This dissertation contributes to a better understanding of concepts, theories, and applications of classical and quantum information in various large-scale systems. The dissertation is structured in two parts: \n \nThe first half is concerned with classical systems: \n \nFirst, we study a basic-yet-never-fully-appreciated property of many real-world complex networks---the scale-free (SF) property---which was often considered to be only represented by the degree distribution. We define a new fundamental quantity, however, the degree--degree distance, which can better represent the SF property by showing statistically more significant power laws and better explain the evolution of real-world networks, e.g., Wikipedia Webpages. \n \nSecond, we study brain tractography of a healthy subject by diffusion-weighted magnetic resonance imaging (dMRI) data and find that the dependence of dMRI signal on the interpulse time can decode the smaller-than-resolution brain structure and might unravel how information transmits. This finding is confirmed by Monte-Carlo simulation of water-molecule diffusion which let us understand how to optimally measure the thickness of axon sheets in the brain. \n \nThe second half is concerned with quantum systems: \n \nOur first work is to understand how to establish long-distance entanglement transmission in a quantum network where each link has non-zero concurrence---a measure of bipartite entanglement. We introduce a fundamental statistical theory, concurrence percolation theory (ConPT), and find the existence of an entanglement transmission threshold predicted by ConPT which is lower than the known classical-percolation-based results---a “quantum advantage” that is more general and efficient than expected. ConPT also shows a percolation-like universal critical behavior derived by finite-size analysis. \n \nOur second work is to study continuous-time quantum walk as an open system that strongly interacts with the environment where non-Markovianity may significantly speed up the dynamics. We confirm this speed-up by first introducing a general multi-scale perturbation method that works on integro-differential equations and then building the Hamiltonian on regular networks, e.g., star or complete graphs, which can be mapped to an error correction algorithm scheme of practical significance. \n \nOur third work explores the possible use of entanglement entropy (EE) in machine-learning fields. We introduce a new long-short-term-memory-based recurrent neural network architecture using tensorization techniques to forecast chaotic time series, the learnability of which is determined not only by the number of free parameters but also the tensorization complexity---recognized as how EE scales.
This dissertation contributes to a better understanding of concepts, theories, and applications of classical and quantum information in various large-scale systems. The dissertation is structured in two parts: \n \nThe first half is concerned with classical systems: \n \nFirst, we study a basic-yet-never-fully-appreciated property of many real-world complex networks---the scale-free (SF) property---which was often considered to be only represented by the degree distribution. We define a new fundamental quantity, however, the degree--degree distance, which can better represent the SF property by showing statistically more significant power laws and better explain the evolution of real-world networks, e.g., Wikipedia Webpages. \n \nSecond, we study brain tractography of a healthy subject by diffusion-weighted magnetic resonance imaging (dMRI) data and find that the dependence of dMRI signal on the interpulse time can decode the smaller-than-resolution brain structure and might unravel how information transmits. This finding is confirmed by Monte-Carlo simulation of water-molecule diffusion which let us understand how to optimally measure the thickness of axon sheets in the brain. \n \nThe second half is concerned with quantum systems: \n \nOur first work is to understand how to establish long-distance entanglement transmission in a quantum network where each link has non-zero concurrence---a measure of bipartite entanglement. We introduce a fundamental statistical theory, concurrence percolation theory (ConPT), and find the existence of an entanglement transmission threshold predicted by ConPT which is lower than the known classical-percolation-based results---a “quantum advantage” that is more general and efficient than expected. ConPT also shows a percolation-like universal critical behavior derived by finite-size analysis. \n \nOur second work is to study continuous-time quantum walk as an open system that strongly interacts with the environment where non-Markovianity may significantly speed up the dynamics. We confirm this speed-up by first introducing a general multi-scale perturbation method that works on integro-differential equations and then building the Hamiltonian on regular networks, e.g., star or complete graphs, which can be mapped to an error correction algorithm scheme of practical significance. \n \nOur third work explores the possible use of entanglement entropy (EE) in machine-learning fields. We introduce a new long-short-term-memory-based recurrent neural network architecture using tensorization techniques to forecast chaotic time series, the learnability of which is determined not only by the number of free parameters but also the tensorization complexity---recognized as how EE scales.
Understanding and harnessing the power of quantum systems has the potential to transform many domains in science and technology. However, before we can achieve these aspirations, we must first build a better understanding of how quantum systems fundamentally behave. In this thesis, we approach this question through the lens of learning theory to develop new paradigms for learning about quantum systems and understanding their structural properties. We deliver several surprising results, upending previous beliefs about even fundamental laws and giving provably efficient algorithms for learning about quantum systems in settings previously conjectured to be intractable. Typically in quantum many-body systems, the particles in the system interact locally with respect to some geometry as described by a local Hamiltonian. Two key questions are first, understanding equilibrium properties of a system with a given Hamiltonian and second, recovering the Hamiltonian from measurements of the properties of the system. For the first, we prove a universal law that there is a sudden death of entanglement, at a critical temperature depending only on the geometry but not on the system size. For the second, we give the first efficient algorithm for recovering the Hamiltonian at any temperature, breaking a conjectured barrier at low temperatures. Beyond systems with local interactions, we also consider learning and testing properties of general quantum states, focusing on the interplay between statistical complexity and near-term quantum device constraints, only allowing for entangled measurements over a limited number of copies of the state. We characterize the optimal rates for learning and testing with single-copy measurements and for multi-copy measurements in many relevant near-term regimes.
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 . . . . . . . . . . . . .
Establishing long-distance quantum entanglement, i.e., entanglement transmission, in quantum networks (QN) is a key and timely challenge for developing efficient quantum communication. Traditional comprehension based on classical percolation assumes a necessary condition for successful entanglement transmission between any two infinitely distant nodes: they must be connected by at least a path of perfectly entangled states (singlets). Here, we relax this condition by explicitly showing that one can focus not on optimally converting singlets but on establishing concurrence-a key measure of bipartite entanglement. We thereby introduce a new statistical theory, concurrence percolation theory (ConPT), remotely analogous to classical percolation but fundamentally different, built by generalizing bond percolation in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance network analysis, we determine the path connectivity by series and parallel rules and approximate higher-order rules via star-mesh transforms. Interestingly, we find that the entanglement transmission threshold predicted by ConPT is lower than the known classical-percolation-based results and is readily achievable on any series-parallel networks such as the Bethe lattice. ConPT promotes our understanding of how well quantum communication can be further systematically improved versus classical statistical predictions under the limitation of QN locality-a "quantum advantage" that is more general and efficient than expected. ConPT also shows a percolationlike universal critical behavior derived by finite-size analysis on the Bethe lattice and regular two-dimensional lattices, offering new perspectives for a theory of criticality in entanglement statistics.
Although the concept of quantum entanglement has been known for about seventy years, it only recently quit the realms of meta-theoretical discussions when it was discovered how entanglement can be exploited to compute and communicate with an unprecedented power. The primary motivation of the work presented in this thesis has been to contribute to the big e#ort that has been done during the last decade to understand and quantify quantum entanglement. We have developed advanced techniques of linear and multilinear algebra to investigate and classify entangled pure and mixed quantum states, and discussed some novel applications in the field of quantum information theory. The results presented in this thesis are mainly of interest from a fundamental point a view: entanglement is the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought [186]. It is however a real privilege that fundamental research in quantum information theory bears the tools of tomorrow's electrical engineers: the ongoing miniaturization of electronic components will soon reach a scale where quantum mechanical e#ects play a major role.
The performance of distributing entanglement between two distant nodes in a large-scale quantum network (QN) of partially entangled bipartite pure states is generally benchmarked against the classical entanglement percolation (CEP) scheme. Improvements beyond CEP were only achieved by nonscalable strategies for restricted QN topologies. This paper explores and amplifies a new and more effective mapping of a QN, referred to as concurrence percolation theory (ConPT), that suggests using deterministic rather than probabilistic protocols for scalably improving on CEP across arbitrary QN topology. More precisely, we implement ConPT via a deterministic entanglement transmission (DET) scheme that is fully analogous to resistor network analysis, with the corresponding series and parallel rules represented by deterministic entanglement swapping and concentration protocols, respectively. The main contribution of this paper is to establish a powerful mathematical framework, which is applicable to arbitrary d-dimensional information carriers (qudits), that provides different natural optimality metrics in terms of generalized k-concurrences (a family of fundamental entanglement measures) for different QN topologies. In particular, we conclude that the introduced DET scheme (a) is optimal over the well-known nested repeater protocol for distilling entanglement from partially entangled qubits and (b) leads to higher success probabilities of obtaining a maximally entangled state than using CEP. The implementation of the DET scheme is experimentally feasible as tested on IBM's quantum computation platform.
** The primary topic of this dissertation is the study of the relationships between parts and wholes as described by particular physical theories, namely generalized probability theories in a quasi-classical physics framework and non-relativistic quantum theory. ** A large part of this dissertation is devoted to understanding different aspects of four different kinds of correlations: local, partially-local, no-signaling and quantum mechanical correlations. Novel characteristics of these correlations have been used to study how they are related and how they can be discerned via Bell-type inequalities that give non-trivial bounds on the strength of the correlations. ** The study of quantum correlations has also prompted us to study a) the multi-partite qubit state space with respect to its entanglement and separability characteristics, and b) the differing strength of the correlations in separable and entangled qubit states. Results include a novel classification of multipartite (partial) separability and entanglement, strong constraints on the monogamy of entanglement and of non-local correlations, and many new entanglement detection criteria that are directly experimentally accessible. ** Because of the generality of the investigation these results also have strong foundational as well as philosophical repercussions for the different sorts of physical theories as a whole; notably for the viability of hidden variable theories for quantum mechanics, for the possibility of doing experimental metaphysics, for the question of holism in physical theories, and for the classical vs. quantum dichotomy.