The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

Abstract This chapter seeks to outline a few basic problems in quantum statistical physics, where recent experimental advances from the atomic physics community offer the hope of dramatic progress. The focus is on nonequilibrium situations, where the powerful concepts and methods of equilibrium statistical physics and “linear response” theory (for small deviations from equilibrium) are not applicable. The problems discussed here are chosen in part because they have a high degree of “universality” or generality across different microscopic situations, as the major challenge in nonequilibrium statistical physics, both quantum and classical, has been to find principles as general as the basic principles of equilibrium statistical physics or linear response.

When subject to a pair-breaking perturbation, the pairing susceptibility of a disordered superconductor exhibits substantial long-ranged mesoscopic fluctuations. Focusing on a thin film subject to a parallel magnetic field, it is proposed that the quantum phase transition to the bulk superconducting condensate may be pre-empted by the formation of a glass-like phase with multi-fractal correlations of a complex order parameter. Although not universal, we argue that such behaviour may be a common feature of quantum critical phenomena in disordered environments.

Mechanisms of quantum phase coherence heavily influence spectral and transport properties of weakly disordered normal conductors. Such effects are manifest in weak and strong localization effects, and characteristic fluctuation phenomena. Over the past thirty years, theoretical progress in elucidating the mechanisms of quantum phase coherence in weakly disordered structures has been substantial: By now a consistent theory of weakly interacting disordered structures has been developed (For a review, see e.g., Refs. [1-3]).