categorical quantum mechanicspersonal.strath.ac.uk/ross.duncan/talks/2012/catqm-pt1.pdf ·...
TRANSCRIPT
Ross Duncan ● Brussels 2012
Categorical Quantum Mechanics
Ross DuncanLaboratoire d’Information Quantique
Université Libre de Bruxelles
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Background and Motivations
"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more."
--John von Neumann, letter to G. Birkhoff, 1935
Friday 17 February 2012
Ross Duncan ● Brussels 2012
What is the structure?
Mathematically, QM is formalised over complex Hilbert spaces, an 80 year old formalism that was thought flawed by its creator John von Neumann.
Categorical QM aims to reconstruct quantum mechanics in terms of minimal algebraic structures:
• discover the mathematical heart of the theory
• suggest constraints on successor theories e.g. quantum gravity
• clarify the relationship between quantum and classical computation
• suggest new primitives for the design and analysis of quantum algorithms
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
λx.λy.λz.xz(yz)
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
Hilbert space, unitary transforms, self-adjoint operators.... ?
λx.λy.λz.xz(yz)
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
Hilbert space, unitary transforms, self-adjoint operators....
λx.λy.λz.xz(yz)
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The need for abstraction:
D ~ 2^1764
This is an 8-bit adder
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Motivation & Context
New structural axiomatisation of QM:
• Hilbert space formalism not a perfect fit for QM
• Can also study alternative quantum-like theories
• emphasis on composition
We use Monoidal Categories:
• Very general framework, including Hilbert spaces
• Well studied in literature
• Beautiful diagrammatic presentation
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Motivations
1. Better Theory
• Usual formulation of QM has lots of “junk”
• Which axioms make quantum mechanics quantum?
• Are there other models which have some of these properties?
2. Better Tools
• Reason more easily about quantum processes
- formal tools!
• Synthesise quantum programs more easily
• Type systems for quantum programs?
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According to wikipedia, category theory:
• “deals in an abstract way with mathematical structures and relationships between them.”
Study QM using the tools of category theory to find which structures are responsible for quantum phenomena.
• find other models of those structures
• use these structures as a translation between different quantum settings
• manipulate these structures directly for easier calculations
- diagrammatic notation!
Categorical approach
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Monoidal CategoriesCategorical view point:
• don’t think about states
• do think about processes
Monoidal categories allow us to
• combine processes sequentially
• ... and in parallel
Examples:
• Sets and functions
• Sets and relations
• Finite dimensional Hilbert spaces and linear maps
• ... and many many more
− ◦ −−⊗−
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What needs to be added?
†-Monoidal Categories
Hilbert Space QM Too much stuffToo little structure
Too few objectsToo few equations
Friday 17 February 2012
Ross Duncan ● Brussels 2012
What needs to be added?
†-Monoidal Categories
Hilbert Space QM Too much stuffToo little structure
Too few objectsToo few equations
Add more algebraic gadgetry
Friday 17 February 2012
Ross Duncan ● Brussels 2012
What needs to be added?
†-Monoidal Categories
Hilbert Space QM Too much stuffToo little structure
Too few objectsToo few equations
Add more algebraic gadgetry
Unbiasedness(Phase groups)
Observables(copying and deleting)
Complementarity(Bialgebras)
.
.
.
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What needs to be added?
†-Monoidal Categories
Hilbert Space QM Too much stuffToo little structure
Too few objectsToo few equations
Add more algebraic gadgetry
Expressive enough to do real work
Unbiasedness(Phase groups)
Observables(copying and deleting)
Complementarity(Bialgebras)
.
.
.
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DISCLAIMER
There are going to be a LOT of definitions in this talk.
sorry.
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Monoidal Categories & Diagrams
Pessimistic Diagram Convention
PAST / HEAVEN
FUTURE / HELL
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Why Diagrams?
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Why Diagrams?
12
�11
�⊗
1 0 0 00 1 0 00 0 1 00 0 0 eiγ
⊗�
0 11 0
�
◦
��1 0 0 00 0 0 1
�⊗
�1 11 −1
��◦
�0 11 0
�⊗
1 0 0 00 0 1 00 1 0 00 0 0 1
◦
1 0 0 00 1 0 00 0 0 10 0 1 0
⊗�
1 00 1
�
⊗�
1 00 1
�
◦
1 0 0 00 1 0 00 0 0 10 0 1 0
◦��
cos π6
i sin π6
�⊗
�1 00 eiβ
��
⊗
1 0 0 00 1 0 00 0 1 00 0 0 eiα
= ?
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Historical Examples
Circuits (classical and quantum) :
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Historical Examples
Feynmann Diagrams:
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Historical Examples
“Applications of Negative Dimensional Tensors”
(Penrose)
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Historical Examples
“Applications of Negative Dimensional Tensors”
(Penrose)
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Historical Examples
Proofs-Nets in Linear Logic
(Girard, Danos-Regnier, Tortoro de Falco, many others)
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Historical Examples
Proofs-Nets in Linear Logic
(Blute-Cocket-Seely-Trimble, 1996)
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Historical Examples
Interaction Combinators
(Lafont, 1997)
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Historical Examples
“Coherence for Compact Closed Categories”
(Kelly-Laplaza, 1980)
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Historical Examples
“Coherence for Compact Closed Categories”
(Kelly-Laplaza, 1980)
?Friday 17 February 2012
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Historical Examples
“Geometry of Tensor Calculus I”
(Joyal-Street, 1991)
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Historical Examples
“Traced Monoidal Categories”
(Joyal-Street-Verity, 1996)
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Diagrams
D
A B
E
j
C
a.k.a. Monoidal categories
j : A⊗B → C ⊗D ⊗ E
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Input Systems
Diagrams
D
A B
E
j
C
a.k.a. Monoidal categories
j : A⊗B → C ⊗D ⊗ E
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Input Systems
Diagrams
D
A B
E
j
C
Interaction, or process, or state change, or ...
a.k.a. Monoidal categories
j : A⊗B → C ⊗D ⊗ E
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Input Systems
Diagrams
D
A B
E
j
C
Output Systems
Interaction, or process, or state change, or ...
a.k.a. Monoidal categories
j : A⊗B → C ⊗D ⊗ E
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Input Systems
Diagrams
D
A B
E
j
C
Output Systems
Interaction, or process, or state change, or ...
“Time” a.k.a. Monoidal categories
j : A⊗B → C ⊗D ⊗ E
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A hierarchy of categoriesCategories
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A hierarchy of categoriesCategories
Monoidal Categories
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A hierarchy of categoriesCategories
Monoidal Categories
Symmetric Monoidal Categories
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A hierarchy of categoriesCategories
Monoidal Categories
Symmetric Monoidal Categories
Compact Categories
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A hierarchy of categoriesCategories
Monoidal Categories
Symmetric Monoidal Categories
Compact Categories
†-Categories
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A hierarchy of categoriesCategories
Monoidal Categories
Symmetric Monoidal Categories
Compact Categories
†-Categories
†-Compact Categories
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A hierarchy of categoriesCategories
Monoidal Categories
Symmetric Monoidal Categories
Compact Categories
†-Categories
†-Compact Categories
F.D. Hilbert spacesSets and RelationsStrongly self-dual convex cones
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A category consists of objects etc, and arrows between them:
f : A→ B g : B → C h : C → D
CategoriesA,B, C,
f gA
B
B
C
C
h
D
f : A→ B g : B → C h : C → D
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A category consists of objects etc, and arrows between them:
CategoriesA,B, C,
f
g
A
B
C
C
h
D
f : A→ B g : B → C h : C → D
g ◦ f : A → C
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Ross Duncan ● Brussels 2012
A category consists of objects etc, and arrows between them:
CategoriesA,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Friday 17 February 2012
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A category consists of objects etc, and arrows between them:
CategoriesA,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Friday 17 February 2012
Ross Duncan ● Brussels 2012
A category consists of objects etc, and arrows between them:
CategoriesA,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
Friday 17 February 2012
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A category consists of objects etc, and arrows between them:
CategoriesA,B, C,
f
g
A
B
C
h
D
h ◦ g ◦ f : A → D
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idA : A→ A
Categories
idA : A→ A
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Categories
idA : A→ A
B
f
f ◦ idA : A → B
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Categories
idA : A→ A
B
f
idB ◦ f : A → B
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Categories
idA : A→ A
B
f
f : A→ B
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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)
f ⊗ h : A⊗ C → B ⊗D
Monoidal Categories
f h
D
A
B
C
gf
A
B
h
D
C
C
B
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and identities:
The tensor product is bifunctorial, meaning that it preserves composition:
idA⊗B = idA ⊗ idB
(g ◦ f)⊗ (k ◦ h) = (g ⊗ k) ◦ (f ⊗ h)
Monoidal Categories
f h
D
A
B
C
g k
C E
A B
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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
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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
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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
Friday 17 February 2012
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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
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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
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d : I → A∗ ⊗A e : A⊗A∗ → I
Compact Closure
cut elimination in MLL proof nets
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d : I → A∗ ⊗A e : A⊗A∗ → I
Compact Closure
cut elimination in MLL proof nets
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d : I → A∗ ⊗A e : A⊗A∗ → I
Compact Closure
=
cut elimination in MLL proof nets
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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
fhD
A
B
C
g
k
C
E
c
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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
fhD
A
B
C
g
k
C
E
c
= c
f
A
B
C
D
g
C
k
E
h
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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�
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The Category FRel
FRel is the category of finite relations. It is †-compact 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 �= ∅
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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}iA
A∗
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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}iA
A∗
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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}iA
A∗
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
Bennett et al 1993; Abramsky & Coecke 2004
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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
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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
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
Bennett et al 1993; Abramsky & Coecke 2004
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f
Bennett et al 1993; Abramsky & Coecke 2004
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f
Bennett et al 1993; Abramsky & Coecke 2004
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f
Bennett et al 1993; Abramsky & Coecke 2004
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
f†
f
Bennett et al 1993; Abramsky & Coecke 2004
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Example: Quantum Teleportation
ψ : I → A φ† : A → I φ† ◦ ψ : I → I
Bob
Aleks
Bennett et al 1993; Abramsky & Coecke 2004
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Example: Preparing a Bell state
H
∧X
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|0� : I → Q
Example: Preparing a Bell state
H
∧X
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|0� : I → Q|0� : I → Q
Example: Preparing a Bell state
H
∧X
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|0� : I → Q|0� : I → Q
H : Q→ Q
Example: Preparing a Bell state
H
∧X
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|0� : I → Q|0� : I → Q
H : Q→ Q
∧X : Q⊗Q→ Q⊗Q
Example: Preparing a Bell state
H
∧X
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Classical Structure
Copying, deleting, and all that
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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�
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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)
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“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.
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Classical Structuresδ = δ† = �† =
=
=
Comonoid Laws
= =
� =
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Classical Structuresδ = δ† = �† =
=
=
= =
Monoid Laws
� =
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Classical Structuresδ = δ† = �† =
Frobenius LawIsometry Law
= = =
� =
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Classical Structuresδ = δ† = �† =� =
In other words: a classical structure is a special commutative †-Frobenius algebra
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Classical Structures
Given any finite dimensional Hilbert space we can define a classical structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Example:
define a classical 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.
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Classical Structures
Given any finite dimensional Hilbert space we can define a classical structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Example:
define a classical 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.
=
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Classical Structures
Given any finite dimensional Hilbert space we can define a classical structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Theorem: in FDHilb, classical structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]
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Each (well behaved) observable defines a basis, hence each observable defines a classical structure!
Classical Structures
Given any finite dimensional Hilbert space we can define a classical structure by
δ : A→ A⊗A :: ai �→ ai ⊗ ai
� : A→ I ::�
i
ai �→ 1
Theorem: in FDHilb, classical structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]
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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
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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
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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
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Classical Structure begets Compact Structure
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Frobenius
Classical Structure begets Compact Structure
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Classical Structure begets Compact Structure
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Unit Law
Unit Law
Classical Structure begets Compact Structure
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Classical Structure begets Compact Structure
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Classical Structure begets Compact Structure
==
Each classical structure induces a self-dual compact structure.
Self duality is addressed in Coecke, Paquette, & Perdrix 2008
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Quantum Observables
fundamental : incompatible observables* pos vs mom* X vs Z spins in fd
What is “classical” anyway?
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Quantum Observables
http://dresdencodak.com
fundamental : incompatible observables* pos vs mom* X vs Z spins in fd
What is “classical” anyway?
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Quantum Observables
Each observable quantity O is represented as self-adjoint operator :
•The possible values of O are the eigenvalues of
•When we observe O for a system in state , there is probability of observing .
•If is the outcome of the measurement, the system is then in state .
In general, measuring then will give a different answer than measuring first!
Not every quantum observable is well defined at the same time.
O =�
i λi |ei� �ei|λi O
|ψ��ei | ψ�2 λi
λi|ei�
O1 O2
O2
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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�
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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
✲
Friday 17 February 2012
Ross Duncan ● Brussels 2012
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
✲
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Complementary Observables
a0
a1
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Complementary Observables
a0
a1
Friday 17 February 2012
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Complementary Observables
a0
a1
Friday 17 February 2012
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Complementary Observables
a0
a1
Friday 17 February 2012
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Complementary Observables
a0
a1
b1 b0
Friday 17 February 2012
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Complementary Observables
a0
a1
b1 b0
**
Friday 17 February 2012
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Complementary Observables
a0
a1
b1 b0
**
Friday 17 February 2012
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Complementary Observables
a0
a1
b1 b0
**
|�ai | bj�| =1√D
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Phase Maps
Spinning around an observable
α
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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)
Friday 17 February 2012
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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
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Example: qubits
|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q
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Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q
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Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�
|ψ� =�
ψ1
ψ2
�
|φ� =�
φ1
φ2
�|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q
Friday 17 February 2012
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Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�
|ψ� ⊗ |φ� =
ψ1φ1
ψ1φ2
ψ2φ1
ψ2φ2
|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Example: qubits
|ψ� : I → Q|φ� : I → Q
|ψ� ⊙ |φ� := δ†Z ◦ (|ψ� ⊗ |φ�) =�
ψ1φ1
ψ2φ2
�
δ†Z : Q⊗Q → Q
Friday 17 February 2012
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Example: qubits
|ψ� : I → Q|φ� : I → Q
|ψ� ⊙ |φ� := δ†Z ◦ (|ψ� ⊗ |φ�) =�
ψ1φ1
ψ2φ2
�
a.k.a. the Schur product or convolution product
δ†Z : Q⊗Q → Q
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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:
Friday 17 February 2012
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Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�|ψ� =
�ψ1
ψ2
�
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Example: qubits
δ†Z =�
1 0 0 00 0 0 1
�|ψ� =
�ψ1
ψ2
�
id⊗ |ψ� =
ψ1 0ψ1 00 ψ2
0 ψ2
Friday 17 February 2012
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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
�
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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
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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
Friday 17 February 2012
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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
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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
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A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
α
Unbiased Points
|ψ�Λ(ψ)δ
α=
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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=
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Ross Duncan ● Brussels 2012
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
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Ross Duncan ● Brussels 2012
A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .
Q: When is unitary?
Prop: 1. the unbiased points for form an abelian group w.r.t. 2. the arrows generated by the unbiased points form an abelian group w.r.t. composition.
⊙(δ, �)
Unbiased Points
|ψ�Λ(ψ)δ
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Example: qubits
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Example: qubits
|0�
|1�
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Example: qubits
π
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Example: qubits
Unbiased pointsπ
|0� + eiα |1�
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Example: qubits
Unbiased pointsπ
α
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Example: qubits
Unbiased pointsπ
α
�1 00 eiα
�=
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Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�
1 0 00 eiα 00 0 eiβ
Friday 17 February 2012
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Example: qutrits
∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�
Unbiased points
1 0 00 eiα 00 0 eiβ
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Example: FRel
D† : (x, x) ∼ x,∀x ∈ X
D† ◦ (id⊗ ψ) is unitary iff ψ = X
Hence the phase group is trivial.
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Example: FRelSuppose that we are working with a two element set = {0,1}. Then:
D : 0 ∼ {(0, 0), (1, 1)}1 ∼ {(0, 1), (1, 0)}
E : 0 ∼ ∗Defines a classical structure whose phase group is Z_2
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Example: FRel
Theorem (Pavlovic) : The classical structures of FRel are exactly biproducts of Abelian groups.
It’s easy to check that in this gives a phase group which is just the product group
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Complementary Observables
A very general theory of interference
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Two kinds of pointsδ = δ† = �† =� =
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Two kinds of pointsδ = δ† = �† =
Classical Points
� =
i
Those points which can be copied by
δ
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Two kinds of pointsδ = δ† = �† =
Classical Points
� =
i= i i
Those points which can be copied by
δ
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* to keep the pictures
tidy, a scalar factor
has been omitted (and everywhere
from here on)
Two kinds of pointsδ = δ† = �† =
Classical Points
� =
i= i i
Those points which can be copied by
δ
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Two kinds of pointsδ = δ† = �† =
Unbiased PointsClassical Points
� =
i= i i
Those points which can be copied by
δ
α
Friday 17 February 2012
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Two kinds of pointsδ = δ† = �† =
Unbiased PointsClassical Points
� =
i= i i
Those points which can be copied by
δ
=α
α
Friday 17 February 2012
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Two kinds of pointsδ = δ† = �† =
Unbiased PointsClassical Points
� =
i= i i
Those points which can be copied by
δ
=
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Complementary Classical Structures
δX = �X =δZ = �Z =
|i� �→ |ii� |±� �→ |±±�|+� �→ 1 |0� �→ 1
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i= i i ⇒
i
i=
Complementary Classical Structures
δX = �X =δZ = �Z =
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i= ⇒
i
i=i i
Complementary Classical Structures
δX = �X =δZ = �Z =
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Complementary Classical Structures
δX = �X =δZ = �Z =
= =
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Complementary Classical Structures
Classical points
Unbiased pointsπ
π
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Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� =
�1 00 eiα
�=!
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Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� =
�1 00 eiα
�=!
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Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� =
�1 00 eiα
�=!
=�
cos α2 i sin α
2i sin α
2 cos α2
�! !
|+� =
|−� =
Friday 17 February 2012
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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
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Classical points are eigenvectors
αi
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Classical points are eigenvectors
α i
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Classical points are eigenvectors
α ii
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Complementary Classical Structures form a Hopf Algebra
Theorem:
(* if there are enough classical points)
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Complementary Classical Structures form a Hopf Algebra
= =
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Closedness Property
δX = �X =δZ = �Z =
Defn: complentary classical structures are called closed when...
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Closedness Property
δX = �X =δZ = �Z =
⇒i
= i i
j= j j
=i⊙ j i⊙ j i⊙ j
Friday 17 February 2012
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Closedness Property
δX = �X =δZ = �Z =
⇒i
= i i
j= j j
=i⊙ j i⊙ j i⊙ j
Friday 17 February 2012
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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.
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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
Friday 17 February 2012
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Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� ==
�1 00 eiα
�!
=�
cos α2 i sin α
2i sin α
2 cos α2
�! !
|+� =
|−� =
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Complementary Classical Structures
Classical points
Unbiased pointsπ
π!
|0� =
|1� ==
= !
|+� =
|−� =
�0 11 0
�!
�1 00 −1
�!
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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
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The Following are Equivalent:
ii
δX = �X =δZ = �Z =
2. The classical maps are comonoid homomorphisms
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The Following are Equivalent:
i i
δX = �X =δZ = �Z =
2. The classical maps are comonoid homomorphisms
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The Following are Equivalent:
δX = �X =δZ = �Z =
3. The classical maps satisfy canonical commutation relations
ij i
j ij
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The Following are Equivalent:
δX = �X =δZ = �Z =
4. The classical structures form a bialgebra
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
αii
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-α
α
-i-i
ii
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-α
α
-i-i
i
i
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-α
α
-i
i
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
-i
i
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
Friday 17 February 2012
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
Friday 17 February 2012
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
!! !!
! !!
!!
! ! !! !
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Automorphism Action
Theorem: The classical maps are group automorphisms of the unbiased points:
Classical points are symmetries of the phase group.
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Ross Duncan ● Brussels 2012
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
Friday 17 February 2012
Ross Duncan ● Brussels 2012
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
Friday 17 February 2012
Ross Duncan ● Brussels 2012
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
}
Friday 17 February 2012
Ross Duncan ● Brussels 2012
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
}
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Example: qutrits
0 1 00 0 11 0 0
·
1
eiα
eiβ
=
eiα
eiβ
1
�
1
ei(β−α)
e−iβ
Friday 17 February 2012
Ross Duncan ● Brussels 2012
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)
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Why care about phase groups?
The phase group seems like a rather random structure. Why do we care? The phase group:
• ... determines the underlying arithmetic
• ... fixes the possibility for interference
• ... determines the possible results of CHSH-type experiments, and hence the possibility for non-locality.(I will discuss this more in Part 3 of this series.)
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The ZX-calculus
Quantum processes, diagrammatically
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Z and X ObservablesThe Pauli Z and X observables are strongly complementary, with some additional features:
• The phase group is [0,2π)
• Dimension 2 implies two classical points
• The action of the classical group is α �→ −α
π
π
π
π
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Z and X Observables
• Both observables generate the same compact structure
- can just treat the diagram as an undirected graph
• the Z and X are related by a definable unitary- gives rise to colour change rule
= =
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Generators for diagrams
Defn: A diagram is an undirected open graph generated by the above vertices.
Defn: let be the dagger compact category of diagrams s.t.
(·)† : α �→ −α
D
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Equations
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Equations
· · ·
· · ·
π2
α+ nπ2
π2
π2
π2
· · ·
· · ·
−π2
α
−π2 −π
2
−π2
=
(colour change)
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Semantics for diagramsDefn: Let be the traced monoidal functor such and that and define its action on generators by:
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Representing Qubits
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Representing Paulis
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Representing CNot
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∧X =
1 0 0 00 1 0 00 0 0 10 0 1 0
Representing Logic Gates
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∧X =
1 0 0 00 1 0 00 0 0 10 0 1 0
Representing Logic Gates
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∧X =
1 0 0 00 1 0 00 0 0 10 0 1 0
Representing Logic Gates
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∧X =
1 0 0 00 1 0 00 0 0 10 0 1 0
Representing Logic Gates
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∧X =
1 0 0 00 1 0 00 0 0 10 0 1 0
Representing Logic Gates
ππ
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∧X =
1 0 0 00 1 0 00 0 0 10 0 1 0
Representing Logic Gates
ππ
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Representing Hadamard
π2
π2
π2
H = 1√2
�1 11 −1
�= � �
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More on the Hadamard
π2
π2
π2
H :=
Friday 17 February 2012
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More on the Hadamard
π2
π2
π2
H :=
− π2
π2
π2
π2
π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
π2
π2
π2
H :=
− π2
π2
π2
π2
π2
=
− π2
π
π2
π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
π2
π2
π2
H :=
− π2
π2
π2
π2
π2
=
− π2
π
π2
π2
=
− π2
− π2
π
π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
π2
π2
π2
H =
− π2
− π2
− π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
π2
π2
π2
H =
− π2
− π2
− π2
=
− π2
− π2
− π2
=
colour change
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
π2
π2
π2
H =
− π2
− π2
− π2
=
− π2
− π2
− π2
= =
π2
π2
π2
colour change colour change
Friday 17 February 2012
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More on the Hadamard
Corollary 1:
H
H
=H
†H =
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More on the Hadamard
Corollary 2:
H H
H H
· · ·
· · ·
α
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More on the Hadamard
Corollary 2:
H H
H H
· · ·
· · ·
α
· · ·
· · ·
α+ nπ2
π2
π2
π2
π2
π2
π2
π2
π2
=
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More on the Hadamard
Corollary 2:
H H
H H
· · ·
· · ·
α
· · ·
· · ·
α+ nπ2
π2
π2
π2
π2
π2
π2
π2
π2
=
· · ·
· · ·
α
π2
π2
π2
π2
−π2 −π
2
−π2 −π
2
=
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More on the Hadamard
Corollary 2:
H H
H H
· · ·
· · ·
α
· · ·
· · ·
α+ nπ2
π2
π2
π2
π2
π2
π2
π2
π2
=
· · ·
· · ·
α
π2
π2
π2
π2
−π2 −π
2
−π2 −π
2
=
· · ·
· · ·
α=
Corollary 2.1 :total symmetry between red and green
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More on the Hadamard
Corollary 3:
π2
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More on the Hadamard
Corollary 3:
π2
π2
−π2
π2=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 3:
π2
π2
−π2
π2=
−π2
−π2=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 3:
π2
π2
−π2
π2=
−π2
−π2= −π
2=
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Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 3:
π2
π2
−π2
π2=
−π2
−π2= −π
2=
−π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 3:π2 −π
2
=−π
2π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 3:π2 −π
2
=−π
2π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 3:π2 −π
2
=−π
2π2
=
Friday 17 February 2012
Ross Duncan ● Brussels 2012
More on the Hadamard
Corollary 4: Van den Nest’s theorem
Van den Nest, M., Dehaene, J. and De Moor, B. “Graphical description of the action of local clifford transformations on graph states”, Phys. Rev. A 69 (2004) 022316. arXiv:quant-ph/0411115
Duncan, R. and Perdrix, S. “Graph States and the necessity of Euler Decomposition”, Proc. CiE09 (2009), LNCS 5635. arXiv:0902.0500
Friday 17 February 2012
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∧Z =
1 0 0 00 1 0 00 0 1 00 0 0 −1
Representing Logic Gates
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Example: 2 CZs
H
H
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Example: 2 CZs
H
H
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Example: 2 CZs
HHH
H
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Example: 2 CZs
H
H
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Ross Duncan ● Brussels 2012
Example: 2 CZs
H
H
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Ross Duncan ● Brussels 2012
Example: 2 CZs
H
H
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Example: 2 CZs
H
H
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Example: 2 CZs
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Example: Preparing a Bell state
∧X
H
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Example: Preparing a Bell state
H
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Example: Preparing a Bell state
H
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Ross Duncan ● Brussels 2012
Example: Preparing a Bell state
H
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Example: Preparing a Bell state
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Ross Duncan ● Brussels 2012
Example: Preparing a Bell state
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controlledgate
Zπ/2
j1 = |1�
j0 = |0�
input qubits
Example: 2-Qubit Quantum Fourier Transform
π
H−π/4
H
π/4
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Example: 2-Qubit Quantum Fourier Transform
π
H−π/4
H
π/4
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Example: 2-Qubit Quantum Fourier Transform
π
H−π/4
ππH
π/4
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Example: 2-Qubit Quantum Fourier Transform
π
H−π/4
ππ
H
π/4
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Example: 2-Qubit Quantum Fourier Transform
H−π/4
π
π
ππ/4
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Example: 2-Qubit Quantum Fourier Transform
−π/4π
π
ππ/4
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Example: 2-Qubit Quantum Fourier Transform
−π/4π
π
π/4
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Example: 2-Qubit Quantum Fourier Transform
−π/4π
π
π/4
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Example: 2-Qubit Quantum Fourier Transform
π
π/4 π/4
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Example: 2-Qubit Quantum Fourier Transform
π
π/2
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output qubits
j0 = |0� + eiπ2 |1�
j1 = |0� + eiπ |1�
Example: 2-Qubit Quantum Fourier Transform
π
π/2
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Does the ZX-calculus decide the word problem for the Clifford group?
Ross DuncanLaboratoire d’Information Quantique
Université Libre de Bruxelles
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The Pauli Group
Pauli Group:
We don’t care about phase so Y = XZ
X =
�0 11 0
�Y =
�0 −ii 0
�Z =
�1 00 −1
�
Pi = 1⊗ · · ·⊗ 1⊗ P ⊗ 1⊗ · · ·⊗ 1
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The Clifford Group
The Clifford group is the stabilizer of the Pauli group
CP = P �C
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The Clifford Group
The Clifford group is generated by the following matrices:
Zπ/2 =
�1 00 i
�H =
1
2
�1 11 −1
�
CNOT =
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
Friday 17 February 2012
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The Clifford Group
The Clifford group is generated by the following matrices:
CNOT =
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
Zπ/2 =
�1 00 i
�Xπ/2 =
�ω ωω ω
�
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The Clifford Group
• The Clifford group preserves the the eigenstates of the Pauli operators -- the stabilizer formalism.
• The Clifford group is the basis of quantum error correcting codes
• Clifford circuits can be classically simulated in linear time
• Measurement-based quantum computations with only Pauli measurements can be performed in one layer (i.e. no feed-forward)
• Add almost any gate to the Clifford group to get (approximate) universal QC
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The Clifford Group
The Clifford group gets big fast...
|C1| = 24
|C2| = 11520
|C3| = 92827280
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The Word Problem
Given two Clifford circuits, are they equal?
• or: given one circuit, does it compute the identity?
Obviously this is decidable by just working out the matrix of the circuit, but this may be very large.
• No presentation by generators and relations is known.
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The Completeness Question
How powerful is the ZX-calculus?
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The Completeness Question
Conjecture: the ZX-calculus decides the word problem for Cn
How powerful is the ZX-calculus?
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The Completeness Question
Conjecture: the ZX-calculus decides the word problem for Cn
Conjecture: the ZX-calculus decides the word problem for SU(2)
How powerful is the ZX-calculus?
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The Completeness Question
Conjecture: the ZX-calculus decides the word problem for Cn
Conjecture: the ZX-calculus decides the word problem for SU(2)
Conjecture: the ZX-calculus decides the word problem for SU(2n)
How powerful is the ZX-calculus?
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The Completeness Question
Conjecture: the ZX-calculus decides the word problem for Cn
Conjecture: the ZX-calculus decides the word problem for SU(2)
Conjecture: the ZX-calculus decides the word problem for SU(2n)
How powerful is the ZX-calculus?
Friday 17 February 2012
Ross Duncan ● Brussels 2012
The Clifford Group C1
The Clifford group on 1 qubit (aka C1) is generated by the unitaries:
Z π2
=�
1 00 i
�H = 1√
2
�1 11 −1
�
π2
π2
π2
π2
Friday 17 February 2012
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The Clifford Group C1
Or equivalently:
π2
π2
Z π2
=�
1 00 i
�X π
2= 1√
2
�ω ωω ω
�
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The Clifford Group C1
0 vertices:
1 vertex: π2 π − π
2π2 π − π
2
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The Clifford Group C1
2 vertices: π
π= π
π
ππ2
=− π
2
π
π
− π2
=π2
π
π
− π2
=π2
π
ππ2
=− π
2
π
π2π2
− π2
π2
− π2
− π2
π2
− π2
π2
− π2
− π2
− π2
− π2
π2
π2π2
Friday 17 February 2012
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The Clifford Group C1
3 vertices:
π2π2π2
=π2π2π2
=− π
2
− π2
− π2
=− π
2
− π2
− π2
π2
− π2
π2
=π2
− π2
π2
=− π
2π2
− π2
=− π
2π2
− π2
− π2
− π2
π2
=− π
2
− π2
π2
=π2π2
− π2
=π2π2
− π2
− π2
π2π2
=− π
2π2π2
=π2
− π2
− π2
=π2
− π2
− π2
Friday 17 February 2012
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The Clifford Group C1
Theorem: the ZX-calculus decides the word problem for C1
Proof: the 1 + 6 + 13 + 4 = 24 elements shown are all distinct; any combination of n > 3 vertices can be reduced to smaller number via the spider rule and commutation with Paulis.
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CNOT and TONC
CNOT = TONC =
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CNOT and TONC
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TONC and Swap
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TONC and Swap
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TONC and Swap
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The Clifford Group C2
Several classes of 2-qubit cliffords:
This gives 24 * 24 * ( 1 + 1 + 9 + 9) = 11520 diagrams
Check mechanically: they are all different. Nice!
A ∈ {1, , } B ∈ {1, , }
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C2
IDEA: compose an elements of C2 with a generator of the group, show that you get another one of the diagrams already shown
Friday 17 February 2012
Ross Duncan ● Brussels 2012
C2
IDEA: compose an elements of C2 with a generator of the group, show that you get another one of the diagrams already shown
Friday 17 February 2012
Ross Duncan ● Brussels 2012
C2
IDEA: compose an elements of C2 with a generator of the group, show that you get another one of the diagrams already shown
Friday 17 February 2012
Ross Duncan ● Brussels 2012
Theorem: the ZX-calculus decides the word problem for C2
Friday 17 February 2012