely, suppos i id. Then: and by applying, So we have a Frobenius structure, defined by a nondegenerate form. Is it a dagger Frobenius structure? 24 / 41

78 Frobenius law from way of the dagger Proof. (continued.) Conversely, suppos i id. Then: and by applying, So we have a Frobenius structure, defined by a nondegenerate form. Is it a dagger Frobenius structure? The condition that i preserves multiplication gives: ( ) So the form definition gives rise to the correct comultiplication. 24 / 41

79 Classification in FHilb Theorem: special dagger Frobenius structures in FHilb are of the form M n1 M nk. 25 / 41

80 Classification in FHilb Theorem: special dagger Frobenius structures in FHilb are of the form M n1 M nk. Proof: Cayley: dagger Frobenius structure on H embeds into H H H H isomorphic to M dim(h) so H involutive subalgebra of M dim(h) : C*-algebra Artin-Wedderburn: must be of form M n1 M nk 25 / 41

81 Classification in FHilb Theorem: special dagger Frobenius structures in FHilb are of the form M n1 M nk. Proof: Cayley: dagger Frobenius structure on H embeds into H H H H isomorphic to M dim(h) so H involutive subalgebra of M dim(h) : C*-algebra Artin-Wedderburn: must be of form M n1 M nk Corollary: classical structurn FHilb copy orthonormal bases Proof: must be of form C C. 25 / 41

82 Orthogonal bases Frobenius structure that copies basis is dagger Frobenius if and only if basis is orthogonal. 26 / 41

83 Orthogonal bases Frobenius structure that copies basis is dagger Frobenius if and only if basis is orthogonal. Proof. For nonzero copyable states: x x y x y x y x y y x x x x x x x x x x 26 / 41

84 Orthogonal bases Frobenius structure that copies basis is dagger Frobenius if and only if basis is orthogonal. Proof. For nonzero copyable states: x x y x y x y x y y x x x x x x x x x x If x y 0, then this is satisfied. 26 / 41

85 Orthogonal bases Frobenius structure that copies basis is dagger Frobenius if and only if basis is orthogonal. Proof. For nonzero copyable states: x x y x y x y x y y x x x x x x x x x x If x y 0, then this is satisfied. If x y 0, this implies x x x y. Similarly y x y y. 26 / 41

86 Orthogonal bases Frobenius structure that copies basis is dagger Frobenius if and only if basis is orthogonal. Proof. For nonzero copyable states: x x y x y x y x y y x x x x x x x x x x If x y 0, then this is satisfied. If x y 0, this implies x x x y. Similarly y x y y. Hence x y x y x x x y y x + y y 0, so x y. 26 / 41

87 Orthogonal bases and morphisms In FHilb, morphism between two commutative dagger Frobenius structures acts as function on copyable states if and only if it is comonoid homomorphism. 27 / 41

88 Orthogonal bases and morphisms In FHilb, morphism between two commutative dagger Frobenius structures acts as function on copyable states if and only if it is comonoid homomorphism. Proof. Suffices to see about basis of copyable states { }. f f f f f Hence f( ) copyable. 27 / 41

89 Classification in Rel Theorem: Special dagger Frobenius structures in Rel correspond exactly to groupoids. 28 / 41

90 Classification in Rel Theorem: Special dagger Frobenius structures in Rel correspond exactly to groupoids. Proof. Write A A M A for multiplication, U A for unit. 28 / 41

91 Classification in Rel Theorem: Special dagger Frobenius structures in Rel correspond exactly to groupoids. Proof. Write A A M A for multiplication, U A for unit. M is single-valued: by speciality a(m M )b iff a b: b b c d a a So: if (c, d)ma and (c, d)mb, must have a b. May simply write ab for unique c with (a, b)mc. Remember: ab not always defined! 28 / 41

92 Classification in Rel Proof. (continued) Associativity: (ab)c a(bc) ab bc a b c a b c So ab and (ab)c defined exactly when bc and a(bc) are defined, and then (ab)c a(bc). 29 / 41

93 Classification in Rel Proof. (continued) Unitality: for units x, y U b b b x y a a a So: a, b allow x U with xa b iff a b. And: a, b allow y U with ay b iff a b. If z U then xz x for some x U. But then x z! Units idempotent; multiplication of different ones undefined. If xa a x a, then a xa x(x a) (xx )a, so x x. So every element has unique left/right identity. 30 / 41

94 Classification in Rel Proof. (continued) Category: U set of objects, A set of morphisms. If fg defined and gh defined, want (fg)h f(gh) defined too: fg h fg h g f(gh) (fg)h f gh f gh If fg and gh defined then LHS defined, so RHS defined too. 31 / 41

95 Classification in Rel Proof. (continued) Inverses: for f A with left unit x and right unit y: x f x f g f f y f y 32 / 41

96 Phases Let (A,, ) be Frobenius structurn a monoidal dagger category. State I a A is called phase when: a a a a 33 / 41

97 Phases Let (A,, ) be Frobenius structurn a monoidal dagger category. State I a A is called phase when: a a a a Its (right) phase shift is the following morphism A A: a a 33 / 41

98 Example phases For classical structurn FHilb copying basis { }, vector a a 1 e 1 + a n e n is phasff each a i on unit circle: a i / 41

99 Example phases For classical structurn FHilb copying basis { }, vector a a 1 e 1 + a n e n is phasff each a i on unit circle: a i 2 1. The unit of a Frobenius structurs always a phase. 34 / 41

100 Example phases For classical structurn FHilb copying basis { }, vector a a 1 e 1 + a n e n is phasff each a i on unit circle: a i 2 1. The unit of a Frobenius structurs always a phase. In a compact dagger category, phases for pair of pants (A A,, ) correspond to unitary morphisms. Proof. 34 / 41

101 Example phases For classical structurn FHilb copying basis { }, vector a a 1 e 1 + a n e n is phasff each a i on unit circle: a i 2 1. The unit of a Frobenius structurs always a phase. In a compact dagger category, phases for pair of pants (A A,, ) correspond to unitary morphisms. Proof. The name of an morphism A f A is a phase when: f f f f But this means f f id A ; similarly f f id A. 34 / 41

102 Example phases Phases of Frobenius structure M n in FHilb form set U(n) of n-by-n unitary matrices. Hence phases of M k1 M kn range over U(k 1 ) U(k n ). 35 / 41

103 Example phases Phases of Frobenius structure M n in FHilb form set U(n) of n-by-n unitary matrices. Hence phases of M k1 M kn range over U(k 1 ) U(k n ). Classical structure C n copying basis {e 1,..., e n }. Phases are elements of U(1) U(1); phase shift C n C n is accompanying unitary matrix. 35 / 41

104 Example phases Phases of Frobenius structure M n in FHilb form set U(n) of n-by-n unitary matrices. Hence phases of M k1 M kn range over U(k 1 ) U(k n ). Classical structure C n copying basis {e 1,..., e n }. Phases are elements of U(1) U(1); phase shift C n C n is accompanying unitary matrix. The phases of a Frobenius structurn Rel induced by a group G are elements of that group G itself. 35 / 41

105 Example phases Phases of Frobenius structure M n in FHilb form set U(n) of n-by-n unitary matrices. Hence phases of M k1 M kn range over U(k 1 ) U(k n ). Classical structure C n copying basis {e 1,..., e n }. Phases are elements of U(1) U(1); phase shift C n C n is accompanying unitary matrix. The phases of a Frobenius structurn Rel induced by a group G are elements of that group G itself. Proof. For a subset a G, equation defining phases reads {g 1 h g, h a} {1 G } {hg 1 g, h a}. So if g G, then a {g} is a phase. But if a contains distinct elements g h of G, cannot be phase. Similarly, a not phase. Hence a phase precisely when singleton {g}. 35 / 41

106 Phase group In a monoidal dagger category, the phases for a dagger Frobenius structure form a group, with unit and: a + b a b 36 / 41

107 Phase group In a monoidal dagger category, the phases for a dagger Frobenius structure form a group, with unit and: a + b a b Proof. This is again a well-defined phase: a + b a b a b b a + b a b a b b The flipped equation follows similarly. Associativity is clear, hence phases form a monoid. 36 / 41

108 Phase group Proof. (continued) Left-inverse of phase a is: a a 37 / 41

109 Phase group Proof. (continued) Left-inverse of phase a is: a a Left-inverse of a is a: ( a) + a a a a a 37 / 41

110 Phase group Proof. (continued) Left-inverse of phase a is: a a Left-inverse of a is a: ( a) + a a a a a Similarly thers right-inverse. But in monoids, left and right inverses are equal: l l(xr) (lx)r r. 37 / 41

111 Example phase groups In FHilb, the phase group for the pair of pants Frobenius structurs the unitary group. 38 / 41

112 Example phase groups In FHilb, the phase group for the pair of pants Frobenius structurs the unitary group. Phase addition in the Frobenius structure M k1 M kn in FHilb is entrywise multiplication in U(k 1 ) U(k n ). In particular, phase addition in a classical structurn FHilb is multiplication of diagonal matrices. 38 / 41

113 Example phase groups In FHilb, the phase group for the pair of pants Frobenius structurs the unitary group. Phase addition in the Frobenius structure M k1 M kn in FHilb is entrywise multiplication in U(k 1 ) U(k n ). In particular, phase addition in a classical structurn FHilb is multiplication of diagonal matrices. In Rel, the phase group induced by a group G is the group itself. 38 / 41

114 Phased spider theorem Let (A,, ) be classical structurn braided monoidal dagger category. Any connected morphism A m A n built of finitely many,, id, σ and phases using,, and, equals n {}}{ a }{{} m where a ranges over all the phases used in the diagram. 39 / 41

115 Phased spider theorem Let (A,, ) be classical structurn braided monoidal dagger category. Any connected morphism A m A n built of finitely many,, id, σ and phases using,, and, equals n {}}{ a }{{} m where a ranges over all the phases used in the diagram. Proof. Use braidings to have all phases dangle at the bottom. Apply Spider Theorem. Use phase addition to reduce to single phase a on bottom right. Apply Spider Theorem again. 39 / 41

116 State transfer State transfer protocol: transfer state of Hilbert space H from one system to another, with success probability 1/ dim(h) 2. May be lax in drawing, e.g. projection H H H H: The procedure looks like this: 1 n 1 n measure the first qubit measure both qubits together prepare the second qubit 1 n 1 n Extra challenge: apply phase gate while transferring state 1 n φ φ condition on first qubit measurement projection prepare second qubit 40 / 41

117 Summary Frobenius structures: interacting co/monoid, self-duality Normal forms: spider theorems Frobenius law: justified by coherence Classification: matrix algebras, bases, groupoids Phases: unitary operators, state transfer Next week: interaction between two Frobenius structures 41 / 41