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Complementarity -...
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Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Kind of old news now:
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Phase Groups and Complementarity
Ross DuncanOxford University Computing Laboratory
α
α
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Motivations
A⇒ B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Motivations
A⇒ B !A−◦B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Motivations
A⇒ B !A−◦B
Hilbert space, unitary transforms,
self-adjoint operators.... ?
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Motivations
A⇒ B !A−◦B
Hilbert space, unitary transforms,
self-adjoint operators....
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
What we did:
We reformulated (a large part of) quantum mechanics:
• high level structural approach based on monoidal categories
• axiomatics expressive enough to cover universal quantum computation
- and powerful enough to simulate algorithms and prove equivalence between quantum state and programs
• simple to understand and manipulate graphical calculus
- which is being implemented in semi-automatic GUI based rewriting tool
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables Special †-Frobenius Algebrasaka classical structures
aka basis structuresaka observable structures
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables Special †-Frobenius Algebrasaka classical structures
aka basis structuresaka observable structures
Phase Groups
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables Special †-Frobenius Algebrasaka classical structures
aka basis structuresaka observable structures
Phase Groups
MutuallyUnbiased Bases
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables Special †-Frobenius Algebrasaka classical structures
aka basis structuresaka observable structures
Phase Groups
MutuallyUnbiased Bases
Complementary
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables Special †-Frobenius Algebrasaka classical structures
aka basis structuresaka observable structures
Phase Groups
MutuallyUnbiased Bases
Hopf Algebras &BialgebrasComplementary
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
In This Talk:
Observables Special †-Frobenius Algebrasaka classical structures
aka basis structuresaka observable structures
Phase Groups
MutuallyUnbiased Bases
Hopf Algebras &BialgebrasComplementary
Automorphisms of
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
What is a Phase Group?
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
What is a Phase Group?
• An abelian group of unitary maps on the state space
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
What is a Phase Group?
• An abelian group of unitary maps on the state space
• which leave some observable fixed
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
What is a Phase Group?
• An abelian group of unitary maps on the state space
• which leave some observable fixed
α
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
- Symmetric monoidal categories
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
- Symmetric monoidal categories
• Unitarity:
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
- Symmetric monoidal categories
• Unitarity:
- †-symmetric monoidal categories
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
- Symmetric monoidal categories
• Unitarity:
- †-symmetric monoidal categories
• Observables:
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
- Symmetric monoidal categories
• Unitarity:
- †-symmetric monoidal categories
• Observables:
- Special †-Frobenius algebras
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ingredients of the Theory
To construct phase groups we need:
• Tensor products:
- Symmetric monoidal categories
• Unitarity:
- †-symmetric monoidal categories
• Observables:
- Special †-Frobenius algebras
This is a very general setting including much more than just quantum mechanics.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Why Bother?
• Qubits, qutrits, etc
• Finite relations
• Toy models
• Convex operational theories
• Continuous variable QM
• Stab and Spek
• .....
What can phase groups say about quantum theory?
What can phase groups say about other theories?
QM
GNST
PRbox
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
DISCLAIMER
There are going to be a LOT of definitions in this talk.
sorry.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Quantum Observables
What is “classical” anyway?
fundamental : incompatible observables* pos vs mom* X vs Z spins in fd
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Quantum Observables
What is “classical” anyway?
http://dresdencodak.com
fundamental : incompatible observables* pos vs mom* X vs Z spins in fd
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
b1 b0
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
b1 b0
**
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
b1 b0
**
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
b1 b0
**
|�ai | bj�| =1√D
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
a0
a1
b1 b0
**
|�ai | bj�| =1√D
“Mutually Unbiased Bases”
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Z =�
1 00 −1
�X =
�0 11 0
�
X and Z Spins
α |0�+ β |1�
|0�✛
p=α2
|1�
✛p=
β2
|+�
p=
(α+
β)/
2 2✲
|−�
p=(α−
β)/2 2
✲
We can measure the spin of qubit |ψ� = α |0� + β |1�
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Z =�
1 00 −1
�X =
�0 11 0
�
X and Z SpinsWe can measure the spin of qubit |ψ� = α |0� + β |1�
|0�
|0�✛
p=1
|1�
✛p=
0
|+�
p=
1/2
✲
|−�
p=1/2
✲
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Z =�
1 00 −1
�X =
�0 11 0
�
X and Z SpinsWe can measure the spin of qubit |ψ� = α |0� + β |1�
(|0�+ |1�)/√
2
|0�✛
p=1/
2
|1�
✛p=
1/2
|+�
p=
1✲
|−�
p=0
✲
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
unless and are orthogonal [Wooters & Zurek 1982]
Theorem: There are no unitary operations D such that
D : |ψ� �→ |ψ� ⊗ |ψ�D : |φ� �→ |φ� ⊗ |φ�
No-Cloning and No-Deleting
unless and are orthogonal [Pati & Braunstein 2000]
Theorem: There are no unitary operations E such that
E : |ψ� �→ |0�E : |φ� �→ |0�
|ψ�
|ψ�
|φ�
|φ�
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
then the category collapses [Abramsky 2007]
Theorem: if a †-compact category has natural transformations
δ : − ⇒ −⊗−� : − ⇒ I
No-Cloning and No-Deleting, abstractly
(Translation: in our abstract setting there are no universal cloning or deleting operations, just like in quantum mechanics)
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
“Classical” Quantum States
When can a quantum state be treated as if classical?
• no-go theorems allow copying and deleting of orthogonal states;
In other words:
• A quantum state may be copied and deleted if it is an eigenstate of some known observable.
We’ll use this property to formalise observables in terms of copying and deleting operations.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical Properties
In general, quantum observables are incompatible - not defined at the same time : position and momentum; X and Z spin; etc.
Traditional quantum logic constructs a property lattice for each set of compatible observables; the incompatible properties are simply incomparable in the lattice.
However if we want to compute with quantum mechanics we need to know how these observables relate to each other, i.e. how they interfere.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Our Approach
We aim to extract the positive content from the incompatibility of quantum observables:
• basis of monoidal categories : no-cloning, no-deleting;
• observable structures : axiomatised in terms of a copying operation
• Phase groups : constructed from the observable structures
• incompatible observables : how do classical operations which act on complementary states interact?
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Monoidal Categories
A theory of interacting processes
f h
D
A
B
C
g k
C E
Ross Duncan ● Ottawa 2009
f : A→ B g : B → C h : C → D
A category consists of objects etc, and arrows between them:
Categories
A,B, C,
f gA
B
B
C
C
h
D
f : A→ B g : B → C h : C → D
Ross Duncan ● Ottawa 2009
A category consists of objects etc, and arrows between them:
Categories
A,B, C,
f
g
A
B
C
C
h
D
f : A→ B g : B → C h : C → D
g ◦ f : A → C
Ross Duncan ● Ottawa 2009
A category consists of objects etc, and arrows between them:
Categories
A,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Ross Duncan ● Ottawa 2009
A category consists of objects etc, and arrows between them:
Categories
A,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Ross Duncan ● Ottawa 2009
A category consists of objects etc, and arrows between them:
Categories
A,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Ross Duncan ● Ottawa 2009
A category consists of objects etc, and arrows between them:
Categories
A,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Ross Duncan ● Ottawa 2009
idA : A→ A
Categories
idA : A→ A
Ross Duncan ● Ottawa 2009
Categories
idA : A→ A
B
f
f ◦ idA : A → B
Ross Duncan ● Ottawa 2009
Categories
idA : A→ A
B
f
idB ◦ f : A → B
Ross Duncan ● Ottawa 2009
Categories
idA : A→ A
B
f
f : A→ B
Ross Duncan ● Ottawa 2009
f ⊗ h : A⊗ C → B ⊗D
The tensor is associative:
A strict monoidal category is a category equipped with a tensor product on both objects and arrows:
(f ⊗ g)⊗ h = f ⊗ (g ⊗ h)
Monoidal Categories
f h
D
A
B
C
gf
A
B
h
D
C
C
B
Ross Duncan ● Ottawa 2009
(g ◦ f)⊗ (k ◦ h) = (g ⊗ k) ◦ (f ⊗ h)
and identities:
The tensor product is bifunctorial, meaning that it preserves composition:
idA⊗B = idA ⊗ idB
Monoidal Categories
f h
D
A
B
C
g k
C E
A B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
(idB ⊗ k) ◦ (f ⊗ idD) = (f ⊗ idE) ◦ (idA ⊗ k) = f ⊗ k
In particular we have the following:
Monoidal Categories
DA
B E
f
k
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
(idB ⊗ k) ◦ (f ⊗ idD) = (f ⊗ idE) ◦ (idA ⊗ k) = f ⊗ k
In particular we have the following:
Monoidal Categories
DA
B E
f
k
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
(idB ⊗ k) ◦ (f ⊗ idD) = (f ⊗ idE) ◦ (idA ⊗ k) = f ⊗ k
In particular we have the following:
Monoidal Categories
DA
B E
f k
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
j : A⊗B → C ⊗D ⊗ E
Of course, it’s quite possible to have arrows between tensors of objects which are not tensors themselves, e.g.
Monoidal Categories
D
A B
E
j
C
could be drawn like:
Ross Duncan ● Ottawa 2009
No lines are drawn for I in the graphical notation:
Monoidal categories have a special unit object called I which is a left and right identity for the tensor:
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
I ⊗A = A = A⊗ I
idI ⊗ f = f = f ⊗ idI
Monoidal Categories
ψ : I → A φ† : A → I φ† ◦ ψ : I → Iψ : I → A φ† : A → I φ† ◦ ψ : I → I
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Any arrow can be multiplied by c using the tensor:
The arrows are called scalars and they enjoy some special properties:
c1 ⊗ c2 = c1 ◦ c2
c1 ⊗ c2 = c2 ⊗ c1
Monoidal Categories
c : I → I
fc
A
B
It doesn’t matter where c is drawn.
f : A→ B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Any arrow can be multiplied by c using the tensor:
The arrows are called scalars and they enjoy some special properties:
c1 ⊗ c2 = c1 ◦ c2
c1 ⊗ c2 = c2 ⊗ c1
Monoidal Categories
c : I → I
fc
A
B
It doesn’t matter where c is drawn.
f : A→ B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
σA,B : A⊗B → B ⊗A
Symmetric Monoidal Categories
A
B A
B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
σA,B : A⊗B → B ⊗A
Symmetric Monoidal Categories
A B
A
B A
B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
σA,B : A⊗B → B ⊗A
Symmetric Monoidal Categories
A B
A
B A
B
=
A
BA
B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
σA,B : A⊗B → B ⊗A
Symmetric Monoidal Categories
A
B
Bgf
B
C
C
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
σA,B : A⊗B → B ⊗A
Symmetric Monoidal Categories
A
B
B
g f
B
C
C
Ross Duncan ● Ottawa 2009
A monoidal category is called †-monoidal if it is equipped with an involutive functor, which reverses the arrows while leaving the objects unchanged, which preserves the tensor structure.
f : A→ B f† : B → A
†-Monoidal Categories
(·)†
f
A
B A
Bf
Ross Duncan ● Ottawa 2009
An arrow is called unitary when:
†-Monoidal Categories
f : A→ B f† : B → A
(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A
BA
f
A
f
f
f
B
Ross Duncan ● Ottawa 2009
An arrow is called unitary when:
†-Monoidal Categories
f : A→ B f† : B → A
(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A
BA
A
f
f
B
Ross Duncan ● Ottawa 2009
An arrow is called unitary when:
†-Monoidal Categories
f : A→ B f† : B → A
(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A
BA
A B
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Category FDHilb
FDHilb is the category of finite dimensional complex Hilbert spaces. It is †-monoidal with the following structure.
• Objects: finite dimensional Hilbert spaces, etc
• Arrows: all linear maps
• Tensor: usual (Kronecker) tensor product;
• is the usual adjoint (conjugate transpose)
A linear map picks out exactly one vector. It is a ket and is the corresponding bra.
Hence is the inner product .
A,B, C,
I = Cf†
ψ : I → Aψ† : A→ I
ψ† ◦ φ : I → I �ψ | φ�
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Preparing a Bell state
H
∧X
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
|0� : I → Q
Example: Preparing a Bell state
H
∧X
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
|0� : I → Q|0� : I → Q
Example: Preparing a Bell state
H
∧X
��10
�⊗
�10
��
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
|0� : I → Q|0� : I → Q
H : Q→ Q
Example: Preparing a Bell state
H
∧X
��10
�⊗
�10
���1√2
�1 00 1
�⊗
�1 11 −1
��·
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
|0� : I → Q|0� : I → Q
H : Q→ Q
∧X : Q⊗Q→ Q⊗Q
Example: Preparing a Bell state
H
∧X
��10
�⊗
�10
���1√2
�1 00 1
�⊗
�1 11 −1
��·
1 0 0 00 1 0 00 0 0 10 0 1 0
·
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Category FRel
FDHilb is the category of finite relations. It is †-monoidal with the following structure.
• Objects: finite sets, etc
• Arrows: Relations
• Tensor: Cartesian product;
• is relational converse:
A relation is simply a subset of X ; its converse is also a subset.
Hence is non-empty iff .
f†
R ⊆ X × Y
X, Y, Z,
X ⊗ Y := X × Y, I = {∗}(x, y) ∈ f ⇔ (y, x) ∈ f†
R ⊆ {∗}×X
S† ◦R : I → I R ∩ S �= ∅
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
d : I → A∗ ⊗A e : A⊗A∗ → I
Compact Closure
cut elimination in MLL proof nets
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
d : I → A∗ ⊗A e : A⊗A∗ → I
Compact Closure
cut elimination in MLL proof nets
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
d : I → A∗ ⊗A e : A⊗A∗ → I
Compact Closure
=
cut elimination in MLL proof nets
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Compact Structure of FDHilbIn FDHilb the compact structure is given by the maps:
whenever is a basis for and is the corresponding basis for the dual space
{ai}i {ai}iAA∗
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Compact Structure of FDHilbIn FDHilb the compact structure is given by the maps:
whenever is a basis for and is the corresponding basis for the dual space
d : 1 �→�
i
ai ⊗ ai e : ai ⊗ ai �→ 1
{ai}i {ai}iAA∗
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Compact Structure of FDHilbIn FDHilb the compact structure is given by the maps:
whenever is a basis for and is the corresponding basis for the dual space
|00�+ |11�√2
In the case of the map picks out the Bell state
which is the simplest example of quantum entanglement.
2 d
d : 1 �→�
i
ai ⊗ ai e : ai ⊗ ai �→ 1
{ai}i {ai}iAA∗
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks|00�+ |11�√
2
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks|00�+ |11�√
2
�00| + �11|√2
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f†
f
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
Bennett et al 1993; Abramsky & Coecke 2004
Ross Duncan ● Ottawa 2009
Thm: one diagram can be deformed to another if and only if their denotations are equal by the structural equations of the category.
Graphical Calculus Theorem
f
hD
A
B
C
g
k
C
E
c
Ross Duncan ● Ottawa 2009
Thm: one diagram can be deformed to another if and only if their denotations are equal by the structural equations of the category.
Graphical Calculus Theorem
f
hD
A
B
C
g
k
C
E
c
= c
f
A
B
C
D
g
C
k
E
h
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
Copying, deleting, and all that
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
δ = δ† = �† =
=
=
Comonoid Laws
= =
� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
δ = δ† = �† =
=
=
= =
Monoid Laws
� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
δ = δ† = �† =� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
δ = δ† = �† =� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
δ = δ† = �† =
Frobenius LawIsometry Law
= = =
� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
Given any finite dimensional Hilbert space we can define an observable structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Example:
define a observable structure over qubits; the standard basis is copied and erased. Note however that:
δ : |0� �→ |00�|1� �→ |11� � : |0�+ |1� �→ 1
δ(|+�) = |00�+|11�√2
showing that not every state can be cloned.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
Given any finite dimensional Hilbert space we can define an observable structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Example:
define a observable structure over qubits; the standard basis is copied and erased. Note however that:
δ : |0� �→ |00�|1� �→ |11� � : |0�+ |1� �→ 1
showing that not every state can be cloned.
=
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Observable Structures
Given any finite dimensional Hilbert space we can define an observable structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Theorem: in FDHilb, observable structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Each (well behaved) observable defines a basis, hence each observable defines an observable structure!
Observable Structures
Given any finite dimensional Hilbert space we can define an observable structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Theorem: in FDHilb, observable structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical Structure begets Compact Structure
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Frobenius Law
Classical Structure begets Compact Structure
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical Structure begets Compact Structure
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Unit Law
Unit Law
Classical Structure begets Compact Structure
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical Structure begets Compact Structure
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical Structure begets Compact Structure
==
Each classical structure induces a self-dual compact structure.
Self duality is addressed in Coecke, Paquette, & Perdrix 2008
Ross Duncan ● Ottawa 2009
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Spider Theorem
Coecke & Paquette 2006
Ross Duncan ● Ottawa 2009
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Spider Theorem
Coecke & Paquette 2006
Ross Duncan ● Ottawa 2009
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Spider Theorem
Coecke & Paquette 2006
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
α
Phase Maps
Spinning around an observable
Ross Duncan ● Ottawa 2009
Let be points of ; we can combine them using the monoid operation
Using the monoid operationψ, φ : I → A A
δ† : A⊗A→ A
ψ ⊙ φ := δ† ◦ (ψ ⊗ φ)
Ross Duncan ● Ottawa 2009
Let be points of ; we can combine them using the monoid operation
Using the monoid operationψ, φ : I → A A
δ† : A⊗A→ A
ψ ⊙ φ := δ† ◦ (ψ ⊗ φ)
θ
θ
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i� |ψ� : I → Q
|φ� : I → Q
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i�
δ†Z =�
1 0 0 00 0 0 1
�|ψ� : I → Q|φ� : I → Q
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i�
δ†Z =�
1 0 0 00 0 0 1
�
|ψ� =�
ψ1
ψ2
�
|φ� =�
φ1
φ2
�|ψ� : I → Q|φ� : I → Q
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i�
δ†Z =�
1 0 0 00 0 0 1
�
|ψ� ⊗ |φ� =
ψ1φ1
ψ1φ2
ψ2φ1
ψ2φ2
|ψ� : I → Q|φ� : I → Q
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i� |ψ� : I → Q
|φ� : I → Q
|ψ� ⊙ |φ� := δ†Z ◦ (|ψ� ⊗ |φ�) =�
ψ1φ1
ψ2φ2
�
Ross Duncan ● Ottawa 2009
Moreover, each point can be lifted to an endomorphism
Using the monoid operationψ : I → A
Λ(ψ) : A→ A
Λ(ψ) := δ† ◦ (ψ ⊗ idA)
This yields a homomorphism of monoids so we have:
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�|ψ� =
�ψ1
ψ2
�
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�|ψ� =
�ψ1
ψ2
�
id⊗ |ψ� =
ψ1 0ψ1 00 ψ2
0 ψ2
Ross Duncan ● Ottawa 2009
Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�|ψ� =
�ψ1
ψ2
�
id⊗ |ψ� =
ψ1 0ψ1 00 ψ2
0 ψ2
ΛZ(ψ) := δ†Z ◦ (id⊗ |ψ�) =�
ψ1 00 ψ2
�
Ross Duncan ● Ottawa 2009
Theorem: any maps constructed from δ, ε, some points and their adjoints, whose graph is connected, is determined by the number of inputs and outputs and the product .
Generalised Spider Theoremψi : I → A
�i ψi
Ross Duncan ● Ottawa 2009
Theorem: any maps constructed from δ, ε, some points and their adjoints, whose graph is connected, is determined by the number of inputs and outputs and the product .
Generalised Spider Theoremψi : I → A
�i ψi
φ
φ
Ross Duncan ● Ottawa 2009
Theorem: any maps constructed from δ, ε, some points and their adjoints, whose graph is connected, is determined by the number of inputs and outputs and the product .
Generalised Spider Theoremψi : I → A
�i ψi
Ross Duncan ● Ottawa 2009
A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
Unbiased Points
|ψ�Λ(ψ)δ
Ross Duncan ● Ottawa 2009
α
A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
Unbiased Points
|ψ�Λ(ψ)δ
α=
Ross Duncan ● Ottawa 2009
A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
Unbiased Points
|ψ�Λ(ψ)δ
α=
Ross Duncan ● Ottawa 2009
A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
Unbiased Points
|ψ�Λ(ψ)δ
= -α
Ross Duncan ● Ottawa 2009
Prop: 1. the unbiased points for form an abelian group w.r.t. to ;2. the arrows generated by the unbiased points form an abelian group w.r.t. composition.
⊙(δ, �)
A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
Unbiased Points
|ψ�Λ(ψ)δ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qubits
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qubits
|0�
|1�
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qubits
π
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qubits
Unbiased pointsπ
|0� + eiα |1�
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qubits
Unbiased pointsπ
α
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qubits
Unbiased pointsπ
α
�1 00 eiα
�=
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�
1 0 00 eiα 00 0 eiβ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�
Unbiased points
1 0 00 eiα 00 0 eiβ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: FRel
D† : (x, x) ∼ x,∀x ∈ X
D† ◦ (id⊗ ψ) is unitary iff ψ = X
Hence the phase group is trivial.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Observables
A very general theory of interference
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Two kinds of points
δ = δ† = �† =� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Two kinds of points
δ = δ† = �† =
Classical Points
� =
i
Those points which can be copied by δ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Two kinds of points
δ = δ† = �† =
Classical Points
� =
i= i i
Those points which can be copied by δ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
* to keep the pictures tidy, a scalar factor has been omitted (and everywhere
from here on)
Two kinds of points
δ = δ† = �† =
Classical Points
� =
i= i i
Those points which can be copied by δ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Two kinds of points
δ = δ† = �† =
Unbiased PointsClassical Points
� =
i= i i
Those points which can be copied by δ
α
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Two kinds of points
δ = δ† = �† =
Unbiased PointsClassical Points
� =
i= i i
Those points which can be copied by δ
=α
α
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Two kinds of points
δ = δ† = �† =
Unbiased PointsClassical Points
� =
i= i i
Those points which can be copied by δ
=
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
δX = �X =δZ = �Z =
|i� �→ |ii� |±� �→ |±±�|+� �→ 1 |0� �→ 1
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
i= i i ⇒
i
i=
Complementary Classical Structures
δX = �X =δZ = �Z =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
i= ⇒
i
i=i i
Complementary Classical Structures
δX = �X =δZ = �Z =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
δX = �X =δZ = �Z =
= =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
Classical points
Unbiased pointsπ
π
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� =
�1 00 eiα
�=!
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� =
�1 00 eiα
�=!
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� =
�1 00 eiα
�=!
=�
cos α2 i sin α
2i sin α
2 cos α2
�! !
|+� =
|−� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�
1 0 00 eiα 00 0 eiβ
∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�
1 + eiα + eiβ 1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ
1 + ωeiα + ωeiβ 1 + eiα + eiβ 1 + ωeiα + ωeiβ
1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ 1 + eiα + eiβ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical points are eigenvectors
αi
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical points are eigenvectors
α i
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical points are eigenvectors
α ii
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Closedness Property
δX = �X =δZ = �Z =
Defn: complentary classical structures are called closed when...
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Closedness Property
δX = �X =δZ = �Z =
⇒i
= i i
j= j j
=i⊙ j i⊙ j i⊙ j
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Closedness Property
δX = �X =δZ = �Z =
⇒i
= i i
j= j j
=i⊙ j i⊙ j i⊙ j
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Closedness Property
δX = �X =δZ = �Z =
In fdHilb it is always possible to construct a pair of mutually unbiased bases with this property...
... but in big enough dimension, it is possible to construct MUBs which are not closed.
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Classical Points form a Subgroup
By the defn of complementarity, we have that:
PROP: If the observable structure is closed, and there are finitely many classical points, then they form a subgroup of the unbiased points, i.e.
In particular, each is a permutation on .
CX ⊆ UZ
CZ ⊆ UX
(CZ ,⊙X) ≤ (UX ,⊙X)(CX ,⊙Z) ≤ (UZ ,⊙Z)
ΛX(zi) CZ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� ==
�1 00 eiα
�!
=�
cos α2 i sin α
2i sin α
2 cos α2
�! !
|+� =
|−� =
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� ==
= !
|+� =
|−� =
�0 11 0
�!
�1 00 −1
�!
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Following are Equivalent:
δX = �X =δZ = �Z =
⇒i
=i i
j
=j j =
i⊙ j i⊙ j i⊙ j
⇒i
=i i
j
=j j =
i⊙ j i⊙ j i⊙ j
1. The classical structures are
closed
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Following are Equivalent:
ii
δX = �X =δZ = �Z =
2. The classical maps are comonoid
homomorphisms
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Following are Equivalent:
i i
δX = �X =δZ = �Z =
2. The classical maps are comonoid
homomorphisms
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Following are Equivalent:
δX = �X =δZ = �Z =
3. The classical maps satisfy canonical
commutation relations
ij i
j ij
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
The Following are Equivalent:
δX = �X =δZ = �Z =
4. The classical structures form a
bialgebra
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
αii
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-α
α
-i-i
ii
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-α
α
-i-i
i
i
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-α
α
-i
i
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-i
i
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
action of red points copied by green on the green phase group
!! !!
! !!
!!
! ! !! !
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
Classical points are symmetries of the phase group.
action of red points copied by green on the green phase group
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�
1 0 00 eiα 00 0 eiβ
∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�
1 + eiα + eiβ 1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ
1 + ωeiα + ωeiβ 1 + eiα + eiβ 1 + ωeiα + ωeiβ
1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ 1 + eiα + eiβ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�
1 0 00 eiα 00 0 eiβ
∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�
1 + eiα + eiβ 1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ
1 + ωeiα + ωeiβ 1 + eiα + eiβ 1 + ωeiα + ωeiβ
1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ 1 + eiα + eiβ
Phase maps are classical when:
α ∈ {0,π
3,2π
3} β = 2π − α
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
CZ = {
1 0 00 1 00 0 1
,
0 1 00 0 11 0 0
,
0 0 11 0 00 1 0
}
CX = {
1 0 00 1 00 0 1
,
1 0 00 ω 00 0 ω
,
1 0 00 ω 00 0 ω
}
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
CZ = {
1 0 00 1 00 0 1
,
0 1 00 0 11 0 0
,
0 0 11 0 00 1 0
}
CX = {
1 0 00 1 00 0 1
,
1 0 00 ω 00 0 ω
,
1 0 00 ω 00 0 ω
}
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
0 1 00 0 11 0 0
·
1
eiα
eiβ
=
eiα
eiβ
1
�
1
ei(β−α)
e−iβ
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
0 1 00 0 11 0 0
·
1
eiα
eiβ
=
eiα
eiβ
1
�
1
ei(β−α)
e−iβ
(α, β) �→ (β − α,−α)
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
0 1 00 0 11 0 0
·
1
eiα
eiβ
=
eiα
eiβ
1
�
1
ei(β−α)
e−iβ
(α, β) �→ (β − α,−α)
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Example: qutrits
0 1 00 0 11 0 0
·
1
eiα
eiβ
=
eiα
eiβ
1
�
1
ei(β−α)
e−iβ
(α, β) �→ (β − α,−α)
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Conclusions
We formalised complementary quantum observables in the language of monoidal categories:
• each observable defines two groups: its classical points and its unbiased points
• Interference between pairs of complementary observables is characterised via a group of automorphisms
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ongoing work 1
Current research directions:
• Investigate connections with multipartite entanglement (with Bill Edwards)
• Algorithmic properties of MBQC (with Simon Perdrix)
• Study toy models of QM based on the automorphism approach.
• Use phase groups to construct MUBs
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Ongoing work 2
Collaborating with Lucas Dixon (Edinburgh) and Aleks Kissinger (Oxford) to automate this calculus using a graphical rewriting system / interactive theorem prover.
- Rewriting properties e.g. normal forms?
- Pattern languages: ellipses and indexed containers?
http://dream.inf.ed.ac.uk/projects/quantomatic/
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
I cited results by these people:
Bob Coecke Samson Abramsky
Dusko Pavlovic
Eric PaquetteSimon Perdrix
Peter Selinger... and Jamie Vicary Use it: http://arxiv.org
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Reference
B. Coecke and R. Duncan, Interacting Quantum Observables, Proceedings of ICALP, 2008
(Long version going onto arxiv.org as soon as we can get the 100s of diagrams past their TeX robot....maybe later today)
Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009
Reference
B. Coecke and R. Duncan, Interacting Quantum Observables, Proceedings of ICALP, 2008
(Long version going onto arxiv.org as soon as we can get the 100s of diagrams past their TeX robot....maybe later today)
Thanks!