This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. In Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices. In order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. In Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. In Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. Additionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs).
This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. In Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices. In order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. In Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. In Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. Additionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs).
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics. The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes. The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory. In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem. Furthermore, we show that answering the question whether one can divide up a random variable X into a sum of n iid random variables Yi, i.e. X=∑i=1nYi, is poly-time computable; relaxing the iid condition renders the problem NP-hard. In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock. We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting. This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions. Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases. We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes.
This thesis presents a collection of projects spanning different sub-fields of quantum science, undertaken with the goal of understanding and advancing quantum technologies. Broadly, it can be divided into two thematic parts — parameter estimation and open system dynamics. Each project addresses inverse and ill-posed problems in Physics, which lie at the heart of developing robust and scalable quantum devices.A central theme in quantum technology is the challenge of inference of microscopic system parameters from limited, indirect observations. The first part of the thesis investigates this through the problem of Hamiltonian reconstruction via local measurements. Indirect estimation of couplings in quantum systems relies on the measurement of the spectrum and the overlap of eigenvectors with some reference state. For a one-dimensional chain, this data can be obtained by local measurement on one end of the chain, thereby reducing the need for full Hamiltonian tomography. Information about couplings is extracted from the temporal data obtained by measuring the expectation values of the time evolution 𝑒−𝑖𝐻𝑡 for a reference state. In Chapter 3, we analyze the robustness of this estimation scheme under perturbations arising from beyond nearest neighbour interactions. We show that the error scales linearly with the strength of such interactions and on average, the error increases with the distance of interaction between sites.Chapter 4 examines whether coupling information can be extracted from local thermal measurements, using expectation values of the Gibbs state at varying temperatures rather than temporal data. We develop a procedure to obtain the coefficients of the Taylor expansion of these expectation values and relate them to the Hamiltonian parameters. For very small chains, this mapping can be carried out analytically, while for larger systems it reveals the algebraic structure that links local measurements to the underlying couplings and highlights the trade-offs between measurement resources, numerical stability, and identifiability. The analysis also clarifies how additional local fields or self-interaction terms enhance the information content of thermal data, suggesting a role for local thermal probes as practical tools for Hamiltonian verification and parameter estimation.Building on the ideas of controlling and inferring information about a quantum system, we now turn to active intervention through adaptive measurement. In Chapter 5, we develop a process tensor driven procedure for cooling the spin bath that is coupled to a nitrogen vacancy (NV) center in diamond. By modeling the sequence of intermediate quantum instruments acting on a correlated system-environment process, we optimize a pair of completely positive (CP) maps to maximize the population in a lower-energy state. Unlike existing approaches that rely solely on positive operator valued measures (POVMs), our method encompasses a broader class of CP maps, including adaptive feed-forward and mixed operations. We explicitly account for initial correlation between system and environment and distinguish between the controllable and uncontrollable parts of the dynamics. Our framework supports non-Markovian operations and benchmarks them against optimal Markovian strategies. The optimization is performed via Semi-definite Programming (SDP), ensuring global optimality. We also assess robustness to Hamiltonian noise and provide a proposal for implementation using NV center in diamonds.Finally, Chapter 6 explore the relaxation dynamics of topological qubits, focusing on a tetron qubit formed from two uncoupled Kitaev chains coupled to a finite fermionic bath. Using the correlation matrix formalism for purely quadratic couplings, we show that the relaxation is fundamentally independent of temperature when either the bath is large, or the zero-mode energy of the system is close to zero. This challenges a key assumption in the field that cooling enhances coherence. Interestingly, we find that introducing glassy disorder into the system slows down relaxation, offering a potential route to prolong qubit lifetimes.Collectively, these studies form a coherent narrative around extraction, control and utilization of quantum information under fundamental constraints. By tackling such inverse and often ill-posed problems, this thesis contributes new tools and insights essential for the realization of future quantum technologies.
Quantum Mechanics was established as the theory of the microscopic world, which allowed to understand processes in atoms and molecules. Its emergence led to a new scientific paradigm that quickly spread to different research fields. Two relevant examples are Quantum Thermodynamics and Quantum Many-Body Theory, where the former aims to characterize thermodynamic processes in quantum systems and the latter intends to understand the properties of quantum many-body systems. In this thesis, we tackle some of the questions in the overlap between these disciplines, focusing on the concepts of temperature and correlations. Specifically, it contains results on the following topics: locality of temperature, correlations in long-range interacting systems and thermometry at low temperature. The problem of locality of temperature is considered for a system at thermal equilibrium and consists in studying whether it is possible to assign a temperature to any of the subsystems of the global system such that both local and global temperatures are equal. We tackle this problem in two different settings, for generic one-dimensional spin chains and for a bosonic system with a phase transition at non-zero temperature. In the first case, we consider generic one-dimensional translation-invariant spin systems with short-range interactions and prove that it is always possible to assign a local temperature equal to the global one for any temperature, including at criticality. For the second case, we consider a three-dimensional discretized version of the Bose-Einstein model at the grand canonical ensemble for some temperature and particle density, and characterize its non-zero-temperature phase transition. Then, we show that temperature is locally well-defined at any temperature and at any particle density, including at the phase transition. Additionally, we observe a qualitative relation between correlations and locality of temperature in the system. Moving to correlations, we consider fermionic two-site long-range interacting systems at thermal equilibrium. We show that correlations between anti-commutative operators at non-zero temperature are upper bounded by a function that decays polynomially with the distance and with an exponent that is equal to the interaction exponent, which characterizes the interactions in the Hamiltonian. Moreover, we show that our bound is asymptotically tight and that the results extend to density-density correlations as well as other types of correlations for quadratic and fermionic Hamiltonians with long-range interactions. Regarding the results on thermometry, we consider a bosonic model and prove that strong coupling between the probe and the system can boost the thermal sensitivity for low temperature. Furthermore, we provide a feasible measurement scheme capable of producing optimal estimates at the considered regime.
I Plenary talks.- An adiabatic theorem without a gap condition.- Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum.- Spectral problems in the theory of photonic crystals.- Optimal eigenvalues for some Laplacians and Schroedinger operators depending on curvature.- The spectral shift operator.- On the scattering operator for the Schroedinger equation with a time-dependent potential.- H_2-construction and some applications.- Scattering with time periodic potentials and cyclic states.- Some geometry related to decay properties of the resolvent of a class of symmetric operators.- On some asymptotic formulas in the strong localization regime of the theory of disordered systems.- Spectral measures and category.- Quantum dots. A survey of rigorous results.- II Session talks.- A simple model of concentrated nonlinearity.- Anomalous electron trapping by magnetic flux tubes and electric current vortices.- On the absolutely continuous energy distribution of a quantum mechanical system in a bounded domain.- Some aspects of generalized contact interaction in one-dimensional quantum mechanics.- Traces and trace norms for semigroup differences...- About a resolvent formula.- The determinant anomaly in low-dimensional quantum systems.- Linear Boltzmann equation as the weak coupling limit of the random Schroedinger equation.- Coexistence of different spectral types for almost periodic Schroedinger equations in dimension one.- Dynamical localization for random Schroedinger operators and an application to the almost Mathieu operator.- On fractal structure of the spectrum for periodic point perturbations of the Schroedinger operator with a uniform magnetic field.- A Weyl-Berry formula for the scattering operator associated to self-similar potentials on the line.- Localization and Lifshitz tails for random quantum waveguides.- Birman-Schwinger analysis for bound states in a pair of parallel quantum waveguides with a semitransparent boundary.- On the absolute continuity of spectra of periodic elliptic operators.- Hardy inequalities for magnetic Dirichlet forms.- Adiabatic curvature, chaos and the deformation of Riemann surfaces.- Operator interpretation of resonances arising in spectral problems for 2 x 2 matrix Hamiltonians.- On the operator-norm convergence of the Trotter-Kato product formula.- A particular case of the inverse problem for the Sturm-Liouville equation with parameter dependent potential.- One-dimensional Schroedinger operators with decaying potentials.- Stability of limiting absorption under singular perturbations.- Existence of averaging integrals for self-adjoint operators.- Monotonicity versus non-monotonicity in random operators.- A model in perturbation theory.- Band gap of the spectrum in periodically curved quantum waveguides.- A list of other talks.- A list of participants.
In this thesis, we consider two topics that involve strong spatial or temporal fluctuations in interacting quantum many-body systems. We first consider far-from equilibrium quantum hydrodynamics in light-driven helical edge states of a 2D topological insulator. We then consider the enhancement of superconducting and magnetic instabilities in fermion systems with random or structured inhomogeneity, driven by the nesting of fractal wave functions. In the first part, we study quantum quenches of helical edge liquids with spin-flip inelastic scattering. Counterpropagating charge packets in helical edges can be created by an ultrashort electric pulse applied across a 2D topological insulator. Localized “hot spots” that form due to scattering enable two types of strongly nonlinear wave dynamics. First, propagating packets develop self-focusing shock fronts. Second, colliding packets with opposite charge can exhibit near-perfect retroreflection, despite strong dissipation. This leads to an interaction-mediated frequency doubling that could be detected experimentally from emitted terahertz radiation. In the second part, we study the enhanced correlation effects in many-fermion systems induced by structured or random inhomogeneities. We consider the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions (“spectrum-wide quantum criticality,” SWQC) in the one-dimensional Aubry-Andr ́e and power-law random-banded matrix models (PRBMs) with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, whereas the pairing amplitude is maximized near this transition in both models. Although increasing inhomogeneity always depletes the superfluid stiffness, we find that superconductivity is still controlled by the fractal-enhanced amplitude near the Anderson transition. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconductor models, can robustly enhance superconductivity. We also study the interplay of magnetic ordering and the spectrum-wide quantum critical wave functions in PRBMs with repulsive Hubbard interaction. We employ self-consistent Hartree-Fock numerics and analytical field theory calculations using a Keldysh Finkel’stein nonlinear sigma model. We find that weak interactions delocalize the system, while strong interaction localizes the system. Ferromagnetism sets in near the Anderson transition, where the ferromagnetic spin susceptibility is also shown to peak. This indicates that ferromagnetic ordering is enhanced by the fractal wave functions. Using the sigma model, we calculate the quantum corrections to the conductance and spin susceptibility. We find that the analytical and numerical results are mutually consistent.