Tensors, !-graphs, and non-commutative quantum structures Aleks - PowerPoint PPT Presentation

Conditions: F1-F2 F1: No directed edge name can appear more than once as these φ can not be plugged together: φ ˆ a ψ ˆ = a ψ

Conditions: F1-F2 F1: No directed edge name can appear more than once as these φ can not be plugged together: φ ˆ a ψ ˆ = a ψ F2: No box can appear more than once to prevent overlap (which B � A � B � � A . . . . . . we do not allow in this formalism): �� = A A . . . . . .

Conditions: F1-F2 F1: No directed edge name can appear more than once as these φ can not be plugged together: φ ˆ a ψ ˆ = a ψ F2: No box can appear more than once to prevent overlap (which B � A � B � � A . . . . . . we do not allow in this formalism): �� = A A . . . . . .

Conditions: C1-C3 C1: An edge entering a !-box can’t be on a node already in that B � B !-Box: � B φ φ [ˆ = a � B a

Conditions: C1-C3 C1: An edge entering a !-box can’t be on a node already in that B � B !-Box: � B φ φ [ˆ = a � B a

Conditions: C1-C3 C1: An edge entering a !-box can’t be on a node already in that B � B !-Box: � B φ φ [ˆ = a � B a C2: Nested !-Boxes around an edge must be nested in the same way B A � A in the rest of the tensor: � A φ ψ φ [[ˆ ψ = a � A � B a

Conditions: C1-C3 C1: An edge entering a !-box can’t be on a node already in that B � B !-Box: � B φ φ [ˆ = a � B a C2: Nested !-Boxes around an edge must be nested in the same way B A � A in the rest of the tensor: � A φ ψ φ [[ˆ ψ = a � A � B a

Conditions: C1-C3 C1: An edge entering a !-box can’t be on a node already in that B � B !-Box: � B φ φ [ˆ = a � B a C2: Nested !-Boxes around an edge must be nested in the same way B A � A in the rest of the tensor: � A φ ψ φ [[ˆ ψ = a � A � B a C3: Bound edges must be in compatible !-boxes: B = φ [ˆ a � B ψ ˇ φ a a ψ a

Conditions: C1-C3 C1: An edge entering a !-box can’t be on a node already in that B � B !-Box: � B φ φ [ˆ = a � B a C2: Nested !-Boxes around an edge must be nested in the same way B A � A in the rest of the tensor: � A φ ψ φ [[ˆ ψ = a � A � B a C3: Bound edges must be in compatible !-boxes: B = φ [ˆ a � B ψ ˇ φ a a ψ a

Operations The operation Kill removes a !-box and all nodes and edges in it:

Operations The operation Kill removes a !-box and all nodes and edges in it: � B �→ 1 , [ e � B �→ ǫ, � e ] B �→ ǫ ] � Kill B := [ G

Operations The operation Kill removes a !-box and all nodes and edges in it: � B �→ 1 , [ e � B �→ ǫ, � e ] B �→ ǫ ] � Kill B := [ G Expanding a !-box creates a new copy of its contents with new names for all new edges/boxes. Write fr ( G ) for G with all names replaced by new ones (choosen by predetermined function fr ).

Operations The operation Kill removes a !-box and all nodes and edges in it: � B �→ 1 , [ e � B �→ ǫ, � e ] B �→ ǫ ] � Kill B := [ G Expanding a !-box creates a new copy of its contents with new names for all new edges/boxes. Write fr ( G ) for G with all names replaced by new ones (choosen by predetermined function fr ). � B �→ � B fr ( G ) , [ e � B �→ [ e � B fr ( e ) , � e ] B �→ � � Exp B := [ G G fr ( e ) � e ] B ]

Table of Contents Introduction Tensor notation Definitions Induction Summary

Spider Node Definition := :=

Spider Node Definition := := Theorem = A A B B

Spider Node Definition := := Theorem = A A B B We would like to do induction on !-box B splitting into a base case (after Kill B ) and an inductive step (proving Exp B from original).

Induction Kill B ( G = H ) ( G = H ) = ⇒ Exp B ( G = H ) (Induction) G = H

Induction Kill B ( G = H ) Fix B ( G = H ) = ⇒ Exp B ( G = H ) (Induction) G = H

Spider Theorem ( Kill B ) Theorem (base) = A A

Spider Theorem ( Kill B ) Theorem (base) = A A Proof. = A A

Spider Theorem ( Kill B ) Theorem (base) = A A Proof. = = A A A

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B Proof. A B

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B Proof. = A A B B

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B Proof. = = A A B B A B

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B Proof. = = = A A A B B A B B

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B Proof. IH = = = = = A A A A B B B A B B

Spider Theorem ( Fix B = ⇒ Exp B ) Theorem ⇒ (step) = = A A A B B A B B Proof. IH = = = = = A A A A A B B B B A B B