Background: PCR has the potential to detect and precisely quantify specific DNA sequences, but it is not yet often used as a fully quantitative method. A number of data collection and processing strategies have been described for the implementation of quantitative PCR. However, they can be experimentally cumbersome, their relative performances have not been evaluated systematically, and they often remain poorly validated statistically and/or experimentally. In this study, we evaluated the performance of known methods, and compared them with newly developed data processing strategies in terms of resolution, precision and robustness.

We investigate entanglement distribution in pure-state quantum networks. We consider the case when non-maximally 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 1D 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 2D networks, finding again counter-intuitive 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.

In recent years, new algorithms and cryptographic protocols based on the laws of quantum physics have been designed to outperform classical communication and computation. We show that the quantum world also opens up new perspectives in the field of complex networks. Already the simplest model of a classical random network changes dramatically when extended to the quantum case, as we obtain a completely distinct behavior of the critical probabilities at which different subgraphs appear. In particular, in a network of N nodes, any quantum subgraph can be generated by local operations and classical communication if the entanglement between pairs of nodes scales as N −2 .On the one hand, complex networks describe a wide variety of systems in nature and society, as chemical reactions in a cell, the spreading of diseases in populations or communications using the Internet [1]. Their study has traditionally been the territory of graph theory, which initially focused on regular graphs, and was extended to random graphs by the mathematicians Paul Erdős and Alfréd Rényi in a series of seminal papers [2,3,4] in the 1950s and 1960s. With the improvement of computing power and the emergence of large databases, these theoretical models have become increasingly important, and in the past few years new properties which seem universal in real networks have been described, as a small-world [5] or a scale-free [6] behavior.On the other hand, quantum networks are expected to be developed in a near future in order to achieve, for instance, perfectly secure communications [7,8]. These networks are based on the laws of quantum physics and will offer us new opportunities and phenomena as compared to their classical counterpart. Recently it has been shown that quantum phase transitions may occur in the entanglement properties of quantum networks defined on regular lattices, and that the use of joint strategies may be beneficial, for example, for quantum teleportation between nodes [9,10]. In this work we introduce a simple model of complex quantum networks, a new class of systems that exhibit some totally unexpected properties. In fact we obtain a completely different classification of their behavior as compared to what one would expect from their classical counterpart.A classical network is mathematically represented by a graph, which is a pair of sets G = (V, E) where V is a set of N nodes (or vertices) and E is a set of L edges (or links) connecting two nodes. The theory of random graphs, aiming to tackle networks with a complex topology, considers graphs in which each pair of nodes i and j are joined by a link with probability p i,j . The simplest and most studied model is the one where this probability is independent of the nodes, with p i,j = p, and the resulting graph is denoted G N,p . The construction of these graphs can be considered as an evolution process:Evolution process of a classical random graph with N = 10 nodes: starting from isolated nodes, we randomly add edges with increasing probability p, to eventually get the co...

Abstract. The concentration and distribution of quantum entanglement is an essential ingredient in emerging quantum information technologies. Much theoretical and experimental effort has been expended in understanding how to distribute entanglement in one-dimensional networks. However, as experimental techniques in quantum communication develop, protocols for multi-dimensional systems become essential. Here, we focus on recent theoretical developments in protocols for distributing entanglement in regular and complex networks, with particular attention to percolation theory and network-based error correction.

We consider the problem of creating a long-distance entangled state between two stations of a network, where neighboring nodes are connected by noisy quantum channels. We show that any two stations can share an entangled pair if the effective probability for the quantum errors is below a certain threshold, which is achieved by a local encoding of the qubits and a global bit-flip correction. In contrast to the conventional quantum-repeater schemes, we do not need to store the qubits in quantum memory for a long time: our protocol is a one-shot process ͑i.e., the elementary entangled pairs are used only once͒ involving one-way classical communication. Furthermore, the overhead of local resources increases only logarithmically with the size of the network, making our proposal favorable to long-distance quantum communication.