Quantum Picturalism Bob Coecke 1 , Chris Heunen 2 , and Aleks - - PowerPoint PPT Presentation

Quantum Picturalism

Bob Coecke1, Chris Heunen2, and Aleks Kissinger3

1University of Oxford 2University of Edinburgh 3Radboud University Nijmegen

Foundations 2016, LSE

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CUP 2016 OUP 2016

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Quantum Picturalism

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Quantum Picturalism

The idea: Describe quantum theory entirely in terms of:

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Quantum Picturalism

The idea: Describe quantum theory entirely in terms of:

f

B A C D

processes

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Quantum Picturalism

The idea: Describe quantum theory entirely in terms of:

f

B A C D

processes connectivity

B A C

g

A

f

D

h

A

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Quantum Picturalism

The idea: Describe quantum theory entirely in terms of:

f

B A C D

processes connectivity

B A C

g

A

f

D

h

A

interaction

=

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Quantum Picturalism

The idea: Describe quantum theory entirely in terms of:

f

B A C D

processes connectivity

B A C

g

A

f

D

h

A

interaction

=

Not in terms of:

• Hilbert space
• self-adjoint operators, unitary transformations
• calculations with matrices/complex numbers
• ....

(though some may be emergent notions)

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Why?

• Simpler!

(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦ (f ⊗ g ⊗ 1 ⊗ 1) ◦ (h ⊗ 1) = (g ⊗ f ) ◦ (1 ⊗ k) ◦ (h ⊗ 1)

vs.

k k h f

=

f g h g

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Why?

• Simpler!

(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦ (f ⊗ g ⊗ 1 ⊗ 1) ◦ (h ⊗ 1) = (g ⊗ f ) ◦ (1 ⊗ k) ◦ (h ⊗ 1)

vs.

k k h f

=

f g h g

• New perspective = new insights

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Why?

• Simpler!

(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦ (f ⊗ g ⊗ 1 ⊗ 1) ◦ (h ⊗ 1) = (g ⊗ f ) ◦ (1 ⊗ k) ◦ (h ⊗ 1)

vs.

k k h f

=

f g h g

• New perspective = new insights
• Reconstruction ⇐ ‘diagrammatic backbone’ + extra assms

e.g. Pavia 2010 and Hardy 2011 Hardy (2010): “we join the quantum picturalism revolution”

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Why?

• Simpler!

(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦ (f ⊗ g ⊗ 1 ⊗ 1) ◦ (h ⊗ 1) = (g ⊗ f ) ◦ (1 ⊗ k) ◦ (h ⊗ 1)

vs.

k k h f

=

f g h g

• New perspective = new insights
• Reconstruction ⇐ ‘diagrammatic backbone’ + extra assms

e.g. Pavia 2010 and Hardy 2011 Hardy (2010): “we join the quantum picturalism revolution”

• A ‘theory playground’

e.g. QT vs. real/boolean-valued/modal QT, stabiliser QT vs. Spekken’s toy theory, OPTs, ...

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Why?

• Simpler!

(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦ (f ⊗ g ⊗ 1 ⊗ 1) ◦ (h ⊗ 1) = (g ⊗ f ) ◦ (1 ⊗ k) ◦ (h ⊗ 1)

vs.

k k h f

=

f g h g

• New perspective = new insights
• Reconstruction ⇐ ‘diagrammatic backbone’ + extra assms

e.g. Pavia 2010 and Hardy 2011 Hardy (2010): “we join the quantum picturalism revolution”

• A ‘theory playground’

e.g. QT vs. real/boolean-valued/modal QT, stabiliser QT vs. Spekken’s toy theory, OPTs, ...

• New calculational tools, applications in quantum

info/computation

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Processes

• A process is anything with zero or more inputs and zero or

more outputs

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Processes

• A process is anything with zero or more inputs and zero or

more outputs

• For example, this function:

f (x, y) = x2 + y

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Processes

• A process is anything with zero or more inputs and zero or

more outputs

• For example, this function:

f (x, y) = x2 + y ...is a process when takes two real numbers as input, and produces a real number as output.

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Processes

• A process is anything with zero or more inputs and zero or

more outputs

• For example, this function:

f (x, y) = x2 + y ...is a process when takes two real numbers as input, and produces a real number as output.

• We could also write it like this:

f

R R R

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Processes

• A process is anything with zero or more inputs and zero or

more outputs

• For example, this function:

f (x, y) = x2 + y ...is a process when takes two real numbers as input, and produces a real number as output.

• We could also write it like this:

f

R R R

• The labels on wires are called system-types or just types

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More processes

• Similarly, computer programs are processes

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More processes

• Similarly, computer programs are processes
• For example, a program that sorts lists might look like this:

quicksort

lists lists

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More processes

• Similarly, computer programs are processes
• For example, a program that sorts lists might look like this:

quicksort

lists lists

• These are also perfectly good processes:

binoculars

light light light light

cooking

bacon breakfast eggs food

baby

love poo noise

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Diagrams

• We can combine simple processes to make more complicted
• nes, described by diagrams:

g f h

D A C B A A

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Diagrams

• We can combine simple processes to make more complicted
• nes, described by diagrams:

g f h

D A C B A A

• The golden rule: only connectivity matters!

k k h f

=

f g h g

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Types and Process Theories

• Connections are only allowed where the types match
• Ill-typed diagrams are undefined:

noise love

baby

poo food

quicksort

?

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Types and Process Theories

• Connections are only allowed where the types match
• Ill-typed diagrams are undefined:

noise love

baby

poo food

quicksort

?

• In fact, these processes don’t ever make sense to plug together

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Types and Process Theories

• Connections are only allowed where the types match
• Ill-typed diagrams are undefined:

noise love

baby

poo food

quicksort

?

• In fact, these processes don’t ever make sense to plug together
• A family of processes which do make sense together is called a

process theory

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Process Theory: Definition

A process theory consists of:

• a set T of system-types,
• a set P of processes

which are:

• closed under forming diagrams:

g f h

D A C B A

→

A A

d

C A

∈ P

A

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Special processes: states and effects

• Processes with no inputs are called states:

ψ

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Special processes: states and effects

• Processes with no inputs are called states:

ψ

Interpret as: preparing a system in a particular configuration, where we don’t care what came before.

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Special processes: states and effects

• Processes with no inputs are called states:

ψ

Interpret as: preparing a system in a particular configuration, where we don’t care what came before.

• Processes with no outputs are called effects:

π

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Special processes: states and effects

• Processes with no inputs are called states:

ψ

Interpret as: preparing a system in a particular configuration, where we don’t care what came before.

• Processes with no outputs are called effects:

π

Interpret as: testing for a property π, where we don’t care what happens after.

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Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ

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Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ Interpret as: what happens when a state meets an effect

ψ π

effect state number

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Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ Interpret as: what happens when a state meets an effect

ψ π

effect state probability

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Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ Interpret as: what happens when a state meets an effect

ψ π

effect state possibility

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Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ Interpret as: what happens when a state meets an effect

ψ π

effect state number This is called the (generalised) Born rule

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Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ Interpret as: what happens when a state meets an effect

ψ π

effect state number This is called the (generalised) Born rule

• From properties of diagrams, we get:

λ

·

µ

:=

λ µ

1 :=

Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 11 / 49

Numbers

• A number is a process with no inputs or outputs, written as:

λ

• r just:

λ Interpret as: what happens when a state meets an effect

ψ π

effect state number This is called the (generalised) Born rule

• From properties of diagrams, we get:

λ

·

µ

:=

λ µ

1 :=

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Process theories in general

Q: What kinds of behaviour can we study using just diagrams, and nothing else?

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Process theories in general

Q: What kinds of behaviour can we study using just diagrams, and nothing else? A: (Non-)separability

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Separability for states

• Separable:

ψ =

ψ1 ψ2

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Separability for states

• Separable:

ψ =

ψ1 ψ2

• vs. ‘completely non-separable’:

Definition

A state ψ is called cup-state if there exists an effect φ, called a cap-effect, such that: φ ψ = ψ φ =

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Cup-states

• By introducing some clever notation:

:= ψ := φ

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Cup-states

• By introducing some clever notation:

:= ψ := φ

• Then these equations:

φ ψ = ψ φ =

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Cup-states

• By introducing some clever notation:

:= ψ := φ

• Then these equations:

φ ψ = ψ φ =

• ...look like this:

= =

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Yank the wire!

= =

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Yank the wire!

= =

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A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

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A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

• Proof. Suppose a cup-state separates:

=

ψ1 ψ2

Then for any f :

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A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

• Proof. Suppose a cup-state separates:

=

ψ1 ψ2

Then for any f :

f

=

A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

• Proof. Suppose a cup-state separates:

=

ψ1 ψ2

Then for any f :

f

=

f

=

A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

• Proof. Suppose a cup-state separates:

=

ψ1 ψ2

Then for any f :

f

=

f

=

ψ2 ψ1

f

=

A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

• Proof. Suppose a cup-state separates:

=

ψ1 ψ2

Then for any f :

f

=

f

=

ψ2 ψ1

f

=

ψ1

f

ψ2

=:

A no-go theorem for separability

Theorem

If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics.

• Proof. Suppose a cup-state separates:

=

ψ1 ψ2

Then for any f :

f

=

f

=

ψ2 ψ1

f

=

ψ1

f

ψ2

=:

φ π

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Transpose

f

A B

∼ =

← →

f

B A

=: f T

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Transpose

f

A B

∼ =

← →

f

B A

=: f T

f

=

f

i.e. (f T)T = f

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Tranpose = rotation

A bit of a deformation:

f

• f

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Tranpose = rotation

A bit of a deformation:

f

• f

allows some clever notation: f := f

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Tranpose = rotation

A bit of a deformation:

f

• f

allows some clever notation: f := f

= = = =

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Transpose = rotation

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Adjoint = reflection

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Adjoint = reflection

ψ

†

→ ψ state ψ testing for ψ

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Adjoint = reflection

ψ

†

→ ψ state ψ testing for ψ Extends from states/effects to all processes:

B A

f

†

→ f

A B

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4 kinds of box

f f f f

adjoint adjoint conjugate conjugate transpose A A B B A A B B

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Doubling

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Doubling

If the ‘numbers’ of our process theory are complex numbers (e.g. as in linear maps), then we have a problem:

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Doubling

If the ‘numbers’ of our process theory are complex numbers (e.g. as in linear maps), then we have a problem: ψ φ effect state complex number = probability!

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Doubling

Solution: multiply by the conjugate: ψ φ

• ψ

φ ψ φ

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Doubling

Solution: multiply by the conjugate: ψ φ

• ψ

φ ψ φ (i.e. use the ‘plain old’ Born rule: φ|ψφ|ψ = |φ|ψ|2)

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Doubling

New problem: We lost this:

ψ π

effect state probability

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Doubling

New problem: We lost this:

ψ π

effect state probability ...which was the basis of our interpretation for states, effects, and numbers.

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Doubling

Solution: Make a new process theory with doubling ‘baked in’:

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Doubling

Solution: Make a new process theory with doubling ‘baked in’: ψ ψ :=

• ψ

φ φ :=

• φ

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Doubling

Solution: Make a new process theory with doubling ‘baked in’: ψ ψ :=

• ψ

φ φ :=

• φ

Then: effect state probability ψ ψ φ φ := :=

• ψ
• φ

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Doubling

The new process theory has doubled systems H := H ⊗ H: :=

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Doubling

The new process theory has doubled systems H := H ⊗ H: := and processes: double   f   := =

• f

f f

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Doubling preserves diagrams

f g h = k l = ⇒

• g
• h
• f

=

• k
• l

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...but kills global phases

λ λ

= 1 (i.e. λ = eiα) = ⇒

double   λ f   = f

λ λ

f = f f =

• f

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Discarding

Doubling also lets us do something we couldn’t do before:

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Discarding

Doubling also lets us do something we couldn’t do before: throw stuff away!

• ψ

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Discarding

Doubling also lets us do something we couldn’t do before: throw stuff away!

• ψ

How? Like this: :=

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Discarding

For normalised ψ, the two copies annihilate:

• ψ

= ψ ψ = ψ ψ =

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Quantum maps

Definition

The process theory of quantum maps has as types (doubled) Hilbert spaces H and as processes:

• f

. . .

• . . .

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Causality

A quantum map is called causal if: Φ =

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Causality

A quantum map is called causal if: Φ = If we discard the output of a process, it doesn’t matter which process happened.

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Causality

A quantum map is called causal if: Φ = If we discard the output of a process, it doesn’t matter which process happened. causal ⇐ ⇒ deterministically physically realisable

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Consequence: no signalling

ρ Claire Bob Aleks Ψ Φ

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Consequence: no signalling

ρ Claire Bob Aleks Ψ Φ

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Consequence: no signalling

ρ Claire Bob Aleks Ψ Φ

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Consequence: no signalling

ρ Claire Bob Aleks Ψ

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Consequence: no signalling

ρ Claire Bob Aleks Ψ

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Consequences of doubling + causality

• Impossibility of deterministic teleporation:

Aleks Bob Bob Aleks = ρ

• Purification/Stinespring dilation

Φ =

• f
• Quantum no-broadcasting theorem

∆ = ∆ =

= ⇒

= ρ Φ

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Classical and quantum interaction

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Classical and quantum interaction

   quantum :=   

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Classical and quantum interaction

   quantum :=    =   classical :=  

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Classical and quantum interaction

encode := measure :=

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Quantum teleportation: take 2

ρ Aleks Bob

? ?

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Quantum teleportation: take 2

ρ Aleks Bob

• U

?

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Quantum teleportation: take 2

ρ

• U

Aleks Bob

• U

Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 37 / 49

Quantum teleportation: take 2

ρ

• U

Aleks Bob

• U

Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 37 / 49

Quantum teleportation: take 2

ρ

• U

Aleks Bob

• U

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Quantum teleportation: take 2

ρ Aleks Bob

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Complementarity

=

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Complementarity

=

Interpretation: (encode in ) THEN (measure in ) = (no data flow)

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e.g. Stern-Gerlach

N S

S N

S N

blocked!

=

X-measurement 1st Z-measurement 2nd Z-measurement

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e.g. Quantum Key Distribution

Aleks Bob = Eve Aleks Eve Bob

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Graphical calculus

Complementarity + group structure = ZX-calculus:

α

... =

β

... ... ... ...

α+β

... ... ...

β

... ... ...

α+β

... ...

α

= ... ≈ ... ... .... .... ≈

• π

2 π 2 π 2

• π

2 π 2

• π

2

• π

2

A sound and complete equational theory for stabilizer quantum mechanics.

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Quantum circuit simplification

π 4

• π

4

• π

4 π 4

• π

2 π 4

= =

π 4

• π

4

=

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Measurement-based quantum computation

Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z

α

...

π

= ...

π

=

α

... ... ... ...

α π π α

... ... ... ... ... ... ... ... ... ... ... ...

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Quantum algorithms

• f

:=

• Uf

⇒ simple derivations of Deutsch-Jozsa, quantum seach, and hidden subgroup algorithms.

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GHZ/Mermin non-locality

quantum theory any local theory

π 2 π 2 π 2 π 2 π 2 π 2

π π π π

= =

yA

i

yC

i

zA

i

yB

i

zC

i

zB

i

yA

i

yB

i

yC

i

zC

i

zB

i

zA

i

= =

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Multipartite entanglement

SLOCC-classification of 3 qubits:

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Automation

Quantomatic:

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• Categorical Quantum Mechanics I: Causal Quantum Processes. Coecke

and Kissinger. arXiv:1510.05468

• Categorical Quantum Mechanics II: Classical-Quantum Interaction.

Coecke and Kissinger. arXiv:1605.08617

• Categories of Quantum and Classical Channels. Coecke, Kissinger,
• Heunen. arXiv:1305.3821

Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 48 / 49

Thanks! Joint work with:

...

Abramsky, Backens, Coecke, Duncan, Edwards, Gogioso, Hadzihasanovic, Heunen, Lal, Merry, Pavlovic, Paquette, Perdrix, Quick, Selinger, Vicary, Wang, Zamdzhiev, ...and many more!

http://quantomatic.github.io

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