Quantum communication networks consist of N distant nodes sharing a quantum state.By means of local operation in each node assisted by classical communication, the nodestry to transform the initial state into perfect quantum correlations, that later will beused to perform a quantum information task, such as quantum teleportation or quantumcryptography. Given a network, defined by a geometry of nodes and connections, it iscrucial to understand whether it is possible to establish long-distance quantum correla-tions, in the sense that the correlations between two end points of the network do notdecrease exponentially with the number of intermediate connections. In this contribu-tion, we present our recent findings on the distribution of entanglement through quan-tum networks. In the case of one-dimensional chains of connected quantum systems, theresults are hardly surprising: a non-exponential decay is possible only when the entangle-ment in the connections between nodes is larger than a maximally entangled state of twoqubits. The picture becomes much richer and interesting for networks of dimension largerthan one: long-distance correlations can be established even when the connecting nodesare not maximally entangled. Actually, the problem of establishing maximally entangledstates between nodes is related to classical percolation in statistical mechanics. We show,then, that statistical concepts, such as percolation and phase transitions, can be usedto optimize the entanglement distribution through quantum networks. Remarkably, thequantum features allow going beyond the known results for classical percolation, givingrise to a new type of critical phenomenon that we call entanglement percolation.
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