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

Tensors, !-graphs, and non-commutative quantum structures

Aleks Kissinger David Quick QPL June 2014

Table of Contents

Introduction Tensor notation Definitions Induction Summary

Quanto

Defining Nodes

Given the graph:

Defining Nodes

Given the graph: Could we define:

a c b b a c

:=

Defining Nodes

Given the graph: = Could we define:

a c b b a c

:=

Defining Nodes

Given the graph: = Could we define:

a c b b a c

:=

b a c a b c

=

Defining Nodes

Given the graph: = Could we define:

a c b b a c

:=

c b a

=

b a c a b c

=

Defining Nodes

Given the graph: = Could we define:

a c b b a c

:=

c b a a c b

=

b a c a b c

= =

Defining Nodes

Given the graph: = Could we define:

a c b b a c

:=

c b a a c b

=

b a c a b c

= =

Defining Nodes

Given the graph: = Could we define:

a c b b a c

:=

c b a a c b

=

b a c a b c

= =

Defining Nodes

Given the graph: = = Could we define:

a c b b a c

:=

c b a a c b

=

b a c a b c

= =

Recursively Defining Nodes

Given Σ =

• ,

, ,

• satisfying:

= = = = = =

Recursively Defining Nodes

Given Σ =

• ,

, ,

• satisfying:

= = = = = =

We may want to define nodes of the form:

1 n-1 1

:=

n n-1 n

. . . . . .

Recursively Defining Nodes

0 inputs:

:=

Recursively Defining Nodes

0 inputs:

:=

Given k inputs:

k 1 . . .

Recursively Defining Nodes

0 inputs:

:=

Given k inputs:

k 1 . . .

define k+1 inputs:

k 1

:=

k+1 1 k k+1

. . . . . .

Recursively Defining Nodes

0 inputs:

:=

Given k inputs:

k 1 . . .

define k+1 inputs:

k 1

:=

k+1 1 k k+1

. . . . . .

Recursive definitions suggest proof by induction. To do this we need to be more formal with variable arity nodes.

!-Boxes

Use !-boxes for multiple copies of a section from a graph.

. . . replaces

!-Boxes

Use !-boxes for multiple copies of a section from a graph.

. . . replaces

Then we have operations allowing deletion of !-boxes and creation

• f a new instance of the contents:

!-Boxes

Use !-boxes for multiple copies of a section from a graph.

. . . replaces

Then we have operations allowing deletion of !-boxes and creation

• f a new instance of the contents:

Kill

!-Boxes

Use !-boxes for multiple copies of a section from a graph.

. . . replaces

Then we have operations allowing deletion of !-boxes and creation

• f a new instance of the contents:

Kill Exp

!-Boxes

Use !-boxes for multiple copies of a section from a graph.

. . . replaces

Then we have operations allowing deletion of !-boxes and creation

• f a new instance of the contents:

Kill Exp Kill

!-Boxes

Use !-boxes for multiple copies of a section from a graph.

. . . replaces

Then we have operations allowing deletion of !-boxes and creation

• f a new instance of the contents:

Kill Exp Kill Exp

!-Box Equations

!-Boxes can be used in equations:

!-Box Equations

!-Boxes can be used in equations:

= . . . . . .

is replaced by

=

B B

!-Box Equations

!-Boxes can be used in equations:

= . . . . . .

is replaced by

=

B B

Which represents concrete equations like:

= = = = , , , ,. . .

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

ψ

a b A a’ b’

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

vs.

b

ψ

a A

ψ

a b A a’ b’

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

vs.

b

ψ

a A

ψ

a b A a’ b’

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

vs.

b

ψ

a A

ψ

a b A a’ b’

vs.

b

ψ

a’ b’ a A

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

vs.

b

ψ

a A

vs.

a A

ψ

b

ψ

a b A a’ b’

vs.

b

ψ

a’ b’ a A

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

vs.

b

ψ

a A

vs.

a A

ψ

b

ψ

a b A a’ b’

vs.

b

ψ

a’ b’ a A

!-Box Expansion

This !-box (labelled A) has many different possible expansions:

ψ

a b A

vs.

b

ψ

a A

vs.

a A

ψ

b

ψ

a b A a’ b’

vs.

b

ψ

a’ b’ a A

vs.

a A b’

ψ

a’ b

Table of Contents

Introduction Tensor notation Definitions Induction Summary

Building a Tensor

ψ

f a b

Building a Tensor

ψ

f a b

= ψ

Building a Tensor

ψ

f a b

= ψfab

Building a Tensor

ψ

f a b

= ψˆ

f ˇ aˇ b

Building a Tensor

ψ

f a b

= ψˆ

f ˇ aˇ b

φ

a b c d e

= φˆ

aˆ bˇ c ˆ d ˇ e

Building a Tensor

φ

c d e

ψ

f

= ψˆ

f ˇ aˇ bφˆ aˆ bˇ c ˆ d ˇ e a a b b

Building a Tensor

φ

c d e

ψ

f

= ψˆ

f ˇ aˇ bφˆ aˆ bˇ c ˆ d ˇ e b b

Building a Tensor

φ

c d e

ψ

f

= ψˆ

f ˇ aˇ bφˆ aˆ bˇ c ˆ d ˇ e

Building a Tensor

φ

c d e

ψ

f

= ψˆ

f ˇ aˇ bφˆ aˆ bˇ c ˆ d ˇ e

Building a Tensor

φ

c d e

ψ

f

= ψˆ

f ˇ aˇ bφˆ aˆ bˇ c ˆ d ˇ e

= ψˆ

f ˇ x ˇ yφˆ x ˆ y ˇ c ˆ d ˇ e

Building a Tensor

φ

c d e

ψ

f

= ψˆ

f ˇ aˇ bφˆ aˆ bˇ c ˆ d ˇ e

B

Building a Tensor

φ

c d e

ψ

f

=

• ψˆ

f ˇ aˇ b

Bφˆ

aˆ bˇ c ˆ d ˇ e

B

Building a Tensor

φ

c d e

ψ

f

=

• ψˆ

f ˇ aˇ b

Bφˆ

aˆ bˇ c ˆ d ˇ e

B

Building a Tensor

φ

c d e

ψ

f

=

• ψˆ

f ˇ aˇ b

Bφˆ

a]B ˆ bˇ c ˆ d ˇ e

B

Building a Tensor

φ

c d e

ψ

f

=

• ψˆ

f ˇ aˇ b

Bφˆ

a]B ˆ bˇ c ˆ d ˇ e

B

Building a Tensor

φ

c d e

ψ

f

=

• ψˆ

f ˇ aˇ b

Bφˆ

a]B[ˆ bB ˇ c ˆ d ˇ e

B

Building a Tensor

φ

c d e

ψ

f

=

• ψˆ

f ˇ aˇ b

Bφˆ

a]B[ˆ bB ˇ c ˆ d ˇ e

B

An Example

φ ψ

B

An Example

= ψ[ˆ

aBφˇ a]B

φ ψ

B

An Example

= ψ[ˆ

aBφˇ a]B

• 1

B φ ψ

B

An Example

= ψ[ˆ

aBφˇ a]B

• 1

B φ ψ

B

φ ψ

,

φ ψ

,

φ ψ

,

φ ψ

, . . .

Twisted Example

= ψ[ˆ

aBφ[ˇ aB

• 1

B φ ψ

B

Twisted Example

= ψ[ˆ

aBφ[ˇ aB

• 1

B φ ψ

B

φ ψ

,

φ ψ

,

φ ψ

,

φ ψ

, . . .

Nesting Example (Inner)

φ

a

ξ

B A

=

• ξˆ

bˇ a

Bφˇ

b]B

A

Nesting Example (Inner)

φ

a

ξ

B A

=

• ξˆ

bˇ a

Bφˇ

b]B

A φ

a

ξ

B A a’

ξ φ

A

KillB ExpB

Nesting Example (Inner)

φ

a

ξ

B A

=

• ξˆ

bˇ a

Bφˇ

b]B

A φ

a

ξ

B A a’

ξ φ

A

KillB ExpB

• φ

A

• ξˆ

bˇ a

Bξˆ

b′ˇ a′φˇ b′ˇ b]B

A

Nesting Example (Outer)

φ

a

ξ

B A

=

• ξˆ

bˇ a

Bφˇ

b]B

A

Nesting Example (Outer)

φ

a

ξ

B A

=

• ξˆ

bˇ a

Bφˇ

b]B

A φ

a

ξ

B A

φ ξ

a’ B’

ExpA KillA

Nesting Example (Outer)

φ

a

ξ

B A

=

• ξˆ

bˇ a

Bφˇ

b]B

A φ

a

ξ

B A

φ ξ

a’ B’

ExpA KillA 1

• ξˆ

bˇ a

Bφˇ

b]B

A ξˆ

b′ˇ a′

B′ φˇ

b′]B′

Table of Contents

Introduction Tensor notation Definitions Induction Summary

Edgeterms

Definition

Fix disjoint, infinite sets E and B of edge names and !-box names,

• respectively. The set of edgeterms Te is defined recursively as

follows:

• ǫ ∈ Te

(i.e empty)

• ˇ

a, ˆ a ∈ Te a ∈ E

• [eA, e]A ∈ Te

e ∈ Te, A ∈ B

• ef ∈ Te

e, f ∈ Te

Edgeterms

Definition

Fix disjoint, infinite sets E and B of edge names and !-box names,

• respectively. The set of edgeterms Te is defined recursively as

follows:

• ǫ ∈ Te

(i.e empty)

• ˇ

a, ˆ a ∈ Te a ∈ E

• [eA, e]A ∈ Te

e ∈ Te, A ∈ B

• ef ∈ Te

e, f ∈ Te Two edgeterms are equivalent if one can be transformed into the

• ther by:

ǫe ≡ e ≡ eǫ e(fg) ≡ (ef )g [ǫA ≡ ǫ ≡ ǫ]A

!-Tensors

Definition

The set of all !-tensor expressions TΣ for a signature Σ is defined recursively as:

• 1 ∈ TΣ

(empty tensor)

• 1ˆ

aˇ b ∈ TΣ

a, b ∈ E

• φe ∈ TΣ

e ∈ Te, φ ∈ Σ

• G

A ∈ TΣ G ∈ TΣ, A ∈ B

• GH ∈ TΣ

G, H ∈ TΣ Satisfying conditions F1-2, C1-3

Conditions: F1-F2

F1: No directed edge name can appear more than once as these can not be plugged together:

φ ψ φˆ

aψˆ a

=

Conditions: F1-F2

F1: No directed edge name can appear more than once as these can not be plugged together:

φ ψ φˆ

aψˆ a

=

Conditions: F1-F2

F1: No directed edge name can appear more than once as these can not be plugged together:

φ ψ φˆ

aψˆ a

=

F2: No box can appear more than once to prevent overlap (which we do not allow in this formalism):

. . . . . .

• . . .

AB . . . A = A

B A

Conditions: F1-F2

F1: No directed edge name can appear more than once as these can not be plugged together:

φ ψ φˆ

aψˆ a

=

F2: No box can appear more than once to prevent overlap (which we do not allow in this formalism):

. . . . . .

• . . .

AB . . . A = A

B A

Conditions: C1-C3

C1: An edge entering a !-box can’t be on a node already in that !-Box:

φ

a

• φ[ˆ

aB

B =

B B

Conditions: C1-C3

C1: An edge entering a !-box can’t be on a node already in that !-Box:

φ

a

• φ[ˆ

aB

B =

B B

Conditions: C1-C3

C1: An edge entering a !-box can’t be on a node already in that !-Box:

φ

a

• φ[ˆ

aB

B =

B B

C2: Nested !-Boxes around an edge must be nested in the same way in the rest of the tensor:

φ

a

φ[[ˆ

aAB

• ψ

A =

B A

ψ

A

Conditions: C1-C3

C1: An edge entering a !-box can’t be on a node already in that !-Box:

φ

a

• φ[ˆ

aB

B =

B B

C2: Nested !-Boxes around an edge must be nested in the same way in the rest of the tensor:

φ

a

φ[[ˆ

aAB

• ψ

A =

B A

ψ

A

Conditions: C1-C3

C1: An edge entering a !-box can’t be on a node already in that !-Box:

φ

a

• φ[ˆ

aB

B =

B B

C2: Nested !-Boxes around an edge must be nested in the same way in the rest of the tensor:

φ

a

φ[[ˆ

aAB

• ψ

A =

B A

ψ

A

C3: Bound edges must be in compatible !-boxes:

φ

a

φ[ˆ

aBψˇ a

=

B

ψ

a

Conditions: C1-C3

C1: An edge entering a !-box can’t be on a node already in that !-Box:

φ

a

• φ[ˆ

aB

B =

B B

C2: Nested !-Boxes around an edge must be nested in the same way in the rest of the tensor:

φ

a

φ[[ˆ

aAB

• ψ

A =

B A

ψ

A

C3: Bound edges must be in compatible !-boxes:

φ

a

φ[ˆ

aBψˇ a

=

B

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: KillB := [

• G

B → 1, [eB → ǫ, e]B → ǫ]

Operations

The operation Kill removes a !-box and all nodes and edges in it: KillB := [

• G

B → 1, [eB → ǫ, e]B → ǫ] 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: KillB := [

• G

B → 1, [eB → ǫ, e]B → ǫ] 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). ExpB := [

• G

B →

• G

B fr(G), [eB → [eB fr(e), e]B → fr(e)e]B]

Table of Contents

Introduction Tensor notation Definitions Induction Summary

Spider Node

Definition

:= :=

Spider Node

Definition

:= :=

Theorem

B A B A

=

Spider Node

Definition

:= :=

Theorem

B A B A

=

We would like to do induction on !-box B splitting into a base case (after KillB) and an inductive step (proving ExpB from original).

Induction

KillB(G = H) (G = H) = ⇒ ExpB(G = H) G = H (Induction)

Induction

KillB(G = H) FixB(G = H) = ⇒ ExpB(G = H) G = H (Induction)

Spider Theorem (KillB)

Theorem

A A

=

(base)

Spider Theorem (KillB)

Theorem

A A

=

(base)

Proof.

A

=

A

Spider Theorem (KillB)

Theorem

A A

=

(base)

Proof.

A

=

A

=

A

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Proof.

B A

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Proof.

B A

=

B A

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Proof.

B A

=

B A A B

=

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Proof.

B A

=

B A A B

= =

B A

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Proof.

B A

=

B A A B

= =

B A

= IH

A B

=

Spider Theorem (FixB = ⇒ ExpB)

Theorem

B A B A

=

⇒

B A B A

=

(step)

Proof.

B A

=

B A A B

= =

B A

= IH

A B

=

B A

Anti-Homomorphism

=

Anti-Homomorphism

= = . . . . . .

Anti-Homomorphism

= = . . . . . . =

B B

Table of Contents

Introduction Tensor notation Definitions Induction Summary

Summary

Summary

• Diagrammatic language:

φ

c d e

ψ

f B

Summary

• Diagrammatic language:

φ

c d e

ψ

f B

• Tensor notation:
• ψˆ

f ˇ aˇ b

Bφˆ

a]B[ˆ bB ˇ c ˆ d ˇ e

Summary

• Diagrammatic language:

φ

c d e

ψ

f B

• Tensor notation:
• ψˆ

f ˇ aˇ b

Bφˆ

a]B[ˆ bB ˇ c ˆ d ˇ e

• Recursive definitions:

:= :=

Summary

• Diagrammatic language:

φ

c d e

ψ

f B

• Tensor notation:
• ψˆ

f ˇ aˇ b

Bφˆ

a]B[ˆ bB ˇ c ˆ d ˇ e

• Recursive definitions:

:= :=

• !-Box Induction:

KillB(G = H) FixB(G = H) = ⇒ ExpB(G = H) G = H (Induction)