Interacting Frobenius Algebras are Hopf R. Duncan Summary - PowerPoint PPT Presentation

Phases & Unbiased Points | 0 � | 0 � | 0 � + e i α | 1 � | 1 � | 1 � ✓ ◆ 1 0 Z α = e i α 0

Incompatible Observables a 0 b 0 b 1 a 1

Complementary Observables a 0 b 1 b 0 a 1 1 ⌅ | � a i | b j ⇥ | = D

Complementary Observables a 0 b 1 b 0 a 1 1 D “Mutually Unbiased Bases” ⌅ | � a i | b j ⇥ | =

Complementarity Two observables A and B are complementary if 1 � p | h a i | b j i | = 8 i, j � d � (aka mutually unbiased) � { U b i } i They are strongly complementary if is a subgroup of the phase group . Φ A

Strongly Complementary Observables are Hopf algebras Theorem 3 : Two observables are strongly complementary iff they form a Hopf algebra � � ” ‘ ÷ µ � � � ÷ ” ‘ µ Coecke and Duncan, “Interacting Quantum Observables: categorical algebra and diagrammatics”, NJP 13(043016), 2011, arXiv:0906.4725.

Strongly Complementary Observables are Hopf algebras Theorem 3 : Two observables are strongly complementary iff they form a Hopf algebra � � ” ‘ ÷ Frobenius µ � � � ÷ ” ‘ Frobenius µ Coecke and Duncan, “Interacting Quantum Observables: categorical algebra and diagrammatics”, NJP 13(043016), 2011, arXiv:0906.4725.

Strongly Complementary Observables are Hopf algebras Theorem 3 : Two observables are strongly complementary iff they form a Hopf algebra � � ” ‘ ÷ µ � � � ÷ ” ‘ µ Hopf Hopf Coecke and Duncan, “Interacting Quantum Observables: categorical algebra and diagrammatics”, NJP 13(043016), 2011, arXiv:0906.4725.

Theorem : 
 In fdHilb , strongly complementary observables exist for every dimension. ... but in big enough dimension, it is possible to construct MUBs which are not strongly complementary.

Bialgebras Defn : A bialgebra is 4-tuple: where: is a monoid � is a comonoid � jointly satisfying the commutation laws: � � = = = = � �

The antipode Defn: A Hopf algebra is a bialgebra with a map 
 s : A ⟶ A 
 satisfying s = (H)

Hopf Algebra Axioms is a monoid is a comonoid = = = = s =

Standard Example | 0 i , . . . , | d � 1 i Given an orthonormal basis. ; | n Í if n = m :: | n Í ‘æ | n Í ¢ | n Í :: | n Í ¢ | m Í ‘æ ” µ 0 otherwise = q = q n œ Z D È n | n œ Z D | n Í ‘ ÷ q | m Í ¢ | m Õ Í :: | n Í ¢ | m Í ‘æ | n + m Í :: | n Í ‘æ ” µ m + m Õ = n = | 0 Í = È 0 | ÷ ‘


 Standard Example | 0 i , . . . , | d � 1 i Given an orthonormal basis. ; | n Í if n = m :: | n Í ‘æ | n Í ¢ | n Í :: | n Í ¢ | m Í ‘æ ” � µ 0 otherwise Proposition 4 : In complex Hilbert space all strongly complementary = q = q n œ Z D È n | n œ Z D | n Í ‘ ÷ pairs are group algebras of abelian groups 
 Coecke, Duncan, Kissinger, Wang, “Strong Complementarity and Non-locality in Categorical Quantum Mechanics”, 
 q | m Í ¢ | m Õ Í :: | n Í ¢ | m Í ‘æ | n + m Í :: | n Í ‘æ ” Proc. LiCS 2012, arXiv:1203.4988. µ m + m Õ = n = | 0 Í = È 0 | ÷ ‘

Normalisation Oh uh: � = � � This is false in the usual model. We want: � 1 � p | h a i | b j i | = 8 i, j d

Scaled Bialgebras Defn : A scaled bialgebra is 4-tuple: where: is a † SCFA � is a † SCFA � jointly satisfying the commutation laws: � � = = = (B) � �

Scaled Bialgebras Defn : A scaled bialgebra is 4-tuple: where: is a † SCFA � is a † SCFA � jointly satisfying the commutation laws: � � = = = (B) � � = Not this one:

The antipode The antipode can be defined as: � s = := � � Then we have: Theorem: the scaled bialgebra is Hopf iff et ( ” , ‘ , µ , ÷ ) i .e. � � = = (+)

Frobenius-Hopf Algebra Axioms is a † SCFA is a † SCFA = = = = =

Frobenius-Hopf Algs Proposition : Let s be the antipode of a commutative Hopf algebra; then � 1. s is the unique map satisfying (H); 
 2. s is a bialgebra morphism; 
 3. s is an involution; 
 4. s commutes with f : H → K for any bialgebra morphism f between Hopf algebras H and K � See, e.g. Street, Quantum Groups , CUP, 2007.

Frobenius-Hopf Algs Corollary : 
 (1) s is a self-adjoint unitary 
 (2) 2. s = s = s (3) s is the antipode of op (4) = f ve f . �

Frobenius-Hopf Algs Corollary : 
 (1) s is a self-adjoint unitary 
 (2) 2. s = s = s (3) s is the antipode of op (4) = f ve f . � Proof(4): = ( s ⊗ m ) � f � ( s † ⊗ n ) = ( s † ⊗ m ) � f � ( s ⊗ n ) = f f .

Frobenius-Hopf Algs Lemma : if f commutes with s then 
 hen f = f ; � f � = = = f f f �

The convolution C We can use the bialgebra to define a convolution: � � f + f Õ := � f Õ 0 := f � � � This gives a commutative monoid.

The convolution := n + 1 = n 0 Lemma : Let � � n � Then = nm and n m = n + m m � � bialgebra morphism for ( ) . Further, is a bialgebra morphism. n

The Integers: In 3       1 0 0 1 0 0 1 0 0 0 = 1 0 0 1 = 0 1 0 2 = 0 0 1       1 0 0 0 0 1 0 1 0

The convolution Lemma : Let f be a bialgebra morphism; then we have: – f ◦ (g + h) = (f ◦ g) + (f ◦ h), 
 – (g + h) ◦ f = (g ◦ f) + (h ◦ f), and 
 – f + (s ◦ f) = 0 . � Hence the bialgebra morphisms of form a unital ring R , and the bialgebra morphisms of do too! op �

f f f f � ( g + h ) = = = ( f � g ) + ( f � h ) g g h h 2. g h g h ( g + h ) � f = = = ( g � f ) + ( h � f ) f f f 3. f + ( f � s ) = = = = = 0 f f f f

The Convolution Lemma : Let a , b ∈ such that ab = 1 . Then b = a † . � Proof: basic idea is show that ( -b ) + a † = 0 � f 0 f 0 f 0 f 0 f f � f f f 0 = = f = = f f f � � f f f f f f = = = = = =

Set-like elements Defn . A point h : I ⟶ A is - set-like if it satisfies is -r of the � h � h h = � � Prop. The set-like elements of a commutative Hopf algebra form an abelian group, with h -1 = sh � is -r Corollary : if the -set-like elements are -unbiased, of the they are a subgroup of the phase group.

Example: qubits Classical points Unbiased points π

Example: qubits | 0 � = � ⇥ 1 0 = α e i α 0 π | 1 � = Classical points Unbiased points π

Example: qubits | 0 � = � ⇥ 1 0 = α e i α 0 π | 1 � = π Classical points Unbiased points π

Example: qubits | 0 � = � ⇥ 1 0 = α e i α 0 π | 1 � = π Classical points Unbiased points π

Example: qubits | 0 � = � ⇥ 1 0 = α e i α 0 π | 1 � = π Classical points Unbiased points π | + � = � ⇥ cos α i sin α = α 2 2 i sin α cos α π 2 2 | �⇥ =

Classical points are eigenvectors i α

Classical points are eigenvectors i α

Classical points are eigenvectors α i i

Classical Maps By definition every -set-like element in determines a - is -r of the phase map: we call these - classical maps. is -r of the � Lemma : let k : 1 ⟶ 1 be -classical; then is -r of the (1) k is a coalgebra morphism for (2) k † = sks

The Following are Equivalent: δ Z = δ X = � Z = � X = i = i i i � j i � j i � j ⇒ = j = j j 1. The classical structures are closed i = i i i � j i � j i � j ⇒ = j = j j

The Following are Equivalent: δ Z = δ X = � Z = � X = 2. The classical i maps are comonoid homomorphisms

The Following are Equivalent: δ Z = δ X = � Z = � X = 2. The classical maps are comonoid homomorphisms i i

The Following are Equivalent: δ Z = δ X = � Z = � X = 3. The classical maps j i i satisfy canonical commutation relations j j i

The Following are Equivalent: δ Z = δ X = � Z = � X = 4. The classical structures form a bialgebra