[PPT] - Quantum teleportation, diagrams, and the one-time pad A. Kissinger PowerPoint Presentation

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum teleportation, diagrams, and the

ne-time pad
A. Kissinger

Digital Security Group Institute for Computing and Information Sciences Radboud University Nijmegen

23rd November 2016

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 1 / 38

SLIDE 2

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Outline

Process theories Non-separability One-time pad Quantum teleportation

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 2 / 38

SLIDE 3

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Outline

Process theories Non-separability One-time pad Quantum teleportation

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 3 / 38

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Processes

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

more outputs

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Processes

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

more outputs

For example, this function:

f (x, y) = x2 + y

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

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.

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

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

A. Kissinger

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SLIDE 8

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

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

⇑

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

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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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

More processes

Similarly, a computer programs are processes
A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

More processes

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

quicksort

lists lists

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

More processes

Similarly, a 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

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

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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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

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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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Diagrams

Special cases are parallel composition:

       

g f

A B D E

        ⊗           

a c b

C F

           :=

g f c b a

A B C D E F

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Diagrams

...and sequential composition:

      

g f

E F D C

      



 

a b

A B C D 

  =

b a f g

E F A B

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types

Connections are only allowed where the types match
A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types

Connections are only allowed where the types match, e.g.:

A

h g

B D

A

C

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types

Connections are only allowed where the types match, e.g.:

A

h g

B D

A

C A C

h g

B D

D
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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types

Connections are only allowed where the types match, e.g.:

A

h g

B D

A

C A C

h g

B D

D

h

A B

☠

D

g

A = C D

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types and Process Theories

Types tell us when it makes sense to plug processes together
A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types and Process Theories

Types tell us when it makes sense to plug processes together
Ill-typed diagrams are undefined:

noise love

baby

poo food

quicksort

?

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 10 / 38

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types and Process Theories

Types tell us when it makes sense to plug processes together
Ill-typed diagrams are undefined:

noise love

baby

poo food

quicksort

?

In fact, these processes don’t ever sense to plug together
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 10 / 38

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Types and Process Theories

Types tell us when it makes sense to plug processes together
Ill-typed diagrams are undefined:

noise love

baby

poo food

quicksort

?

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

process theory

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

In the process theory of relations:

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

In the process theory of relations:

system-types are sets
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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

In the process theory of relations:

system-types are sets
processes are relations

{x, y, z}

R

{a, b, c}

=      a → x a → y b → z

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

In the process theory of relations:

system-types are sets
processes are relations

{x, y, z}

R

{a, b, c}

=      a → x a → y b → z

...which we can think of as non-deterministic computations:

{x, y, z}

R

{a, b, c}

=      a → {x, y} b → z c → ∅

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

Relations compose in sequentially just like you learned in school:

c1 c2 c3 b1 b3 b2 b4 a1 a2 a3 c3 c1 a2 a1 c2 a3

R S S ◦ R

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

...and they compose in parallel via the cartesian product.

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

...and they compose in parallel via the cartesian product.

that is, systems compose like this:

A B

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

...and they compose in parallel via the cartesian product.

that is, systems compose like this:

A B

:= {(a, b) | a ∈ A, b ∈ B}

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example: relations

...and they compose in parallel via the cartesian product.

that is, systems compose like this:

A B

:= {(a, b) | a ∈ A, b ∈ B}

so relations compose like this:

S R

:: (a, b) → (c, d) ⇐ ⇒    R :: a → c and

S

:: b → d   

A. Kissinger

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SLIDE 34

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

‘no wire’ is a one-element set:

:= {•}

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

‘no wire’ is a one-element set:

:= {•}

...because:

A

= {(a, •) | a ∈ A} ∼ = A =

A

A. Kissinger

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SLIDE 36

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

‘no wire’ is a one-element set:

:= {•}

...because:

A

= {(a, •) | a ∈ A} ∼ = A =

A

processes from ‘no wire’ represent (non-deterministic) states
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 14 / 38

SLIDE 37

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

‘no wire’ is a one-element set:

:= {•}

...because:

A

= {(a, •) | a ∈ A} ∼ = A =

A

processes from ‘no wire’ represent (non-deterministic) states,

e.g. for a bit: =

→ 0
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 14 / 38

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

‘no wire’ is a one-element set:

:= {•}

...because:

A

= {(a, •) | a ∈ A} ∼ = A =

A

processes from ‘no wire’ represent (non-deterministic) states,

e.g. for a bit: =

→ 0

1

=

→ 1
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 14 / 38

SLIDE 39

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

‘no wire’ is a one-element set:

:= {•}

...because:

A

= {(a, •) | a ∈ A} ∼ = A =

A

processes from ‘no wire’ represent (non-deterministic) states,

e.g. for a bit: =

→ 0

1

=

→ 1

∗

=

→ {0, 1}
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 14 / 38

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

...whereas processes to ‘no wire’ are called effects.
A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

...whereas processes to ‘no wire’ are called effects.These test

for the given state(s): =

0 → •
A. Kissinger

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SLIDE 42

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

...whereas processes to ‘no wire’ are called effects.These test

for the given state(s): =

0 → •

1

=

1 → •
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 15 / 38

SLIDE 43

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

...whereas processes to ‘no wire’ are called effects.These test

for the given state(s): =

0 → •

1

=

1 → •

∗

=

{0, 1} → •
when state meets effect, there are two possibilities:

S T

=

→ •

S T

= ∅

A. Kissinger

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SLIDE 44

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Some processes in relations

...whereas processes to ‘no wire’ are called effects.These test

for the given state(s): =

0 → •

1

=

1 → •

∗

=

{0, 1} → •
when state meets effect, there are two possibilities:

S T

=

→ •

S T

= ∅ These stand for true and false.

A. Kissinger

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SLIDE 45

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

States on two systems

States on two systems are more interesting
A. Kissinger

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SLIDE 46

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

States on two systems

States on two systems are more interesting, e.g.:

ψ :=

∗ → {(0, 0), (1, 1)}
A. Kissinger

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SLIDE 47

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

States on two systems

States on two systems are more interesting, e.g.:

ψ :=

∗ → {(0, 0), (1, 1)}

Interpretation: “I don’t know what bit I have, but I know its the same as yours”

A. Kissinger

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SLIDE 48

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

States on two systems

States on two systems are more interesting, e.g.:

ψ :=

∗ → {(0, 0), (1, 1)}

Interpretation: “I don’t know what bit I have, but I know its the same as yours”

States of the two systems no longer have their own, separate

identities

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 16 / 38

SLIDE 49

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

States on two systems

States on two systems are more interesting, e.g.:

ψ :=

∗ → {(0, 0), (1, 1)}

Interpretation: “I don’t know what bit I have, but I know its the same as yours”

States of the two systems no longer have their own, separate

identities

Hence we get...
A. Kissinger

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SLIDE 50

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Outline

Process theories Non-separability One-time pad Quantum teleportation

A. Kissinger

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Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Separable states

A state ψ on two systems is separable if there exist ψ1, ψ2

such that: ψ =

ψ1 ψ2

A. Kissinger

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SLIDE 52

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Separable states

A state ψ on two systems is separable if there exist ψ1, ψ2

such that: ψ =

ψ1 ψ2

Intuitively: the properties of the system on the left are

independent from those on the right

A. Kissinger

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SLIDE 53

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Separable states

A state ψ on two systems is separable if there exist ψ1, ψ2

such that: ψ =

ψ1 ψ2

Intuitively: the properties of the system on the left are

independent from those on the right

In the deterministic-land, all states to separate...
A. Kissinger

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SLIDE 54

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Characterising non-separability

...which is why non-separable states are way more interesting!
A. Kissinger

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SLIDE 55

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Characterising non-separability

...which is why non-separable states are way more interesting!
But, how do we know we’ve found one?
A. Kissinger

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SLIDE 56

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Characterising non-separability

...which is why non-separable states are way more interesting!
But, how do we know we’ve found one?
i.e. that there do not exist states ψ1, ψ2 such that:

ψ =

ψ1 ψ2

A. Kissinger

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SLIDE 57

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Characterising non-separability

...which is why non-separable states are way more interesting!
But, how do we know we’ve found one?
i.e. that there do not exist states ψ1, ψ2 such that:

ψ =

ψ1 ψ2

Problem: Showing that something doesn’t exist is hard.
A. Kissinger

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SLIDE 58

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Characterising non-separability

Solution: Replace a negative property with a postive one:

A. Kissinger

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SLIDE 59

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Characterising non-separability

Solution: Replace a negative property with a postive one:

Definition

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

A. Kissinger

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SLIDE 60

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Cup-states

By introducing some clever notation:

:= ψ := φ

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SLIDE 61

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Cup-states

By introducing some clever notation:

:= ψ := φ

Then these equations:

φ ψ = ψ φ =

A. Kissinger

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SLIDE 62

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Cup-states

By introducing some clever notation:

:= ψ := φ

Then these equations:

φ ψ = ψ φ =

...look like this:

= =

A. Kissinger

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SLIDE 63

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Yank the wire!

= =

A. Kissinger

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SLIDE 64

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Yank the wire!

= =

A. Kissinger

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SLIDE 65

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example

In relations, there is an obvious choice of cup-state:

:=

∗ → {(0, 0), (1, 1)}
A. Kissinger

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SLIDE 66

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example

In relations, there is an obvious choice of cup-state:

:=

∗ → {(0, 0), (1, 1)}
The associated cap-effect corresponds to “checking if two bits

are the same”: :=

{(0, 0), (1, 1)} → ∗
A. Kissinger

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SLIDE 67

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Example

In relations, there is an obvious choice of cup-state:

:=

∗ → {(0, 0), (1, 1)}
The associated cap-effect corresponds to “checking if two bits

are the same”: :=

{(0, 0), (1, 1)} → ∗
This, plus NOT...

NOT

:=

0 → 1

1 → 0 ...gives us enough to start building interesting stuff.

A. Kissinger

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SLIDE 68

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Outline

Process theories Non-separability One-time pad Quantum teleportation

A. Kissinger

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SLIDE 69

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

An incredibly sophisticated security protocol

Suppose Aleks and Bob each have an envelope with the same

(random) bit sealed inside

A. Kissinger

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SLIDE 70

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

An incredibly sophisticated security protocol

Suppose Aleks and Bob each have an envelope with the same

(random) bit sealed inside

Aleks wants to send a bit to Bob, but is paranoid (as usual)
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 25 / 38

SLIDE 71

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

An incredibly sophisticated security protocol

Suppose Aleks and Bob each have an envelope with the same

(random) bit sealed inside

Aleks wants to send a bit to Bob, but is paranoid (as usual)
He opens his envelope, and tells Bob if the bit inside is the

same as the one he wants to send

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 25 / 38

SLIDE 72

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

An incredibly sophisticated security protocol

Suppose Aleks and Bob each have an envelope with the same

(random) bit sealed inside

Aleks wants to send a bit to Bob, but is paranoid (as usual)
He opens his envelope, and tells Bob if the bit inside is the

same as the one he wants to send

Bob opens his envelope, and:
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 25 / 38

SLIDE 73

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

An incredibly sophisticated security protocol

Suppose Aleks and Bob each have an envelope with the same

(random) bit sealed inside

Aleks wants to send a bit to Bob, but is paranoid (as usual)
He opens his envelope, and tells Bob if the bit inside is the

same as the one he wants to send

Bob opens his envelope, and:
if the bits matched before, Bob now has Aleks’ bit,
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 25 / 38

SLIDE 74

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

An incredibly sophisticated security protocol

Suppose Aleks and Bob each have an envelope with the same

(random) bit sealed inside

Aleks wants to send a bit to Bob, but is paranoid (as usual)
He opens his envelope, and tells Bob if the bit inside is the

same as the one he wants to send

Bob opens his envelope, and:
if the bits matched before, Bob now has Aleks’ bit,
otherwise he flips the bit.
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 25 / 38

SLIDE 75

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad with relations

we can represent the envelopes with the shared random bit as

a cup-state: :=

∗ → {(0, 0), (1, 1)}
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 26 / 38

SLIDE 76

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad with relations

we can represent the envelopes with the shared random bit as

a cup-state: :=

∗ → {(0, 0), (1, 1)}
then checking whether two bits are the same is a

‘measurement’ that Aleks can perform on his systems

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 26 / 38

SLIDE 77

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad with relations

we can represent the envelopes with the shared random bit as

a cup-state: :=

∗ → {(0, 0), (1, 1)}
then checking whether two bits are the same is a

‘measurement’ that Aleks can perform on his systems

There are two possible outcomes:

     := “the same” ,

NOT

:= “NOT the same”     

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 26 / 38

SLIDE 78

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad with relations

...which we can write as:

    

Ui

    

i∈{0,1}

U0

:= &

U1

:=

NOT

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 27 / 38

SLIDE 79

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad with relations

...which we can write as:

    

Ui

    

i∈{0,1}

U0

:= &

U1

:=

NOT

Then, the Ui satisfy:

Ui Ui

=

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 27 / 38

SLIDE 80

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad diagram

So, the OTP protocol looks like this:

Ui

b

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 28 / 38

SLIDE 81

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

One-time pad diagram

So, the OTP protocol looks like this: Bob

Ui

Aleks

ψ

envelope 2 envelope 1 Bob’s fix Aleks’ “measurement” Aleks’ bit

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 28 / 38

SLIDE 82

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Bob

Ui

Aleks

b

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 29 / 38

SLIDE 83

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Ui Ui

Aleks Bob

b

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 29 / 38

SLIDE 84

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

b

Ui

Bob

Ui

Aleks

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 29 / 38

SLIDE 85

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Aleks

b

Bob

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 29 / 38

SLIDE 86

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Outline

Process theories Non-separability One-time pad Quantum teleportation

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 30 / 38

SLIDE 87

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum bits

We go from classical to quantum by changing the process

theory: relations ⇒ quantum maps

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 31 / 38

SLIDE 88

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum bits

We go from classical to quantum by changing the process

theory: relations ⇒ quantum maps

The quantum analogue to a bit is a qubit, which represents

the state of the simplest non-trivial quantum system

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 31 / 38

SLIDE 89

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum bits

We go from classical to quantum by changing the process

theory: relations ⇒ quantum maps

The quantum analogue to a bit is a qubit, which represents

the state of the simplest non-trivial quantum system

Example: polarization of a photon
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 31 / 38

SLIDE 90

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum bits

The state space of a bit consists of two points: 0 and 1
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 32 / 38

SLIDE 91

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum bits

The state space of a bit consists of two points: 0 and 1
...whereas qubits, it forms a sphere:

ψ 1

α θ

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 32 / 38

SLIDE 92

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum bits

The state space of a bit consists of two points: 0 and 1
...whereas qubits, it forms a sphere:

ψ 1

α θ

“Plain old” bits live at the North Pole and the South Pole.
A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 32 / 38

SLIDE 93

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum entanglement

In quantum-land, we can realise a ‘cup’ using quantum

entanglement ⇐ = “Bell state”

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 33 / 38

SLIDE 94

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum entanglement

In quantum-land, we can realise a ‘cup’ using quantum

entanglement ⇐ = “Bell state”

Even though this thing is (slightly) more complicated to

describe, it acts just like before

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 33 / 38

SLIDE 95

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum measurement

We also have a quantum analogue for Aleks’ measurement:

    

Ui

    

i∈{0,1,2,3}

⇐ = “Bell measurement”

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 34 / 38

SLIDE 96

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Quantum measurement

We also have a quantum analogue for Aleks’ measurement:

    

Ui

    

i∈{0,1,2,3}

⇐ = “Bell measurement” where there are now three different ways to “NOT”: U0 := U1 := U2 := U3 :=

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 34 / 38

SLIDE 97

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

OTP ⇒ quantum teleportation

Bob

Ui

Aleks

ψ

envelope 2 envelope 1 Bob’s fix Aleks’ “measurement” Aleks’ bit

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 35 / 38

SLIDE 98

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

OTP ⇒ quantum teleportation

Bob

Ui

Aleks

ψ

Bell state Bob’s fix Bell measurement quantum state

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 35 / 38

SLIDE 99

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Bob

Ui

Aleks

ψ

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 36 / 38

SLIDE 100

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Bob

Ui

Aleks

ψ

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 36 / 38

SLIDE 101

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Bob

Ui

Aleks

ψ

Ui

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 36 / 38

SLIDE 102

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

...and it works

Bob Aleks

ψ

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 36 / 38

SLIDE 103

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Two for the price of one

The moral: In both OTP and teleporation, Aleks must send

Bob i, otherwise the whole thing fails

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 37 / 38

SLIDE 104

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Two for the price of one

The moral: In both OTP and teleporation, Aleks must send

Bob i, otherwise the whole thing fails

By using a shared resource:

:= shared random bit := Bell state

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 37 / 38

SLIDE 105

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Two for the price of one

The moral: In both OTP and teleporation, Aleks must send

Bob i, otherwise the whole thing fails

By using a shared resource:

:= shared random bit := Bell state

Aleks can send one kind of thing:

i ∈ {0, 1} := public data i ∈ {0, 1, 2, 3} := classical data

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 37 / 38

SLIDE 106

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Two for the price of one

The moral: In both OTP and teleporation, Aleks must send

Bob i, otherwise the whole thing fails

By using a shared resource:

:= shared random bit := Bell state

Aleks can send one kind of thing:

i ∈ {0, 1} := public data i ∈ {0, 1, 2, 3} := classical data

...and Bob gets another kind of thing:

b

:= private data

ψ

:= quantum state

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 37 / 38

SLIDE 107

Process theories Non-separability One-time pad Quantum teleportation

Radboud University Nijmegen

Thanks!

A. Kissinger

23rd November 2016 Quantum teleportation, diagrams, and the one-time pad 38 / 38