'90 '91 '92 '93 '94 '95 '96 '97 '98 '99 '00

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What is being plotted?

'90 '91 '92 '93 '94 '95 '96 '97 '98 '99 '00

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20

30

40

50

Answer: Number of paperswith “quantum entanglement”

in title or abstract

N. D. Mermin, Phys. Rev. Lett. (1990)

Entanglement is a physical resource:Bennett, DiVincenzo, Smolin and Wootters,Phys. Rev. A (November, 1996)

Michael A. Nielsen

University of Queensland

Entanglement

Goals: 1. To explain why we regard entanglement as a

physical resource, like energy or mass.2. To explain how entanglement can be quantified.3. To explain how the quantitative theory of

entanglement can be used to gain insight into quantum

information processing, and into other physical processes.

Entanglement revisited

Alice Bob

00 11

2

a b

Schroedinger (1935): “I would not call[entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”

Entanglement and classicality

Bell (1964) and Aspect (1982): Entanglement can beused to show that no “locally realistic” (that is, classical) theory of the world is possible.

Further reading: Asher Peres, “Quantum theory: concepts and methods”, Kluwer (1993).

Using entanglement to do stuff

entanglement-based quantum cryptographysuperdense coding

quantum teleportation

Entanglement is a useful resource that can be used to accomplish tasks that would otherwise bedifficult or impossible.

quantum computing

Given an information-processing goal, we canalways ask “What would I gain by throwing someentanglement into the problem?”

Representation independence of entanglement

Properties independent ofphysical representation

00 11

2

Electron spin: 2

HH VVPhoton polarization:

2

etcetera

Qualitative equivalence of different entangled states

1 1 12 22

equivalen2 copies of 00 11 22 is to

00 113 copies of

t

!2

Summary

1. Entanglement is not classical.2. Entanglement is a resource that can be used to do interesting things.3. Entanglement has properties independent of physical representation.4. Different entangled states are qualitatively equivalent to one another.

Can we develop a quantitative theory of entanglement?

What might we get out of such a theory?

(Figure taken from Boston University’s 1999 PY105 class.)

Thermodynamics is a setof high-level principlesgoverning the behaviourof energy.

We hope that the theoryof entanglement will be asimilarly powerful set of high-level principles governingentanglement.

How massive is a given object?

How massive is a given object?

How massive is a given object?

How massive is a given object?

number of standard masses

Mass limnumber of copies of object

A standard unit for entanglement

Alice Bob

00 11

2

Question: Why use the Bell state as the standard unit?

Answer: “Because it’s there” – we’ll do so because it’sclearly an important state, and in the spirit of exploration.

Answer: Later on, we’ll see that choosing the Bell stateleads to some interesting connections with other problems.

How can we “balance” entanglement?

m n

Entanglement limmn

What it means for one state to be“at least as entangled” as another

Alice Bob

Hello Bob … Hello Alice

Alice Bob

o 1o

Entanglement Entanglement

What it means for one state to be“at least as entangled” as another

can be converted to by an ("local operationsand classical communication") prot

LOCl.

Coco

Alice Bob

3 100 11

4 4

o 1o

00 11 (50% of the time)

2

An example of an LOCC protocol

Such a protocol will let us copies of 3 1

00 11 into Bell pairs

dis

4 2

ll

4

i

.

t nn

How the protocol works

Consider the circuit

3 100 11

4 4

0

0 1 U

measure m

'm

2

0

Show that2

Pr 0 1 and

Exerc

0 1 .3

ise:

3

Find a circuit of controlled-nots and single-qubitunitaries to implEx

emercise:

ent .U

1 233

00 00 11

01 01

U

U

Consider the circuit

3 100 11

4 4

0

0 1 U

measure m

'm

Distillation procedure:

0U

measure 03 1

00 114 4

0

3/ 4 1' 00 11

43

00 11 1 w. p.

22

Thus copies of Bell pairs2n

n

1 233

00 00 11

01 01

U

U

How the protocol works

Back to balancing entanglement

m

n

m

n

Not possible in general!

m n

Entanglement limmn

LOCC LOCC

How to balance entanglement

m

n

(1 )m

n

For any 0 and suffi ciently large and :m n

Entanglement limmn

E( ) Bell statesn n

E( ) is the numberof Bell states

maximal

distille that can be

, per copy od f .

E( ) Bell statesn

n

Bell statesn k Show that by local operations

and classical communication, Alice and Bob can't increase the number of Bellpairs the

Exercis

y sh

e:

are.

How much entanglement?

Alice Bob

trA B trAB

E A BS S

That is, Bell states.An nS

Example

Alice Bob

cos 00 sin 11

2Suppose .

3AS

Bell

Bell

Schumacher

compressteleport

Bobcompletes

teleportation

0

Schumacherdecompress

How to go f rom Bell states to copies of , by LOCC

AnSn

Entanglement can only decrease underlocal operations and classical communication

n

n

E( ) Bell statesn

E( ) E( )

An entangled analogue to thesecond law of thermodynamics

E( ) is the numberof Bell states

maximal

distille that can be

, per copy od f .

Approximate teleportation

Alice Bob

00 11

2

Approximate teleportation

Alice Bob

01

The original teleportation protocol

Alice Bob

01

Teleporting entanglement

Alice Bob

Teleporting entanglement

Alice Bob

01 01

The ability to teleport anarbitrary state implies the ability

to teleport entanglement

Approximate teleportation

Alice Bob

E ebits (E < 1)1 ebit

Total initial entanglement between Alice and Bob at most E ebits.

If Alice and Bob only do local operations and classical communicationthen the final entanglement between their systems cannot be morethan when it started.

Approximate teleportation

Alice Bob

At most E ebits

Since the final entanglement is not 1 ebit, some states must beimperfectly teleported.

Approximate teleportation

Alice Bob

E ebits (E < 1)

min minF

min

11 1

3F E

Back to the “Why Bell states?” question

Alice and Bob share a large number ofcopies of , and can do unlimited classical communication,as well as arbitrary operations on th

Phys

eir

ical resourc

local sys

:

.

e

tems

Alice wI nf ormat ants to ion processing t send qubits to ask: Bob.

The qubit communication should takeplace with fi deCriterion f o

lity approacr succ

hing ess:

one.

How many copies of are needed to reliably communicatea qubit f rom Alice to Bob?

Teleportation: shared entanglement and classical communicationenables the communication of qubits.

max # of qubits that can be communicatedEntanglement

copy of

Alice Bob