Quantum Mechanical Vistas on the Road to Quantum Gravity

In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.</p

Learning Identifiable Representations: Independent Influences and Multiple Views

Intelligent systems, whether biological or artificial, perceive unstructured information from the world around them: deep neural networks designed for object recognition receive collections of pixels as inputs; living beings capture visual stimuli through photoreceptors that convert incoming light into electrical signals. Sophisticated signal processing is required to extract meaningful features (e.g., the position, dimension, and colour of objects in an image) from these inputs: this motivates the field of representation learning. But what features should be deemed meaningful, and how to learn them? We will approach these questions based on two metaphors. The first one is the cocktail-party problem, where a number of conversations happen in parallel in a room, and the task is to recover (or separate) the voices of the individual speakers from recorded mixtures—also termed blind source separation. The second one is what we call the independent-listeners problem: given two listeners in front of some loudspeakers, the question is whether, when processing what they hear, they will make the same information explicit, identifying similar constitutive elements. The notion of identifiability is crucial when studying these problems, as it specifies suitable technical assumptions under which representations are uniquely determined, up to tolerable ambiguities like latent source reordering. A key result of this theory is that, when the mixing is nonlinear, the model is provably non-identifiable. A first question is, therefore, under what additional assumptions (ideally as mild as possible) the problem becomes identifiable; a second one is, what algorithms can be used to estimate the model. The contributions presented in this thesis address these questions and revolve around two main principles. The first principle is to learn representation where the latent components influence the observations independently. Here the term “independently” is used in a non-statistical sense—which can be loosely thought of as absence of fine-tuning between distinct elements of a generative process. The second principle is that representations can be learned from paired observations or views, where mixtures of the same latent variables are observed, and they (or a subset thereof) are perturbed in one of the views—also termed multi-view setting. I will present work characterizing these two problem settings, studying their identifiability and proposing suitable estimation algorithms. Moreover, I will discuss how the success of popular representation learning methods may be explained in terms of the principles above and describe an application of the second principle to the statistical analysis of group studies in neuroimaging