Entanglement Trajectory and its Boundary

In this article, we offer a novel approach to examining the entanglement that arises from quantum computing. We analyze the reduced density matrices at various points during the execution of a quantum algorithm, and represent the dominant eigenvalue and von Neumann entropy on a graph to form an "entanglement trajectory." To establish the trajectory's limits, we employ random matrix theory. Through examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate how the entanglement trajectory remains within the boundaries that we have established, generating a unique feature for each example. Furthermore, we demonstrate that these boundaries and features can be extended to trajectories defined by other measures of entanglement. Numerical simulations are available through open access

Riemannian Flows for Supervised and Unsupervised Geometric Image Labeling

In this thesis we focus on the image labeling problem, which is used as a subroutine in many image processing applications. Our work is based on the assignment flow which was recently introduced as a novel geometric approach to the image labeling problem. This flow evolves over time on the manifold of row-stochastic matrices, whose elements represent label assignments as assignment probabilities. The strict separation of assignment manifold and feature space enables the data to lie in any metric space, while a smoothing operation on the assignment manifold results in an unbiased and spatially regularized labeling. The first part of this work focuses on theoretical statements about the asymptotic behavior of the assignment flow. We show under weak assumptions on the parameters that the assignment flow for data in general position converges towards integral probabilities and thus ensures unique assignment decisions. Furthermore, we investigate the stability of possible limit points depending on the input data and parameters. For stable limits, we derive conditions that allow early evidence of convergence towards these limits and thus provide convergence guarantees. In the second part, we extend the assignment flow approach in order to impose global convex constraints on the labeling results based on linear filter statistics of the assignments. The corresponding filters are learned from examples using an eigendecomposition. The effectiveness of the approach is numerically demonstrated in several academic labeling scenarios. In the last part of this thesis we consider the situation in which no labels are given and therefore these prototypical elements have to be determined from the data as well. To this end we introduce an additional flow on the feature manifold, which is coupled to the assignment flow. The resulting flow adapts the prototypes in time to the assignment probabilities. The simultaneous adaptation and assignment of prototypes not only provides suitable prototypes, but also improves the resulting image segmentation, which is demonstrated by experiments. For this approach it is assumed that the data lie on a Riemannian manifold. We elaborate the approach for a range of manifolds that occur in applications and evaluate the resulting approaches in numerical experiments