We present a percolation strategy based on multipartite measurements to propagate entanglement in quantum networks. We consider networks spanned on regular lattices whose bonds correspond to pure but nonmaximally entangled pairs of qubits, with any quantum operation allowed at the nodes. Despite significant effort in the past, improvements over standard (classical) percolation have been found for only a few lattices, often with restrictions on the initial amount of entanglement in the bonds. In contrast, multipartite entanglement percolation outperform the classical percolation protocols, as well as all previously known quantum ones, over the entire range of initial entanglement and for every lattice that we considered. Finally, we briefly show that our ideas also find application in noisy networks.
We present a percolation strategy based on multipartite measurements to propagate entanglement in quantum networks. We consider networks spanned on regular lattices whose bonds correspond to pure but nonmaximally entangled pairs of qubits, with any quantum operation allowed at the nodes. Despite significant effort in the past, improvements over standard (classical) percolation have been found for only a few lattices, often with restrictions on the initial amount of entanglement in the bonds. In contrast, multipartite entanglement percolation outperform the classical percolation protocols, as well as all previously known quantum ones, over the entire range of initial entanglement and for every lattice that we considered. Finally, we briefly show that our ideas also find application in noisy networks.
We study strategies for establishing long-distance entanglement in quantum networks. Specifically, we consider networks consisting of regular lattices of nodes, in which the nearest neighbors share a pure but nonmaximally entangled pair of qubits. We look for strategies that use local operations and classical communication. We compare the classical entanglement percolation protocol, in which every network connection is converted with a certain probability to a singlet, with protocols in which classical entanglement percolation is preceded by measurements designed to transform the lattice structure in a way that enhances entanglement percolation. We analyze five examples of such comparisons between protocols, and point out certain rules and regularities in their performance as a function of degree of entanglement and choice of operations.
We investigate entanglement distribution in pure-state quantum networks. We consider the case when nonmaximally entangled two-qubit pure states are shared by neighboring nodes of the network. For a given pair of nodes, we investigate how to generate the maximal entanglement between them by performing local measurements, assisted by classical communication, on the other nodes. We find optimal measurement protocols for both small and large one-dimensional networks. Quite surprisingly, we prove that Bell measurements are not always the optimal ones to perform in such networks. We generalize then the results to simple small two-dimensional (2D) networks, finding again counterintuitive optimal measurement strategies. Finally, we consider large networks with hierarchical lattice geometries and 2D networks. We prove that perfect entanglement can be established on large distances with probability one in a finite number of steps, provided the initial entanglement shared by neighboring nodes is large enough. We discuss also various protocols of entanglement distribution in 2D networks employing classical and quantum percolation strategies.
Quantum networks (QNs) are expected to play a key role in next-generation networks. A QN is comprised of quantum nodes and links (or bonds) representing entangled qubit pairs (EQPs) shared between two nodes. The probability of establishing a pure maximally entangled pair, known as Einstein-Podolsky-Rosen (EPR) pair, between two neighboring nodes represents the linking probability (or bond occupation probability). If such probability exceeds the so-called percolation threshold, the QN exhibits a percolation phenomenon characterized by the emergence of a giant cluster spanning the entire network. In the presence of such a giant cluster, any two arbitrary nodes in the system are connected by a chain of links with a non-zero probability, thus establishing EPR pairs at remote nodes. This work introduces a procedure for obtaining QNs from initially isolated quantum nodes. In the proposed procedure, two copies of EPR pairs are prepared at a source node and two qubits, one from each pair, are sent to a neighboring node through a noisy quantum channel. Then, a pair of mixed quantum states is established between the source node and the neighboring node. By performing local operations and classical communication, an EPR pair can be established between these two nodes with non-zero probability. Repeating this procedure for every pair of source node and neighboring node, a QN exhibiting a percolation phenomenon can be obtained.
Entanglement percolation aims at generating maximal entanglement between any two nodes of a quantum network by utilizing strategies based solely on local operations and classical communication between the nodes. As it happens in classical percolation theory, the topology of the network is crucial, but also the entanglement shared between the nodes of the network. In a network of identically partially entangled states, the network topology determines the minimum entanglement needed for percolation. In this work, we generalize the protocol to scenarios where the initial entanglement shared between any two nodes of the network is not the same but has some randomness. In such cases, we find that for classical entanglement percolation, only the average initial entanglement is relevant. In contrast, the quantum entanglement percolation protocol (within the q-swap framework) degrades under these more realistic conditions as the width of the distribution increases, suggesting that Random CEP may become the optimal LOCC strategy in sufficiently heterogeneous quantum networks.
This Thesis contributes to the theory of entanglement distribution in quantum networks, analyzing the generation of long-distance entanglement in particular.We consider that neighboring stations share one partially entangled pair of qubits, which emphasizes the difficulty of creating remote entanglement in realistic settings.The task is then to design local quantum operations at the stations, such that the entanglement present in the links of the whole network gets concentrated between few parties only, regardless of their spatial arrangement.First, we study quantum networks with a two-dimensional lattice structure, where quantum connections between the stations (nodes) are described by non-maximally entangled pure states (links).We show that the generation of a perfectly entangled pair of qubits over an arbitrarily long distance is possible if the initial entanglement of the links is larger than a threshold.This critical value highly depends on the geometry of the lattice, in particular on the connectivity of the nodes, and is related to a classical percolation problem.We then develop a genuine quantum strategy based on multipartite entanglement, improving both the threshold and the success probability of the generation of long-distance entanglement.Second, we consider a mixed-state definition of the connections of the quantum networks.This formalism is well-adapted for a more realistic description of systems in which noise (random errors) inevitably occurs.New techniques are required to create remote entanglement in this setting, and we show how to locally extract and globally process some error syndromes in order to create useful long-distance quantum correlations.Finally, we turn to networks that have a complex topology, which is the case for most real-world communication networks such as the Internet for instance.Besides many other characteristics, these systems have in common the small-world feature, stating that any two nodes are separated by a few links only.Based on the theory of random graphs, we propose a model of quantum complex networks, which exhibit some totally unexpected properties compared to their classical counterparts.5.1 Propagation of a large GHZ state through the lattice . . .5.2 Bit-flip error correction . . . . . . . . . . . . .
Quantum communication demands efficient distribution of quantum entanglement across a network of connected partners. The search for efficient strategies for the entanglement distribution may be based on percolation theory, which describes evolution of network connectivity with respect to some network parameters. In this framework, the probability to establish perfect entanglement between two remote partners decays exponentially with the distance between them before the percolation transition point, which unambiguously defines percolation properties of any classical network or lattice. Here we introduce quantum networks created with local operations and classical communication, which exhibit non-classical percolation transition points leading to striking communication advantages over those offered by the corresponding classical networks. We show, in particular, how to establish perfect entanglement between any two nodes in the simplest possible network-the 1D chain-using imperfectly entangled pairs of qubits.