Eleni Diamanti
Laboratoire Traitement et Communication de l’Information
CNRS – Télécom ParisTech
Quantum entanglement: a unique
resource for communication and
computation tasks
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ICNFP, Kolymbari, Crete, Greece
August 28 – September 5, 2013
Quantum information
The beautiful and weird features of quantum mechanics were considered
for a long time interesting from a merely philosophical point of view
It is only recently that we realized that they are also a precious
resource for useful applications
for example, interfering with a quantum signal changes it and this
change can be detected
 this is the basis of quantum cryptography!
Quantum information is changing the way we conceive and implement
calculations by proposing methods that have no classical equivalent
quantum computing
It offers the perspective of future communications of higher security
and unprecedented capacities
quantum communication
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Quantum non-locality
We have seen that quantum non-locality is an incredibly powerful tool
Bell inequalities render the non-local properties of quantum theory
accessible to experimental verification
The experimental results contradict predictions derived by
local hidden variable models and confirm the quantum
mechanical predictions
→ reality is non-local!
Bell constructed a correlation test that cannot be passed by local
hidden variable theories but is passed by the quantum state
 
1
2
0
0 1 1

3
How general is this property of quantum states?
Definition of entanglement
The state
 
1
2
0
0 1 1

is an entangled state
A state  is entangled if and only if it cannot be written
   
in a
state product form :
1
Indeed, if  
2
0
then, by expanding as
we find  
1
2
0
0  1 1   
  0   1 ,   0  1
0  1 1    0 0   0 1   1 0   1 1
which leads to a contradiction(  1 /
2,   0,   0,   1 / 2)
4
Entanglement and non-locality
States that can be written as products, or else separable states, can
always be described by models using local hidden variables
Entangled states are incompatible with all theories of local realism and are
required to violate Bell inequalities
Bell test
Separable
There is always an experimental
test, or else entanglement
witness, which can prove that an
entangled state is not separable
The exact nature of the relation between entanglement and
observable non-locality is complex and subject of on-going research
Not all entangled states violate Bell inequalities but they all exhibit
some ‘hidden’ non-locality
[Liang et al, Phys. Rev. A 2012]
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Entanglement as a resource
In addition to their capacity to violate Bell inequalities,
entangled states are the basis of many modern applications of quantum
mechanics
What can two parties, who share an entangled state,
do?
1


0 A 0 B1A1B
Alice
Bob
AB
2


 Quantum teleportation
entanglement is used as a channel to transfer information encoded
on a state perfectly
 Quantum cryptography
entanglement is used to establish unconditionally secure secret
keys, which is known as quantum key distribution (QKD)
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These are tasks impossible to achieve by classical
means!
The unit of quantum information
In quantum information, information is encoded on the qubit
quantum analog to the classical bit
it can be any physical system that can exist in two states
ions, atoms, electrons, photons
The qubit is a vector in a 2-dimensional Hilbert space, spanned by two
basis states
0 ,1
  0  1
   1
This is a superposition state, for which
→ the probabilities of measuring the state in the two basis states add up
to 1
2
2
A qubit is useful when we can
• prepare it in a well defined initial state
• manipulate it by applying well-controlled
operations
• measure it
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Encoding information on entangled
states
Information can be encoded on entangled qubits
 
1
2
0
0 1 1

This is a linear superposition of the state where both particles are in 0
state and the state where they are both in1 state
This is also a maximally entangled state
Apart from witnessing entanglement, we can also quantify it, for
example, by measuring how close to a mixed state is Alice’s system
measured alone
Alice

AB
 0
A
0 B  1
A
1B
Bob
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Quantum teleportation
Alice
Bob

  0  1
AB

1
0

2

Alice has a qubit in an unknown state
A
0
B
1
A
1
B

that she wants to send to Bob
but they don’t have a quantum channel (for example, an optical
fiber)
just a phone line…
she cannot measure the qubit
→ this would destroy it without revealing all the necessary
information
What can Alice do?
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Alice
  0  1
1.
Bob

AB

1
0

2
A
0
B
1
A
1
B

Alice measures her qubit and her half of the entangled pair in a specific
basis

this measurement destroys the entanglement
it randomly projects Bob’s half onto some state (rotation of
)
this state depends on the measurement outcome
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[Bennett et al, Phys. Rev. Lett. 1993]
Alice
  0  1
Bob

AB

1
0

2
A
0
B
1
A
1
B

1.
Alice measures her qubit and her half of the entangled pair in a specific
basis

this measurement destroys the entanglement
it randomly projects Bob’s half onto some state (rotation of
)
this state depends on the measurement outcome
2.
Alice calls Bob and tells him the measurement result
Bob knows which state he has collapsed onto
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[Bennett et al, Phys. Rev. Lett. 1993]
Alice
  0  1
Bob

AB

1
0

2
A
0
B
1
A
1
B

1.
Alice measures her qubit and her half of the entangled pair in a specific
basis

this measurement destroys the entanglement
it randomly projects Bob’s half onto some state (rotation of
)
this state depends on the measurement outcome
2.
Alice calls Bob and tells him the measurement result
Bob knows which state he has collapsed onto
3.

Bob performs an appropriate correction
operation
Bob obtains the original state
[Bennett et al, Phys. Rev. Lett. 1993]
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Quantum teleportation needs an entangled state, used as an off-line
resource, and classical communication
No need for a high quality channel at the time of the communication
The initial unknown state is destroyed at Alice’s domain
The classical bits of information that contain the measurement result do not
reveal information on , 
→ quantum teleportation does not violate the no-cloning theorem!
If the initial state is part of an entangled pair, this process leads
to
entanglement swapping
Charlie

Alice
AC
Bob

AB

BC
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Quantum key distribution
Alice
Bob
Eve
Alice has a very important messageM
Bob,
such that evil Eve cannot hear it
that she wants to share with
Solution: one-time pad (or private key encryption)
if Alice and Bob can establish a secret random keyK
that they both know
(but not Eve!),
then Alice can publicly send the encrypted messageM '  M  K
K
Eve cannot find M if she does not knowK , while Bob, knowing
can very simply calculate M  M ' K
,
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One-time pad offers unconditional security!
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The quantum solution to the key distribution problem is profoundly
linked to entanglement
[Bennett and Brassard 1984, Ekert 1991]
To establish a random secret key, Alice and Bob
share an entangled state  AB
perform local measurements and obtain correlated random outcomes
check if their outcomes violate Bell inequality
Any attempt by Eve to obtain information on the key, for example, by
attempting to entangle herself with Alice and Bob, will inevitably
introduce errors that can be detected
Security derived by monogamy of entanglement
→ only two parties can share perfect correlations, if Alice and Bob share a
maximally entangled state, Eve cannot be correlated with them too
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Entanglement distillation
If we have many copies of a weakly entangled state (and no quantum
channel),
can we get fewer copies of a maximally entangled state
for use in quantum teleportation or quantum key distribution?
Alice
Bob
Bob
Alice
?
m
n<m


AB
 0
A
0 B  1
A
1B
AB
1

0

2
A
0
B
1
YES!
By doing a series of local
measurements and using the
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phone
A
1
B
16

Towards future quantum networks
If future computation and communication systems rely on quantum
resources, then we will need to use connected structures between multiple
users over long distances and with practical components
Quantum states are fragile and the precious correlations vanish when in
contact with the surrounding environment
How can we fight losses, noise, decoherence?
Qubits cannot be amplified and cannot be cloned!
For quantum communications over long distances, an elegant solution
quantum repeaters
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Quantum repeaters
Local
processing &
measurement
Alice
Quantum
memory
Bob
Entanglement distillation
Quantum
memory
…
Entangled pair
Entanglement swapping
A quantum repeater is a small quantum computer!
The capacity of developed systems to preserve quantum states in a hostile
environment will be crucial for quantum communications and quantum
computing
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[Briegel et al, Phys. Rev. Lett. 1998]
Multipartite entangled resources
Multipartite entangled states are less understood than bipartite states
Both witnessing and quantifying the degree of entanglement is
challenging
Such states will be useful for communication and computation tasks in
future quantum networks
For example, measurement based quantum computing (MBQC)
Initial entangled resource state n
Perform computation U with
single qubit measurements
local corrections
n
[Raussendorf and Briegel, Phys. Rev. Lett. 2001]
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Inpu
t
in
Outpu
t
20
U in
Is this all close to becoming a reality?
Quantum teleportation
[Vienna1997]
[UIUC1995]
With modern optical techniques it is
‘relatively’ easy to generate the
state
1
 
H H V V 
2
[IBM – Montréal 1992]
21
Elementary quantum
repeater using Rubidium
atomic ensembles [Hefei
2009]
Entanglement distribution
over 144 km in 2007
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QKD networks
[SECOQC 2008, Tokyo 2010]
EU SPACEQUEST Project in
2015?
MBQC with 4-qubit photon
states and 7-qubit trapped ion
states
in 2013
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Integrated circuit technology for
scalability
Discussion
 Quantum information processing is multidisciplinary
theoretical and experimental physics, computer science, information
theory, engineering,…
.It gives insight to fundamental features of quantum mechanics
But also leads to the development of practical applications based on the
emerging quantum information and communication technologies
 The quantum internet will probably arrive before the quantum laptop!
 Entanglement is a ubiquitous resource in quantum information
 Efficient techniques for generating, characterizing, verifying
and using
entangled states are at the center of research in the field
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Discussion
 Entanglement is also present in other fields!
 Observation of macroscopic entanglement under exploration
[Sekatski et al, Phys. Rev. A 2012]
 Under what conditions quantum coherence can survive in strongly
interacting environments, e.g., in biological tissues?
[Arndt et al, quant-ph arXiv:0911.0155]
 Links with high-energy physics too!
Development of framework for analyzing Heisenberg uncertainty
principle
and Bell inequalities for decaying two-state systems, such as oscillating
meson-antimeson systems [Di Domenico et al, Foundations of Physics 2012]
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