Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum networks are used to describe a wide range of phenomena, such as phase transitions, intricate aspects of many-body quantum systems, or even characteristic features of a future quantum internet. Random quantum networks and their associated directed graphs are employed for capturing statistically dominant features of complex quantum systems. Here, we develop an efficient iterative method capable of evaluating the probability of a graph being strongly connected. It is proven that random directed graphs with constant edge-establishing probability are typically strongly connected, i.e., any ordered pair of vertices is connected by a directed path. This typical topological property of directed random graphs is exploited to demonstrate universal features of the asymptotic evolution of large random qubit networks. These results are independent of our knowledge of the details of the network topology. These findings suggest that other highly complex networks, such as a future quantum internet, may also exhibit similar universal properties.
Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum networks are used to describe a wide range of phenomena, such as phase transitions, intricate aspects of many-body quantum systems, or even characteristic features of a future quantum internet. Random quantum networks and their associated directed graphs are employed for capturing statistically dominant features of complex quantum systems. Here, we develop an efficient iterative method capable of evaluating the probability of a graph being strongly connected. It is proven that random directed graphs with constant edge-establishing probability are typically strongly connected, i.e., any ordered pair of vertices is connected by a directed path. This typical topological property of directed random graphs is exploited to demonstrate universal features of the asymptotic evolution of large random qubit networks. These results are independent of our knowledge of the details of the network topology. These findings suggest that other highly complex networks, such as a future quantum internet, may also exhibit similar universal properties.
We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of random graphs known as complex networks. In the case of classical random graphs, we derive an analytic expression for the averaged entanglement entropy while for general complex networks we rely on numerics. For a large number of nodes n we find a scaling where both the prefactor c and the sub-leading O (1) term g e are characteristic of the different classes of complex networks. In particular, g e encodes topological features of the graphs and is named network topological entropy. Our results suggest that quantum entanglement may provide a powerful tool for the analysis of large complex networks with non-trivial topological properties.
The practical utility of a quantum network depends on its ability to establish entanglement between arbitrary node pairs with quality sufficient to execute entanglement enabled tasks. This capability can be assessed globally, through aggregate performance over all node pairs, as well as locally, at the level of individual nodes. Since entanglement-based connections form a layer above the underlying physical topology, quantum connectivity is not adequately captured by classical topological connectivity metrics. To enable characterisation of the quantum connectivity at the level of the network (or its subnetworks), we introduce the quantum connectivity measure (QCM), which quantifies the average connection quality between pairs of network nodes. Further, we describe two quantities, the quantum-connected fraction (QCF) and the quantum clustering coefficient (QCC), naturally derived from the QCM, which capture important features of the functional connectivity of the quantum network at the level of the network and an individual node, respectively. These metrics of quantum connectivity depend crucially on the entanglement distribution protocol and the quantum network parameters in addition to its physical topology. We demonstrate the crucial distinction between topological and quantum connectivity, showing that even a fully connected graph can be functionally disconnected for quantum tasks if average network edge-concurrence falls below a critical threshold. These quantum connectivity metrics thus provide important tools for the design, optimization, and benchmarking of future 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 . . . . . . . . . . . . .
The recent development of small-scale quantum networks poses the question of which network-level mechanisms could enable quantum communication to operate at scale in a future Quantum Internet. The question can be answered within a well-established framework where a quantum network is mapped into a classical weighted graph, and communication reliability is assessed using methods from network theory. Previous work relied on this framework to determine sufficient conditions for viable network-wide communication only on special network topologies, such as regular lattices, and to devise communication protocols applicable only to small- and moderate-sized real networks. Here, we overcome both limitations by introducing a family of quantum communication protocols that scale to networks with hundreds of millions of nodes. We show that the competition between increasing communication distance and path redundancy determines their large-scale behavior, and that degree heterogeneity can provide sufficient redundancy to sustain communication in small-world networks. These conclusions are supported by a systematic analysis of both real network topologies and synthetic networks, and an analytical treatment in the limit of infinitely large random scale-free graphs.
Quantum networks are of great interest of late which apply quantum mechanics to transfer information securely. One of the key properties which are exploited is entanglement to transfer information from one network node to another. Applications like quantum teleportation rely on the entanglement between the concerned nodes. Thus, efficient entanglement distribution among network nodes is of utmost importance. Several entanglement distribution methods have been proposed in the literature which primarily rely on attributes, such as, fidelities, link layer network topologies, proactive distribution, etc. This paper studies the centralities of the network when the link layer topology of entanglements (referred to as entangled graph) is driven by usage patterns of peer-to-peer connections between remote nodes (referred to as connection graph) with different characteristics. Three different distributions (uniform, gaussian, and power law) are considered for the connection graph where the two nodes are selected from the same distribution. For the entangled graph, both reactive and proactive entanglements are employed to form a random graph. Results show that the edge centralities (measured as usage frequencies of individual edges during entanglement distribution) of the entangled graph follow power law distributions whereas the growth in entanglements with connections and node centralities (degrees of nodes) are monomolecularly distributed for most of the scenarios. These findings will help in quantum resource management, e.g., quantum technology with high reliability and lower decoherence time may be allocated to edges with high centralities.
The title of the present dissertation Decoherence in Quantum Network sounds very general and all-inclusive. Indeed it embraces two topics (decoherence and quantum network) from the area of Quantum Mechanics each of which is described in all respects by a huge literature developed in the last three decades [...]. Quantum decoherence, as the name lets it mean, is the mechanism that makes a quantum system loose its coherence properties, and with them the capability of giving rise to interference phenomena or to other interesting quantum effects [...]. The key idea promoted by decoherence is the insight that realistic quantum systems are never isolated, but are immersed in the surrounding environment and interact continuously with it [...]. As an example one may consider a two-level quantum system (i.e. a quantum bit, usually shortened with a terminology from information science to \\qubit" ) in contact with a wide environment. Hence, quantum systems are open systems, and continuously interact or exchange information with an external environment whose degrees of freedom are too numerous to be monitored. The resulting correlation between the system and the environment spoils quantum coherence and brings about the transition from a pure quantum state to a mixture of quantum state resulting a classical state. To describe decoherence different kind of approaches can be used (for example Master equation, random matrix, etc). A quantum network typically consists of a number of quantum objects (e.g., atoms, ions, quantum dots, cavities, etc.), to be referred to hereafter as the sites or the nodes of the network. They can interact and the interactions (or their correlations) will be usually described by the edges of a graph. Quantum networks can address different information processing tasks. For instance a quantum state can be transferred from qubit to qubit down a chain solely due to the interactions, that is according to the laws of quantum physics [...]. Quantum networks offer us new opportunities and phenomena as compared to classical networks. An extension to large scale of the idea of a quantum network could lead to a futurible quantum internet [...]. The study of networks has traditionally been the territory of graph theory [...], also with the advent of their quantum versions. Within simple quantum network model information processing is usually described by assuming perfect control of the underlying graph. However, this is not much realistic since randomness is often present and it leads to decoherence effects [...]. In contrast, the conservation of coherence is essential for any quantum information process [...], hence there is a persistent interest in decoherence effects in quantum networks, which motivate us to study models for describing such noisy effects. We consider a simple model of quantum network, employing qubits (spin-1/2 particles) attached to the nodes of an underlying graph and we study the simplest task, namely information storage (on a single and two qubits), when the graph randomly changes in time. Actually we randomly add edges to an initially disconnected graph according to the Gilbert model characterized by a weighting parameter ex [...] and in an identically and independent way at each time step. We find that by increasing ex the dynamics of relevant quantities like fidelity, entropy or concurrence, gradually transforms from damped to damped oscillatory and finally to purely oscillatory. That leads to the paper [see, Information dissipation in random quantum networks, by U. Farooq and S. Mancini, OSID 21(3), 1450004, 2014]. We also study a system composed by pairs of qubits attached to each node of a linear chain, a model that stems from quantum dot arrays. Here we use the approach of evolution with a stochastic Hamiltonian to describe the noisy effects. We then evaluate the effect of two most common disorders, namely exchange coupling and hyperfine interaction fluctuations, in adiabatic preparation of ground state in such model. We show that the adiabatic ground state preparation is highly robust against these disorders making the chain a good analog simulator. Moreover, we also study the adiabatic information transfer, using singlet-triplet states, across the chain. In contrast to ground state preparation the transfer mechanism is highly affected by disorders. This suggested that for communication tasks across such chains adiabatic evolution is not as effective and quantum quenches would be preferable. That leads to the paper [see, Adiabatic many-body state preparation and information transfer in quantum dot arrays, by U. Farooq, A. Bayat, S. Mancini and S. Bose Phys. Rev. B 91, 134303, 2015]. The present work is organized as follows. In chapter 1, we shall give a survey of the various types of approach which can be employed to analyse the dynamics of open quantum systems that leads to decoherence effects. In chapter 2, we shall give a general description about quantum network and its possible applications. In chapter 3, we shall discuss the problem of quantum state transfer in qubit network and shall give a brief overview of some scheme that enable nearly prefect state transfer. In chapter 4, we shall discuss singlet-triplet networks, that is networks having on each site a pair of (generally entangled) qubit. Then within this framework we propose a model stemming from quantum dot array. There we shall address the problem of ground state preparation and state transfer. Finally we shall describe the inherent entanglement of the ground state of strongly correlated systems can be exploited for both classical and quantum communications. In chapter 5, we shall propose a decoherence model for qubit networks based on edges representing XY interactions randomly added to a disconnected graph accordingly to a suitable probability distribution. In this way we shall describe dissipation of information initially localize in single or two qubits all over the network. In chapter 6 we shall model the noisy effects in the quantum dot array introduced in chapter 4 and investigate their consequences on the preparation of ground state and quantum state transfer mechanism. Finally we shall draw conclusions.