Complementarity -...

Ross Duncan ● Categories Quanta & Concepts ● Perimeter Institute 2009

Kind of old news now:

Phase Groups and Complementarity

Ross DuncanOxford University Computing Laboratory

Motivations

A⇒ B

Motivations

A⇒ B !A−◦B

Motivations

A⇒ B !A−◦B

Hilbert space, unitary transforms,

self-adjoint operators.... ?

Motivations

A⇒ B !A−◦B

Hilbert space, unitary transforms,

self-adjoint operators....

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

In This Talk:

Observables

In This Talk:

Observables Special †-Frobenius Algebrasaka classical structures

aka basis structuresaka observable structures

In This Talk:

Phase Groups

In This Talk:

Phase Groups

MutuallyUnbiased Bases

In This Talk:

Phase Groups

Complementary

In This Talk:

Phase Groups

Hopf Algebras &BialgebrasComplementary

In This Talk:

Phase Groups

Hopf Algebras &BialgebrasComplementary

Automorphisms of

What is a Phase Group?

• An abelian group of unitary maps on the state space

• which leave some observable fixed

Ingredients of the Theory

To construct phase groups we need:

• Tensor products:

- Symmetric monoidal categories

• Unitarity:

- †-symmetric monoidal categories

• Unitarity:

• Observables:

• Unitarity:

• Observables:

- Special †-Frobenius algebras

• Unitarity:

• Observables:

- Special †-Frobenius algebras

This is a very general setting including much more than just quantum mechanics.

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?

DISCLAIMER

There are going to be a LOT of definitions in this talk.

sorry.

Quantum Observables

What is “classical” anyway?

fundamental : incompatible observables* pos vs mom* X vs Z spins in fd

Quantum Observables

What is “classical” anyway?

http://dresdencodak.com

fundamental : incompatible observables* pos vs mom* X vs Z spins in fd

Complementary Observables

|�ai | bj�| =1√D

“Mutually Unbiased Bases”

Z =�

1 00 −1

�X =

�0 11 0

X and Z Spins

α |0�+ β |1�

|0�✛

2 2✲

|−�

p=(α−

β)/2 2

We can measure the spin of qubit |ψ� = α |0� + β |1�

Z =�

1 00 −1

�X =

�0 11 0

X and Z SpinsWe can measure the spin of qubit |ψ� = α |0� + β |1�

|0�✛

|−�

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�)/√

|0�✛

|−�

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�

|φ�

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)

“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.

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.

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?

Monoidal Categories

A theory of interacting processes

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 : A→ B g : B → C h : C → D

Categories

A,B, C,

f : A→ B g : B → C h : C → D

g ◦ f : A → C

Categories

A,B, C,

h ◦ g ◦ f : A → D

Categories

A,B, C,

h ◦ g ◦ f : A → D

Categories

A,B, C,

h ◦ g ◦ f : A → D

Categories

A,B, C,

h ◦ g ◦ f : A → D

idA : A→ A

Categories

idA : A→ A

Categories

idA : A→ A

f ◦ idA : A → B

Categories

idA : A→ A

idB ◦ f : A → B

Categories

idA : A→ A

f : A→ B

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

(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

(idB ⊗ k) ◦ (f ⊗ idD) = (f ⊗ idE) ◦ (idA ⊗ k) = f ⊗ k

In particular we have the following:

Monoidal Categories

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

could be drawn like:

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

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

It doesn’t matter where c is drawn.

f : A→ B

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

It doesn’t matter where c is drawn.

f : A→ B

σA,B : A⊗B → B ⊗A

Symmetric Monoidal Categories

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

(·)†

An arrow is called unitary when:

f : A→ B f† : B → A

(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A

f : A→ B f† : B → A

(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A

f : A→ B f† : B → A

(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A

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�

Example: Preparing a Bell state

|0� : I → Q

|0� : I → Q|0� : I → Q

��10

�⊗

��

|0� : I → Q|0� : I → Q

H : Q→ Q

��10

�⊗

��1√2

�1 00 1

�⊗

�1 11 −1

��·

|0� : I → Q|0� : I → Q

H : Q→ Q

∧X : Q⊗Q→ Q⊗Q

��10

�⊗

��1√2

�1 00 1

�⊗

�1 11 −1

��·

1 0 0 00 1 0 00 0 0 10 0 1 0

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 .

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 �= ∅

d : I → A∗ ⊗A e : A⊗A∗ → I

Compact Closure

cut elimination in MLL proof nets

d : I → A∗ ⊗A e : A⊗A∗ → I

Compact Closure

d : I → A∗ ⊗A e : A⊗A∗ → I

Compact Closure

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∗

d : 1 �→�

ai ⊗ ai e : ai ⊗ ai �→ 1

{ai}i {ai}iAA∗

|00�+ |11�√2

In the case of the map picks out the Bell state

which is the simplest example of quantum entanglement.

d : 1 �→�

ai ⊗ ai e : ai ⊗ ai �→ 1

{ai}i {ai}iAA∗

Example: Quantum Teleportation

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bennett et al 1993; Abramsky & Coecke 2004

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Aleks|00�+ |11�√

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Aleks|00�+ |11�√

�00| + �11|√2

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

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

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

Observable Structures

Copying, deleting, and all that

δ = δ† = �† =

Comonoid Laws

δ = δ† = �† =

Monoid Laws

δ = δ† = �† =� =

δ = δ† = �† =

Frobenius LawIsometry Law

Given any finite dimensional Hilbert space we can define an observable structure by

δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ 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.

δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ 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.

δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ I ::�

ai �→ 1

Theorem: in FDHilb, observable structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]

Each (well behaved) observable defines a basis, hence each observable defines an observable structure!

δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ I ::�

ai �→ 1

Theorem: in FDHilb, observable structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]

Classical Structure begets Compact Structure

Frobenius Law

Unit Law

Each classical structure induces a self-dual compact structure.

Self duality is addressed in Coecke, Paquette, & Perdrix 2008

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

Spider Theorem

Phase Maps

Spinning around an observable

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)

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)

Example: qubits

δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i� |ψ� : I → Q

|φ� : I → Q

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

Example: qubits

δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i�

δ†Z =�

1 0 0 00 0 0 1

|ψ� =�

|φ� =�

�|ψ� : I → Q|φ� : I → Q

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

Example: qubits

δ†Z : Q → Q⊗Q|i� → |i� ⊗ |i� |ψ� : I → Q

|φ� : I → Q

|ψ� ⊙ |φ� := δ†Z ◦ (|ψ� ⊗ |φ�) =�

ψ1φ1

ψ2φ2

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:

Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� =

�ψ1

Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� =

�ψ1

id⊗ |ψ� =

ψ1 0ψ1 00 ψ2

Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� =

�ψ1

id⊗ |ψ� =

ψ1 0ψ1 00 ψ2

ΛZ(ψ) := δ†Z ◦ (id⊗ |ψ�) =�

ψ1 00 ψ2

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

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

Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ

Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ

Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ

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, �)

Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ

Example: qubits

Unbiased pointsπ

|0� + eiα |1�

Example: qubits

Unbiased pointsπ

Example: qubits

Unbiased pointsπ

�1 00 eiα

Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�

1 0 00 eiα 00 0 eiβ

Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�

Unbiased points

1 0 00 eiα 00 0 eiβ

Example: FRel

D† : (x, x) ∼ x,∀x ∈ X

D† ◦ (id⊗ ψ) is unitary iff ψ = X

Hence the phase group is trivial.

A very general theory of interference

Two kinds of points

δ = δ† = �† =� =

Two kinds of points

δ = δ† = �† =

Classical Points

Those points which can be copied by δ

Two kinds of points

δ = δ† = �† =

Classical Points

i= i i

* 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

Two kinds of points

δ = δ† = �† =

Unbiased PointsClassical Points

i= i i

Two kinds of points

δ = δ† = �† =

i= i i

Two kinds of points

δ = δ† = �† =

i= i i

Complementary Classical Structures

δX = �X =δZ = �Z =

|i� �→ |ii� |±� �→ |±±�|+� �→ 1 |0� �→ 1

i= i i ⇒

δX = �X =δZ = �Z =

i= ⇒

δX = �X =δZ = �Z =

Classical points

Unbiased pointsπ

Classical points

Unbiased pointsπ

|0� =

|1� =

�1 00 eiα

Classical points

Unbiased pointsπ

|0� =

|1� =

�1 00 eiα

Classical points

Unbiased pointsπ

|0� =

|1� =

�1 00 eiα

cos α2 i sin α

2i sin α

2 cos α2

�! !

|+� =

|−� =

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

Classical points are eigenvectors

Closedness Property

δX = �X =δZ = �Z =

Defn: complentary classical structures are called closed when...

Closedness Property

δX = �X =δZ = �Z =

j= j j

=i⊙ j i⊙ j i⊙ j

Closedness Property

δX = �X =δZ = �Z =

j= j j

=i⊙ j i⊙ j i⊙ j

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.

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

Classical points

Unbiased pointsπ

|0� =

|1� ==

�1 00 eiα

cos α2 i sin α

2i sin α

2 cos α2

�! !

|+� =

|−� =

Classical points

Unbiased pointsπ

|0� =

|1� ==

|+� =

|−� =

�0 11 0

�1 00 −1

The Following are Equivalent:

δX = �X =δZ = �Z =

=j j =

i⊙ j i⊙ j i⊙ j

=j j =

i⊙ j i⊙ j i⊙ j

1. The classical structures are

closed

δX = �X =δZ = �Z =

2. The classical maps are comonoid

homomorphisms

δX = �X =δZ = �Z =

2. The classical maps are comonoid

homomorphisms

δX = �X =δZ = �Z =

3. The classical maps satisfy canonical

commutation relations

δX = �X =δZ = �Z =

4. The classical structures form a

bialgebra

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

Automorphism Action

! ! !! !

Automorphism Action

Classical points are symmetries of the phase group.

Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�

1 0 00 eiα 00 0 eiβ

∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�

Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�

1 0 00 eiα 00 0 eiβ

∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�

Phase maps are classical when:

α ∈ {0,π

3} β = 2π − α

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

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

Example: qutrits

0 1 00 0 11 0 0

ei(β−α)

e−iβ

Example: qutrits

0 1 00 0 11 0 0

ei(β−α)

e−iβ

(α, β) �→ (β − α,−α)

Example: qutrits

0 1 00 0 11 0 0

ei(β−α)

e−iβ

(α, β) �→ (β − α,−α)

Example: qutrits

0 1 00 0 11 0 0

ei(β−α)

e−iβ

(α, β) �→ (β − α,−α)

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

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

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/

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

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)

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!

Complementarity -...

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