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

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

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?

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

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]

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!

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

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

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?

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]

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]

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]

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

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

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

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

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

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]

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]

Input Output

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

Elementary quantum repeater using Rubidium atomic ensembles [Hefei 2009] Entanglement distribution over 144 km in 2007 QKD networks [SECOQC 2008, Tokyo 2010]

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

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

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]