1. ICNFP, Kolymbari, Crete, Greece August 28 – September 5, 2013
Eleni Diamanti
Laboratoire Traitement et Communication de l’Information
CNRS – Télécom ParisTech
Quantum entanglement: a unique resource for communication and computation tasks
ICNFP, Kolymbari, Crete, Greece
August 28 – September 5, 2013

2. 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

3. How general is this property of quantum states?
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
How general is this property of quantum states?

4. Definition of entanglement
The state is an entangled state
A state is entangled if and only if it cannot be written in a
state product form :
Indeed, if
then, by expanding as
we find
which leads to a contradiction

5. 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
There is always an experimental test, or else entanglement witness, which can prove that an entangled state is not separable
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]

6. These are tasks impossible to achieve by classical means!
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?
Alice
Bob
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)
These are tasks impossible to achieve by classical means!

7. 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
This is a superposition state, for which
→ the probabilities of measuring the state in the two basis states add up to 1
A qubit is useful when we can
prepare it in a well defined initial state
manipulate it by applying well-controlled operations
measure it

8. Encoding information on entangled states
Information can be encoded on entangled qubits
This is a linear superposition of the state where both particles are in state and the state where they are both in 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
Mixed state = statistical mixture of eigenstates
Alice
Bob

9. Quantum teleportation
Alice
Bob
Alice has a qubit in an unknown state 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?

10. this measurement destroys the entanglement
Alice
Bob
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
[Bennett et al, Phys. Rev. Lett. 1993]

11. this measurement destroys the entanglement
Alice
Bob
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
Alice calls Bob and tells him the measurement result
Bob knows which state he has collapsed onto
[Bennett et al, Phys. Rev. Lett. 1993]

12. this measurement destroys the entanglement
Alice
Bob
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
Alice calls Bob and tells him the measurement result
Bob knows which state he has collapsed onto
Bob performs an appropriate correction operation
Bob obtains the original state
[Bennett et al, Phys. Rev. Lett. 1993]

13. 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
Bob

14. Quantum key distribution
Alice
Bob
Eve
Alice has a very important message that she wants to share with Bob,
such that evil Eve cannot hear it
Solution: one-time pad (or private key encryption)
if Alice and Bob can establish a secret random key that they both know
(but not Eve!),
then Alice can publicly send the encrypted message
Eve cannot find if she does not know , while Bob, knowing ,
can very simply calculate
One-time pad offers unconditional security! 14

15. To establish a random secret key, Alice and Bob
The quantum solution to the key distribution problem is profoundly linked to entanglement
To establish a random secret key, Alice and Bob
share an entangled state
perform local measurements and obtain correlated random outcomes
check if their outcomes violate Bell inequality
[Bennett and Brassard 1984, Ekert 1991]
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

16. By doing a series of local measurements and using the phone
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
Alice
Bob ?
n<m m
YES!
By doing a series of local measurements and using the phone 16

17. 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

18. Local processing & measurement
Quantum repeaters
Local processing & measurement
Alice
Bob
Entanglement distillation
Quantum memory
Quantum memory …
Entanglement swapping
Entangled pair
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
[Briegel et al, Phys. Rev. Lett. 1998]

19. 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
Perform computation U with
single qubit measurements
local corrections
[Raussendorf and Briegel, Phys. Rev. Lett. 2001]

20. Input
Output

21. Is this all close to becoming a reality?
Quantum teleportation
[Vienna1997]
[UIUC1995]
[IBM – Montréal 1992]
With modern optical techniques it is ‘relatively’ easy to generate the state

22. Elementary quantum repeater using Rubidium atomic ensembles [Hefei 2009]
Entanglement distribution over km in 2007
QKD networks
[SECOQC 2008, Tokyo 2010]

23. EU SPACE-QUEST Project in 2015?
MBQC with 4-qubit photon states and 7-qubit trapped ion states
in 2013
Integrated circuit technology for scalability

24. 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

25. 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: ]
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]