Perhaps the deepest unanswered question in theoretical physics is how to consistently combine the theory of general relativity with the principles of quantum mechanics. In parallel, the holographic principle is expected to be a fundamental property of quantum gravity that allows the latter to be described by a theory in one dimension lower which does not contain gravity. As an explicit realization of the holographic principle, the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence has established itself as our best non-perturbative approach to quantum gravity. Moreover, it has become clear that concepts of quantum information theory play a fundamental role in this duality. Nonetheless, the extent of validity of the holographic principle has not been fully explored yet. In parallel, the AdS/CFT correspondence is still primarily a conjecture rooted in theoretical arguments and, while awaiting a mathematical proof, it would immensely benefit from experimental tests validating its predictions. Motivated by these aspects, in this thesis I advance the field of \textit{discrete holography} by establishing explicit holographic models on both bulk and boundary of discretizations of hyperbolic space. These models assess the wide range of validity of the holographic principle beyond the continuum formulation of AdS/CFT while also enabling the experimental realization of holographic predictions. Furthermore, I investigate quantum information quantities such as circuit complexity, quantum discord, and the operator algebra description of entanglement, and elucidate their roles in describing certain properties of black holes. In the first part of this thesis I consider regular tilings of hyperbolic space as a discretization of constant-time slices of AdS$_{3}$. From the perspective of the bulk, I show that they provide a sensible description of the continuum by establishing how the Breitenlohner-Freedman stability bound for a free scalar field is realized on these tilings. I follow up on this result by considering an interacting scalar field on hyperbolic tilings and showing how a discrete lattice model can emulate CFT correlation functions at both zero and finite temperature. The latter are obtained from a discretization of a constant-time slice of an AdS black hole. Additionally, I present a universal experimental platform based on electric circuits on which the aforementioned results can be measured. My results show that these architectures bear tremendous untapped potential to realize further holographic predictions in the laboratory, beyond those appraised in this thesis. From a boundary perspective, I establish a large class of explicit Hamiltonians defined at the asymptotic boundary of hyperbolic tilings. These are prime candidates for boundary theories in discrete holographic dualities and consist of aperiodically disordered quantum spin chains with nearest-neighbor interactions. The aperiodic disorder encodes information about the systematic construction of the bulk tiling based on substitution rules. Using real-space renormalization group techniques, I study the ground state of these models in view of their factorized form into two-spin states, their correlation functions, and their entanglement structure. I show how the latter is exactly captured by a tensor network which extends into the bulk, thus providing a geometric bulk dual to the boundary theory. The high degree of analytic tractability of these models enables discrete manifestations of the holographic principle. These are realized by the standard prescription for computing entanglement entropy in tensor networks. In the second part of this thesis I focus on three quantum information-theoretic quantities and study them in view of describing properties of quantum black holes via the AdS/CFT correspondence. Starting from quantum circuit complexity as a conjectured probe for the emergence of space behind the horizon, I study a novel definition of Nielsen operator complexity for finite- and infinite-dimensional systems. My construction relates complexity with the theory of classical ideal hydrodynamics while still exhibiting the features desired for a holographic complexity measure. Then, I study geometric quantum discord as an efficient tool for diagnosing non-factorization. This quantity is easier to calculate than its NP-hard cousin (quantum discord), and I show how the geometric quantum discord for pure states vanishes if and only if the modular partition function factorizes. I apply this to the explicit example of an eternal wormhole in AdS and discuss the results in view of holography. Moreover, I establish a relation between geometric quantum discord and wormhole contributions to the gravitational path integral, thus shedding light on the mechanisms behind the factorization puzzle of AdS/CFT. Finally, I study local operator algebras as a rigorous approach to entanglement entropy which has been recently employed in AdS/CFT to explain the emergence of time behind the black hole horizon. In particular, I consider operator algebras in an interacting but exactly solvable quantum many-body system of Majorana fermions. I find transitions between different types of operator algebras throughout the phase diagram of the model, which contains a quantum phase transition. This model provides an explicit instance where such operator algebra transitions can be studied in a tractable many-body system. The results presented in this thesis are published in my works listed in App.~A.

Protein folding achieves a clear solution structure in a huge parameter space (the so-called protein folding problem). Proteins fold in water, and get by this a highly ordered structure. Finally, inside a protein crystal for structure resolution, you have everywhere the same symmetries as there is everywhere the same unit cell. We apply this to qubit interactions to do fundamental physics: in a modified cosmology, we replace the big bang by a condensation event in an eternal all-encompassing ocean of free qubits. Interactions of qubits in the qubit ocean are quite rare but provide a nucleus or seed for a new universe (domain) as the qubits become decoherent and freeze-out into defined bit ensembles. Second, we replace inflation by a crystallization event triggered by the nucleus of interacting qubits to which rapidly more and more qubits attach (like in everyday crystal growth). The crystal unit cell guarantees same symmetries everywhere inside the crystal. The textbook inflation scenario to explain the same laws of nature in our domain is replaced by the unit cell of the crystal formed. Interacting qubits solidify, quantum entropy decreases (but increases in the ocean around). In a modified inflation scenario, the interacting qubits form a rapidly growing domain where the n**m states become separated ensemble states, rising long-range forces stop ultimately further growth. Then standard cosmology with the hot fireball model takes over. Our theory agrees well with lack of inflation traces in cosmic background measurements. We explain by cosmological crystallization instead of inflation: early creation of large-scale structure of voids and filaments, supercluster formation, galaxy formation, and the dominance of matter: the unit cell of our crystal universe has a matter handedness avoiding anti-matter. We prove initiation of qubit interactions can only be 1,2,4 or 8-dimensional (agrees with E8 symmetry of our universe). Repulsive forces at ultrashort distances result from quantization, long-range forces limit crystal growth. Crystals come and go in the qubit ocean. This selects for the ability to lay seeds for new crystals, for self-organization and life-friendliness. The phase space of the crystal agrees with the standard model of the basic four forces for n quanta. It includes all possible ensemble combinations of their quantum states m, a total of n**m states. Neighbor states reach according to transition possibilities (S-matrix) with emergent time from entropic ensemble gradients. However, in our four dimensions there is only one bit overlap to neighbor states left (almost solid, only below Planck quantum there is liquidity left). The E8 symmetry of heterotic string theory has six curled-up, small dimensions which help to keep the qubit crystal together and will never expand. Mathematics focusses on the Hurwitz proof applied to qubit interaction, a toy model of qubit interaction and repulsive forces of qubits. Vacuum energy gets appropriate low inside the crystal. We give first energy estimates for free qubits vs bound qubits, misplacements in the qubit crystal and entropy increase during qubit decoherence / crystal formation. Scalar fields for color interaction/confinement and gravity are derived from the qubit-interaction field.

Topological phenomena known from solid state physics have been transferred to a variety of other classical and quantum systems. Due to the equivalence of the Hamiltonian matrix describing tight binding models and the grounded circuit Laplacian describing an electrical circuit we can investigate such phenomena in circuits. By implementing different Hermitian topological models general suggestions on designing those types of circuit are worked out with the aim of minimizing unwanted coupling effects and parasitic admittances in the circuit. Here the existence and the spatial profile of topological states as well as the band structure of the model can be determined. Due to the complex nature of electric admittance the investigations can be directly expanded to systems with broken Hermiticity. The particular advantages of the experimental investigation of non-exclusively topological phenomena by means of electric circuits come to light in the realization of non-Hermitian and non-linear models. Here we find limitation of the Hermitian bulk-boundary correspondence principle, purely real eigenvalues in non-Hermitian PT-symmetrical systems and edge localization of all eigenstates in non-Hermitian and non-reciprocal systems, which in literature is termed the non-Hermitian skin effect. When systems obeying non-linear equations are studied, the grounded circuit Laplacian based on the Fourier-transform cannot be applied anymore. By combination of the connectivity of a topological system together with non-linear van der Pol oscillators self-activated and self-sustained topological edge oscillations can be found. These robust high frequency sinusoidal edge oscillations differ significantly from low frequency relaxation oscillations, which can be found in the bulk of the system.

This work introduced the reader to all relevant fields to tap into an ultrasound-based state of charge estimation and provides a blueprint for the procedure to achieve and test the fundamentals of such an approach. It spanned from an in-depth electrochemical characterization of the studied battery cells over establishing the measurement technique, digital processing of ultrasonic transmission signals, and characterization of the SoC dependent property changes of those signals to a proof of concept of an ultrasound-based state of charge estimation. The State of the art & theoretical background chapter focused on the battery section on the mechanical property changes of lithium-ion batteries during operation. The components and the processes involved to manufacture a battery cell were described to establish the fundamentals for later interrogation. A comprehensive summary of methods for state estimation was given and an emphasis was laid on mechanical methods, including a critical review of the most recent research on ultrasound-based state estimation. Afterward, the fundamentals of ultrasonic non-destructive evaluation were introduced, starting with the sound propagation modes in isotropic boundary-free media, followed by the introduction of boundaries and non-isotropic structure to finally approach the class of fluid-saturated porous media, which batteries can be counted to. As the processing of the ultrasonic signals transmitted through lithium-ion battery cells with the aim of feature extraction was one of the main goals of this work, the fundamentals of digital signal processing and methods for the time of flight estimation were reviewed and compared in a separate section. All available information on the interrogated battery cell and the instrumentation was collected in the Experimental methods & instrumentation chapter, including a detailed step-by-step manual of the process developed in this work to create and attach a sensor stack for ultrasonic interrogation based on low-cost off-the-shelf piezo elements. The Results & discussion chapter opened with an in-depth electrochemical and post-mortem interrogation to reverse engineer the battery cell design and its internal structure. The combination of inductively coupled plasma-optical emission spectrometry and incremental capacity analysis applied to three-electrode lab cells, constructed from the studied battery cell’s materials, allowed to identify the SoC ranges in which phase transitions and staging occur and thereby directly links changes in the ultrasonic signal properties with the state of the active materials, which makes this work stand out among other studies on ultrasound-based state estimation. Additional dilatometer experiments were able to prove that the measured effect in ultrasonic time of flight cannot originate from the thickness increase of the battery cells alone, as this thickness increase is smaller and in opposite direction to the change in time of flight. Therefore, changes in elastic modulus and density have to be responsible for the observed effect. The construction of the sensor stack from off-the-shelf piezo elements, its electromagnetic shielding, and attachment to both sides of the battery cells was treated in a subsequent section. Experiments verified the necessity of shielding and its negligible influence on the ultrasonic signals. A hypothesis describing the metal layer in the pouch foil to be the transport medium of an electrical coupling/distortion between sending and receiving sensor was formulated and tested. Impedance spectroscopy was shown to be a useful tool to characterize the resonant behavior of piezo elements and ensure the mechanical coupling of such to the surface of the battery cells. The excitation of the piezo elements by a raised cosine (RCn) waveform with varied center frequency in the range of 50 kHz to 250 kHz was studied in the frequency domain and the influence of the resonant behavior, as identified prior by impedance spectroscopy, on waveform and frequency content was evaluated to be uncritical. Therefore, the forced oscillation produced by this excitation was assumed to be mechanically coupled as ultrasonic waves into the battery cells. The ultrasonic waves transmitted through the battery cell were recorded by piezo elements on the opposing side. A first inspection of the raw, unprocessed signals identified the transmission of two main wave packages and allowed the identification of two major trends: the time of flight of ultrasonic wave packages decreases with the center frequency of the RCn waveform, and with state of charge. These trends were to be assessed further in the subsequent sections. Therefore, methods for the extraction of features (properties) from the ultrasonic signals were established, compared, and tested in a dedicated section. Several simple and advanced thresholding methods were compared with envelope-based and cross-correlation methods to estimate the time of flight (ToF). It was demonstrated that the envelope-based method yields the most robust estimate for the first and second wave package. This finding is in accordance with the literature stating that an envelope-based method is best suited for dispersive, absorptive media [204], to which lithium-ion batteries are counted. Respective trends were already suggested by the heatmap plots of the raw signals vs. RCn frequency and SoC. To enable such a robust estimate, an FIR filter had to be designed to preprocess the transmitted signals and thereby attenuate frequency components that verifiably lead to a distorted shape of the envelope. With a robust ToF estimation method selected, the characterization of the signal properties ToF and transmitted energy content (EC) was performed in-depth. A study of cycle-to-cycle variations unveiled that the signal properties are affected by a long rest period and the associated relaxation of the multi-particle system “battery cell” to equilibrium. In detail, during cycling, the signal properties don’t reach the same value at a given SoC in two subsequent cycles if the first of the two cycles follows a long rest period. In accordance with the literature, a break-in period, making up for more than ten cycles post-formation, was observed. During this break-in period, the mechanical properties of the system are said to change until a steady state is reached [25]. Experiments at different C-rate showed that ultrasonic signal properties can sense the non-equilibrium state of a battery cell, characterized by an increasing area between charge and discharge curve of the respective signal property vs. SoC plot. This non-equilibrium state relaxes in the rest period following the discharge after the cut-off voltage is reached. The relaxation in the rest period following the charge is much smaller and shows little C-rate dependency as the state is prepared by constant voltage charging at the end of charge voltage. For a purely statistical SoC estimation approach, as employed in this work, where only instantaneous measurements are taken into account and the historic course of the measurement is not utilized as a source of information, the presence of hysteresis and relaxation leads to a reduced estimation accuracy. Future research should address this issue or even utilize the relaxation to improve the estimation accuracy, by incorporating historic information, e.g., by using the derivative of a signal property as an additional feature. The signal properties were then tested for their correlation with SoC as a function of RCn frequency. This allowed identifying trends in the behavior of the signal properties as a function of RCn frequency and C-rate in a condensed fashion and thereby enabled to predict the frequency range, about 50 kHz to 125 kHz, in which the course of the signal properties is best suited for SoC estimation. The final section provided a proof of concept of the ultrasound-based SoC estimation, by applying a support vector regression (SVR) to before thoroughly studied ultrasonic signal properties, as well as current and battery cell voltage. The included case study was split into different parts that assessed the ability of an SVR to estimate the SoC in a variety of scenarios. Seven battery cells, prepared with sensor stacks attached to both faces, were used to generate 14 datasets. First, a comparison of self-tests, where a portion of a dataset is used for training and another for testing, and cross-tests, which use the dataset of one cell for training and the dataset of another for testing, was performed. A root mean square error (RMSE) of 3.9% to 4.8% SoC and 3.6% to 10.0% SoC was achieved, respectively. In general, it was observed that the SVR is prone to overestimation at low SoCs and underestimation at high SoCs, which was attributed to the pronounced hysteresis and relaxation of the ultrasonic signal properties in this SoC ranges. The fact that higher accuracy is achieved, if the exact cell is known to the model, indicates that a variation between cells exists. This variation between cells can originate from differences in mechanical properties as a result of production variations or from differences in manual sensor placement, mechanical coupling, or resonant behavior of the ultrasonic sensors. To mitigate the effect of the cell-to-cell variations, a test was performed, where the datasets of six out of the seven cells were combined as training data, and the dataset of the seventh cell was used for testing. This reduced the spread of the RMSE from (3.6 - 10.0)% SoC to (5.9 – 8.5)% SoC, respectively, once again stating that a databased approach for state estimation becomes more reliable with a large data basis. Utilizing self-tests on seven datasets, the effect of additional features on the state estimation result was tested. The involvement of an additional feature did not necessarily improve the estimation accuracy, but it was shown that a combination of ultrasonic and electrical features is superior to the training with these features alone. To test the ability of the model to estimate the SoC in unknown cycling conditions, a test was performed where the C-rate of the test dataset was not included in the training data. The result suggests that for practical applications it might be sufficient to perform training with the boundary of the use cases in a controlled laboratory environment to handle the estimation in a broad spectrum of use cases. In comparison with literature, this study stands out by utilizing and modifying off-the-shelf piezo elements to equip state-of-the-art lithium-ion battery cells with ultrasonic sensors, employing a range of center frequencies for the waveform, transmitted through the battery cell, instead of a fixed frequency and by allowing the SVR to choose the frequency that yields the best result. The characterization of the ultrasonic signal properties as a function of RCn frequency and SoC and the assignment of characteristic changes in the signal properties to electrochemical processes, such as phase transitions and staging, makes this work unique. By studying a range of use cases, it was demonstrated that an improved SoC estimation accuracy can be achieved with the aid of ultrasonic measurements – thanks to the correlation of the mechanical properties of the battery cells with the SoC.

X-ray dark-field imaging allows to resolve the conflict between the demand for centimeter scaled fields of view and the spatial resolution required for the characterization of fibrous materials structured on the micrometer scale. It draws on the ability of X-ray Talbot interferometers to provide full field images of a sample's ultra small angle scattering properties, bridging a gap of multiple orders of magnitude between the imaging resolution and the contrasted structure scale. The correspondence between shape anisotropy and oriented scattering thereby allows to infer orientations within a sample's microstructure below the imaging resolution. First demonstrations have shown the general feasibility of doing so in a tomographic fashion, based on various heuristic signal models and reconstruction approaches. Here, both a verified model of the signal anisotropy and a reconstruction technique practicable for general imaging geometries and large tensor valued volumes is developed based on in-depth reviews of dark-field imaging and tomographic reconstruction techniques. To this end, a wide interdisciplinary field of imaging and reconstruction methodologies is revisited. To begin with, a novel introduction to the mathematical description of perspective projections provides essential insights into the relations between the tangible real space properties of cone beam imaging geometries and their technically relevant description in terms of homogeneous coordinates and projection matrices. Based on these fundamentals, a novel auto-calibration approach is developed, facilitating the practical determination of perspective imaging geometries with minimal experimental constraints. A corresponding generalized formulation of the widely employed Feldkamp algorithm is given, allowing fast and flexible volume reconstructions from arbitrary tomographic imaging geometries. Iterative reconstruction techniques are likewise introduced for general projection geometries, with a particular focus on the efficient evaluation of the forward problem associated with tomographic imaging. A highly performant 3D generalization of Joseph's classic linearly interpolating ray casting algorithm is developed to this end and compared to typical alternatives. With regard to the anisotropic imaging modality required for tensor tomography, X-ray dark-field contrast is extensively reviewed. Previous literature is brought into a joint context and nomenclature and supplemented by original work completing a consistent picture of the theory of dark-field origination. Key results are explicitly validated by experimental data with a special focus on tomography as well as the properties of anisotropic fibrous scatterers. In order to address the pronounced susceptibility of interferometric images to subtle mechanical imprecisions, an efficient optimization based evaluation strategy for the raw data provided by Talbot interferometers is developed. Finally, the fitness of linear tensor models with respect to the derived anisotropy properties of dark-field contrast is evaluated, and an iterative scheme for the reconstruction of tensor valued volumes from projection images is proposed. The derived methods are efficiently implemented and applied to fiber reinforced plastic samples, imaged at the ID19 imaging beamline of the European Synchrotron Radiation Facility. The results represent unprecedented demonstrations of X-ray dark-field tensor tomography at a field of view of 3-4cm, revealing local fiber orientations of both complex shaped and low-contrast samples at a spatial resolution of 0.1mm in 3D. The results are confirmed by an independent micro CT based fiber analysis.