Computational and Theoretical Insights into Multi-Body Quantum Systems

In generality, perfect predictions of the structure and dynamics of multi-body quantum systems are few and far between. As experimental design advances and becomes more refined, experimentally probing the interactions of multiple quantum systems has become commonplace. Predicting this behavior is not a ``one size fits all" problem, and has lead to the inception of a multitude of successful theoretical techniques which have made precise and verifiable predictions through, in many cases, clever approximations and assumptions. As the state-of-the-art pushes the quantum frontier to new experimental regimes, many of the old techniques become invalid, and there is often no tractable methodology to fall back on. This work focuses on expanding the theoretical techniques for making predictions in newly accessible experimental regimes. The transport of quantum information in a room-temperature dipolar spin network is veritably diffusive in nature, but much less is known about the transport properties of such a sample at low temperatures. This work presupposes that diffusion is still a good model for incoherent transport at low temperatures, and proposes a new method to calculate its diffusion coefficient. The diffusion coefficient is reported as a function of the temperature of the ensemble. Further, the interaction of an i.i.d. spin ensemble with a quantized electromagnetic field has long been analyzed via restriction to the Dicke subspace implicit in the Holstein--Primakoff approximation, as well as other within other approximations. This work reanalyzes the conditions under which such a restriction is valid. In regimes where it is shownt that restricting to the Dicke subspace would be invalid, the Hamiltonian structure is thoroughly analyzed. Various predictions can be made by appealing to a reduction in effective dimensionality via a direct sum decomposition. The main theme of the techniques utilized throughout this work is to appeal to a reduction in difficulty via various theoretical tools in order to prepare for an otherwise intractable computational analysis. Computational insights due to this technique have then gone on to motivate directly provable theoretical results, which might otherwise have remained hidden behind the complexity of the structure and dynamics of a multi-body quantum system

Characterizing entanglement and quantum correlations constrained by symmetry

Entanglement and nonlocal correlations constitute two fundamental resources for quantum information processing, as they allow for novel tasks that are otherwise impossible in a classical scenario. However, their elusive characterization is still a central problem in Quantum Information Theory. The main reason why such a fundamental issue remains a formidable challenge lies in the exponential growth in complexity of the Hilbert space, as well as the space of nonlocal correlations. Physical systems of interest, on the other hand, display symmetries that can be exploited to reduce this complexity, opening the possibility that, for such systems, some of these questions become tractable. This PhD Thesis is dedicated to the study and characterization of entanglement and nonlocal correlations constrained under symmetries. It contains original results in these four threads of research: PPT entanglement in the symmetric states, nonlocality detection in many-body systems, the non-equivalence between entanglement and nonlocality and elemental monogamies of correlations. First, we study PPT entanglement in fully symmetric n-qubit states. We solve the open question on the existence of four-qubit PPT entangled states of these kind, providing constructive examples and methods. Furthermore, we develop criteria for separability, edgeness and the Schmidt number of PPT entangled symmetric states. Geometrically, we focus on the characterization of extremal states of this family and we provide an algorithm to find states with such properties. Second, we study nonlocality in many-body systems. We consider permutationally and translationally invariant Bell inequalities consisting of two-body correlators. These constitute the first tools to detect nonlocality in many-body systems in an experimentally friendly way with our current technology. Furthermore, we show how these Bell inequalities detect nonlocality in physically relevant systems such as ground states of Hamiltonians that naturally arise e.g., in nuclear physics. We provide analytical classes of Bell inequalities and we analytically characterize which states and measurements are best suited for them. We show that the method we introduce can be fully generalized to correlators of any order in any Bell scenario. Finally, we provide some feedback from a more experimental point of view. Third, we demonstrate that entanglement and nonlocality are inequivalent concepts in general; a question that remained open in the multipartite case. We show that the strongest form of entanglement, genuinely multipartite entanglement, does not imply the strongest form of nonlocality, genuinely multipartite nonlocality, in any case. We give a constructive method that, starting from a multipartite genuinely multipartite state admitting a K-local model, extends it to a genuinely multipartite entangled state of any number of parties while preserving the degree of locality. Finally, we show that nonlocal correlations are monogamous in a much stronger sense than the typical one, in which the figure of merit compares a Bell inequality violation between two sets of parties. We show that the amount of Bell violation that a set of parties observes limits the knowledge that any external observer may gain on any of the outcomes of any of the parties performing the Bell experiment. We show that this holds even if such observer is not limited by quantum physics, but it only obeys the no-signalling principle. Apart from its fundamental interest, we show how these stronger monogamy relations boost the performance of some device-independent (DI) protocols such as DI quantum key distribution or DI randomness amplification.El entrelazamiento y las correlaciones no-locales constituyen dos recursos fundamentales para el procesamiento cuántico de la información, ya que abren la posibilidad de realizar tareas que serían imposibles en el sentido clásico. Sin embargo, su elusiva caracterización aún representa uno de los problemas más importantes en la teoría cuántica de la información. La razón principal por la que una cuestión tan básica sigue siendo un reto formidable subyace en el incremento exponencial de la complejidad del espacio de Hilbert, así como del espacio de las correlaciones no-locales. Por otro lado, los sistemas físicos de interés muestran simetrías que pueden ser aprovechadas para reducir dicha complejidad, abriendo la posibilidad que, para tales sistemas, algunas de esas cuestiones devengan tratables. La presente tesis doctoral está enfocada al estudio de la caracterización del entrelazamiento cuántico y las correlaciones no-locales bajo simetrías. Contiene resultados originales en las siguientes líneas de investigación: entrelazamiento del tipo PPT en estados simétricos, detección de no-localidad en sistemas de muchos cuerpos, la no equivalencia entre el entrelazamiento cuántico y la no-localidad y las correlaciones monogámicas elementales. En primer lugar, estudiamos el entrelazamiento del tipo PPT en estados totalmente simétricos de n bits cuánticos. Resolvemos el problema abierto referente a la existencia de estados PPT entrelazados de cuatro bits cuánticos de este tipo, proporcionando ejemplos y métodos constructivos. Además, desarrollamos criterios de separabilidad, estados frontera y número de Schmidt para estados PPT entrelazados y simétricos. Nos centramos en la caracterización de estados extremos dentro de esta familia y proporcionamos un algoritmo para encontrar estados cuánticos con tales propiedades. En segundo lugar, estudiamos la no-localidad en sistemas de muchos cuerpos. Consideramos desigualdades de Bell, invariantes bajo permutaciones o traslaciones, que involucran correladores entre dos cuerpos como mucho. Dichas desigualdades constituyen los primeros tests de detección de no-localidad en sistemas de muchos cuerpos que son accesibles experimentalmente, con el presente nivel de tecnología. Además, demostramos cómo esas desigualdades de Bell pueden detectar no-localidad en estados físicamente relevantes, como los estados de mínima energía de hamiltonianos que aparecen en física nuclear. Proporcionamos clases analíticas de desigualdades de Bell y caracterizamos, también analíticamente, qué estados y medidas son los más adecuados para ellas. Vemos que el método que introducimos es generalizable a cualquier escenario de Bell. Finalmente, comentamos aspectos de interés desde un punto de vista experimental. En tercer lugar, demostramos que el entrelazamiento y las correlaciones no-locales son conceptos no equivalentes en general, resolviendo un problema que persistía abierto en el caso multipartito. Probamos que la forma más fuerte de entrelazamiento no implica la forma más fuerte de no-localidad en ningún caso. Para ello, damos un método constructivo que, dado un estado cuántico multipartito genuinamente entrelazado que admite un modelo K-local, lo extiende a un estado consistente en un número de subsistemas arbitrario, genuinamente entrelazado, preservando el mismo grado de localidad. Finalmente, demostramos que las correlaciones no-locales son monógamas en un sentido mucho más estricto que el que se considera típicamente. Vemos que la cantidad de violación que un conjunto de observadores mide impone restricciones fundamentales en la información que puede obtener cualquier observador externo, resultado que se mantiene asumiendo sólo la imposibilidad de transmisión instantánea de la información. Demostramos su aplicación en protocolos cuánticos independientes del dispositivo (ID) tales como la distribución cuántica de llaves ID o bien la amplificación de aleatoriedad ID

Central Compact-Reconstruction WENO Methods

High-order compact upwind schemes produce block-tridiagonal systems due to performing the reconstruction in the characteristic variables, which is necessary to avoid spurious oscillations. Consequently they are less efficient than their non-compact counterparts except on high-frequency features. Upwind schemes lead to many practical drawbacks as well, so it is desirable to have a compact scheme that is more computationally efficient at all wavenumbers that does not require a characteristic decomposition. This goal cannot be achieved by upwind schemes so we turn to the central schemes, which by design require neither a Riemann solver nor a determination of upwind directions by characteristic decomposition. In practice, however, central schemes of fifth or higher order apparently need the characteristic decomposition to fully avoid spurious oscillations. The literature provides no explanation for this fact that is entirely convincing; however, a comparison of two central WENO schemes suggests one. Pursuing that possibility leads to the first main contribution of this work, which is the development of a fifth-order, central compact-reconstruction WENO (CCRWENO) method. That method retains the key advantages of central and compact schemes but does not entirely avoid characteristic variables as was desired. The second major contribution is to establish that the role of characteristic variables is to to make flux Jacobians within a stencil more diagonally dominant, having ruled out some plausible alternative explanations. The CCRWENO method cannot inherently improve the diagonal dominance without compromising its key advantages, so some strategies are explored for modifying the CCRWENO solution to prevent the spurious oscillations. Directions for future investigation and improvement are proposed

Transport Properties of Driven Topological Systems

The work in this thesis is centred around exploring the transport properties induced by periodically driving topologically non-trivial systems. In particular, it focuses upon the signatures associated with Majorana modes in 1D topological superconductors and their adiabatic manipulations, in anticipation of the next generation of experiments pursuing the realization and control of such excitations, which have been stipulated as the potential building blocks of robust quantum computation. We examine two distinct ways by which external driving can influence a system's topology. Firstly, we focus upon systems for which the modulation results in the emergence of additional topological phases, not present in their static counterparts, the classification of which is not well defined by the usual topological invariants associated with the energy spectrum of the bulk system. For such systems, transport properties are vital in identifying non-trivial topological regimes and, with this motivation, we examine the relationship between driven scattering matrix topological invariants and conductance signatures. Secondly, we determine the transport statistics associated with the adiabatic manipulation of topological excitations appearing in static systems, with a specific focus upon a Majorana braiding protocol. In this way, we demonstrate that the topological protection of the operation is reflected in geometric contributions to the heat transport induced by the driving. In addition to providing potential experiential signatures of such manipulations, this analysis also sheds light on the influence of periodic driving upon exchange fluctuation theorems, that govern the thermodynamics of non-equilibrium quantum systems, and also the performance of such protected operations as nanoscale thermal machines