A pedagogical overview of quantum discord

Recent measures of nonclassical correlations are motivated by different notions of classicality and operational means. Quantum discord has received a great deal of attention in studies involving quantum computation, metrology, dynamics, many-body physics, and thermodynamics. In this article I show how quantum discord is different from quantum entanglement from a pedagogical point of view. I begin with a pedagogical introduction to quantum entanglement and quantum discord, followed by a historical review of quantum discord. Next, I give a novel definition of quantum discord in terms of any classically extractable information, a approach that is fitting for the current avenues of research. Lastly, I put forth several arguments for why discord is an interesting quantity to study and why it is of interest to so many researchers in the community.Comment: 17 pages, 6 figures, to appear in special OSID volume of on open system

Broadcasting in Hyper-cylinder graphs

Broadcasting in computer networking means the dissemination of information, which is known initially only at some nodes, to all network members. The goal is to inform every node in the minimal time possible. There are few models for broadcasting; the simplest and the historical model is called the Classical model. In the Classical model, dissemination happens in synchronous rounds, wherein a node may only inform one of its neighbors. The broadcast question is: What is the minimum number of rounds needed for broadcasting, and what broadcast scheme achieves it? For general graphs, these questions are NP-hard, and it is known to be at least 3 - ε inapproximable for any real ε > 0. Even for some very restricted classes of graphs, the questions remain as an NP-hard problem. Little is known about broadcasting in restricted graphs, and only a few classes have a polynomial solution. Parallel and distributed computing is one of the important domains which relies on efficient broadcasting. Hypercube and torus are the most used network topology in this domain. The widespread use is not only due to their simplicity but also is for their efficiency and high robustness (e.g., fault tolerance) while having an acceptable number of links. In this thesis, it is observed that the Cartesian product of a number of path and cycle graphs produces a valuable set of topologies, we called hyper-cylinders, which contain hypercube and Torus as well. Any hyper-cylinder shares many of the beneficial features of hypercube and torus and might be a suitable substitution in some cases. Some hyper-cylinders are also similar to other practically used topologies such as cube-connected cycles. In this thesis, the effect of the Cartesian product on broadcasting and broadcasting of hyper-cylinders under the Classical and Messy models is studied. This will add a valuable class of graphs to the limited classes of graphs which have a polynomially computable broadcast time. In the end, the relation between worst-case originators and diameters in trees is studied, which may help in the broadcast study of a larger class of graphs where any tree is allowed instead of a path in the Cartesian product

Towards reducing the gap between PMEPR of multicarrier and single carrier signals

It has recently been shown that by altering the sign of each subcarrier in a multicarrier system significant reduction in the peak to mean envelope power (PMEPR) can be obtained. In fact, the PMEPR can even be made a constant independent of the number of subcarriers n. However, finding the best sign requires a search over 2^n possible signs which is computationally prohibitive. In this paper, we first propose a greedy algorithm to choose the signs based on p-norm minimization and we prove that it can achieve a PMEPR of order log n. We further decrease the PMEPR by enlarging the search space considered by the greedy algorithm. By ignoring peaks with probability less than l0^-3, simulation results show that the PMEPR of a multicarrier system with 128 subcarriers each one modulated by 64QAM constellations is reduced to 3.4. This implies that at the cost of one bit of information per subcarrier (i.e., not sending information over the sign of each subcarrier) and modest computational complexity in the transmitter, the PMEPR can be reduced from 12.5 to 3.4 which is within 1.6 dB of the PMEPR of a single carrier system with 64QAM modulation

Problems Related to Classical and Universal List Broadcasting

Broadcasting is a fundamental problem in the information dissemination area. In classical broadcasting, a message must be sent from one network member to all other members as rapidly as feasible. Although it has been demonstrated that this problem is NP-Hard for arbitrary graphs, it has several applications in various fields. As a result, the universal lists model, replicating real-world restrictions like the memory limits of nodes in large networks, is introduced as a branch of this problem in the literature. In the universal lists model, each node is equipped with a fixed list and has to follow the list regardless of the originator. In this study, we focus on both classical and universal lists broadcasting. Classical broadcasting is solvable for a few families of networks, such as trees, unicyclic graphs, tree of cycles, and tree of cliques. In this study, we begin by presenting an optimal algorithm that finds the broadcast time of any vertex in a Fully Connected Tree (FCT_n) in O(|V | log log n) time. An FCT_n is formed by attaching arbitrary trees to vertices of a complete graph of size n where |V| is the total number of vertices in the graph. Then, we replace the complete graph with a Hypercube H_k and propose a new heuristic for the Hypercube of Trees (HT_k). Not only does this heuristic have the same approximation ratio as the best-known algorithm, but our numerical results also show its superiority in most experiments. Our heuristic is able to outperform the current upper bound in up to 90% of the situations, resulting in an average speedup of 30%. Most importantly, our results illustrate that it can maintain its performance even if the network size grows, making the proposed heuristic practically useful. Afterward, we focus on broadcasting with universal lists, in which once a vertex is informed, it must follow its corresponding list, regardless of the originator and the neighbor from which it received the message. The problem of broadcasting with universal lists could be categorized into two sub-models: non-adaptive and adaptive. In the latter model, a sender will skip the vertices on its list from which it has received the message, while those vertices will not be skipped in the first model. In this study, we will present another sub-model called fully adaptive. Not only does this model benefit from a significantly better space complexity compared to the classical model, but, as will be proved, it is faster than the two other sub-models. Since the suggested model fits real-world network architectures, we will design optimal broadcast algorithms for well-known interconnection networks such as trees, grids, and cube-connected cycles. We also present an upper bound for tori under the same model. Then we focus on designing broadcast graphs (bg)’s under this model. A bg is a graph with minimum possible broadcast time from any originator. Additionally, a minimum broadcast graph (mbg) is a bg with the minimum possible number of edges. We propose mbg’s on n vertices for n ≤ 10 and sparse bg’s for 11 ≤ n ≤ 14 under the fully-adaptive model. Afterward, we introduce the first infinite families of bg’s under this model, and we prove that hypercubes are mbg under this model. Later, we establish the optimal broadcast time of k−ary trees and binomial trees under the nonadaptive model and provide an upper bound for complete bipartite graphs. We also improved a general upper bound for trees under the same model. We then suggest several general upper bounds for the universal lists by comparing them with the messy broadcasting model. Finally, we propose the first heuristic for this problem, namely HUB-GA: a Heuristic for Universal lists Broadcasting with Genetic Algorithm. We undertake various numerical experiments on frequently used interconnection networks in the literature, graphs with clique-like structures, and synthetic instances in order to cover many possibilities of industrial topologies. We also compare our results with state-of-the-art methods for classical broadcasting, which is proved to be the fastest model among all. Although the universal list model utilizes less memory than the classical model, our algorithm finds the same broadcast time as the classical model in diverse situations

Distributed match-making

In many distributed computing environments, processes are concurrently executed by nodes in a store- and-forward communication network. Distributed control issues as diverse as name server, mutual exclusion, and replicated data management involve making matches between such processes. We propose a formal problem called distributed match-making as the generic paradigm. Algorithms for distributed match-making are developed and the complexity is investigated in terms of messages and in terms of storage needed. Lower bounds on the complexity of distributed match-making are established. Optimal algorithms, or nearly optimal algorithms, are given for particular network topologies

Broadcasting in highly connected graphs

Throughout history, spreading information has been an important task. With computer networks expanding, fast and reliable dissemination of messages became a problem of interest for computer scientists. Broadcasting is one category of information dissemination that transmits a message from a single originator to all members of the network. In the past five decades the problem has been studied by many researchers and all have come to demonstrate that despite its easy definition, the problem of broadcasting does not have trivial properties and symmetries. For general graphs, and even for some very restricted classes of graphs, the question of finding the broadcast time and scheme remains NP-hard. This work uses graph theoretical concepts to explore mathematical bounds on how fast information can be broadcast in a network. The connectivity of a graph is a measure to assess how separable the graph is, or in other words how many machines in a network will have to fail to disrupt communication between all machines in the network. We initiate the study of finding upper bounds on broadcast time b(G) in highly connected graphs. In particular, we give upper bounds on b(G) for k-connected graphs and graphs with a large minimum degree. We explore 2-connected (biconnected) graphs and broadcasting in them. Using Whitney's open ear decomposition in an inductive proof we propose broadcast schemes that achieve an upper bound of ceil(n/2) for classical broadcasting as well as similar bounds for multiple originators. Exploring further, we use a matching-based approach to prove an upper bound of ceil(log(k)) + ceil(n/k) - 1 for all k-connected graphs. For many infinite families of graphs, these bounds are tight. Discussion of broadcasting in highly connected graphs leads to an exploration of dependence between the minimum degree in the graph and the broadcast time of the latter. By using similar techniques and arguments we show that if all vertices of the graph are neighboring linear numbers of vertices, then information dissemination in the graph can be achieved in ceil(log(n)) + C time. To the best of our knowledge, the bounds presented in our work are a novelty. Methods and questions proposed in this thesis open new pathways for research in broadcasting