1. Recent Progress in Many-Body Theories Barcelona, 20 July 2007 Antonio Acín 1,2 J. Ignacio Cirac 3 Maciej Lewenstein 1,2 1 ICFO-Institut de Ciències Fotòniques (Barcelona) 2 ICREA-Institució Catalana de Recerca i Estudis Avançats 3 Max-Planck Institute for Quantum Optics Cryptographic properties of nonlocal correlations Entanglement Percolation in Quantum Networks

2. Quantum Information Theory (QIT) studies how information can be transmitted and processed when encoded on quantum states. New information applications are possible because of quantum features: communication complexity and computational speed-up, secure information transmission and quantum teleportation. The key resource for all these applications is quantum correlations, or entanglement. A pure state is entangled whenever it cannot be written in a product form: A mixed state is entangled whenever it cannot be obtained by mixing product states: Quantum Information Theory

3. Quantum Communication Distant parties aim at establishing maximally entangled two-qubit states. Crypto: If the parties share this state, they know they have no correlations with any third party. By measuring the state they obtain a perfect secret key. More in general, if the parties have this state they can teleport any qubit. Thus, a maximally entangled state is equivalent to a perfect quantum channel.

4. Entanglement Theory Given a quantum state: Is it entangled? If yes, can the parties transform many copies of it into fewer maximally entangled states? What are the optimal procedures? Entanglement Swapping: A B A B By local operations and classical communication (LOCC) at the repeater, the distant parties are able to establish a maximally entangled state between them.

5. Quantum Networks Quantum Network: N distant nodes share a quantum state ρ. ρ The goal is to establish an entangled state between two distant nodes, A and B, by local operations and classical communication (LOCC).

6. Quantum Networks 1D Structures: the nodes are connected by a series of quantum repeaters. One of our main goals is to consider geometries of larger dimension. There exist several possible figures of merit: The maximum probability such that A and B share a two-qubit maximally entangled state, Briegel, Dür, Cirac and Zoller, PRL’98 The averaged concurrence. The worst-case entanglement. The singlet-conversion probability, SCP. The averaged concurrence. The worst-case entanglement. The singlet-conversion probability, SCP.

7. Quantum Networks We focus on a simple version of the problem where (i) the network has a well-defined geometry and (ii) the state connecting the nodes are pure. φ Despite their apparent simplicity, these networks already contain rich and intriguing features. Example:

8. Classical Entanglement Percolation ABAB φ Φ Majorization Theory: Bond Percolation LatticePercolation Threshold Square Triangular Honeycomb 1/2 0.3473 0.6527 The classical entanglement percolation strategy (CEP) defines some bounds for the minimal amount of entanglement for non-exponential SCP. Nielsen & Vidal

9. Entanglement Percolation Is Classical Entanglement Percolation always optimal? If not, does it predict the right asymptotic behaviour? NO The distribution of entanglement though a quantum network defines a new type of phase transition, an entanglement phase transition that we call entanglement percolation.

10. 1D Geometries 1 Repeater A B A AB AB One has, which is better than the CEP strategy. B ES (zz) Worst-case strategy: the goal is to maximize the minimum of the entanglement over the measurement outcomes. The optimal strategy is ES (zx basis) and gives the same entanglement for all i. The intermediate repeater does not imply any loss of SCP! (this property of course does not scale with the number of repeaters)

11. 1D Geometries Asymptotic regime A R1R2RN B 1.The exponential decay of the SCP whenever automatically follows from this result. 2.Most of these results can be translated to arbitrary dimension, especially for one-way communication LOCC strategies. An exponential decay of the entanglement is observed whenever the connection between the repeater does not majorize the singlet. The same result is obtained by CEP. Verstraete, Martín-Delgado and Cirac, PRL’04

12. 2D Geometries AB CEP: Previous strategy: Not surprisingly, CEP is not optimal for finite lattices. Finite-size entanglement percolation AABBAB A singlet can be established with probability one whenever

13. 2D Geometries Using the previous measurement strategy, we already see some differences with the classical case. Many end points can be connected with probability one!

14. 2D Geometries CEP Combining entanglement swapping and CEP, long-distance entanglement can be established in a network where CEP fails.

15. The distribution of entanglement through quantum networks defines a framework where statistical methods and concepts naturally apply. It leads to a novel type of critical phenomenon, an entanglement phase transition that we call entanglement percolation. Is any amount of pure-state entanglement between the nodes sufficient for entanglement percolation? More examples beyond CEP. Mixed states? Conclusions Raussendorf, Bravyi and Harrington, PRA’05

16. Mixed states In this case, it is much easier to obtain lower bounds for long-distance entanglement. ρ Given a mixed state, there exist many different ensembles: If p E (ρ) is smaller than the percolation threshold probability → long-distance entanglement is impossible.

17. Conclusions Quantum Information Theory Many-Body Systems

18. Thanks for your attention! Antonio Acín, J. Ignacio Cirac and Maciej Lewenstein, Entanglement Percolation in Quantum Networks, Nature Phys. 3, 256 (2007).