This is an updated version of my PhD thesis, defended at the University of Waterloo on the 2nd of April 2025, uploaded to the ArXiv with the goal of reaching a wider audience. The thesis is divided into 5 chapters, respectively containing (I) a brief introduction to local quantum field theory (QFT), (II) a description of local probes in QFT, (III) a discussion of entanglement in QFT and how to probe it, (IV) a description of the regimes where QFT interactions can be approximated by direct interactions, and (V) a discussion the information about the geometry of spacetime contained in quantum fields. The partial goal of this thesis is to serve as a guide for students aiming to tackle these different research programs. If the reader is interested in pursuing one or more research projects detailed here, they are encouraged to contact me for collaboration in these topics.
This is an updated version of my PhD thesis, defended at the University of Waterloo on the 2nd of April 2025, uploaded to the ArXiv with the goal of reaching a wider audience. The thesis is divided into 5 chapters, respectively containing (I) a brief introduction to local quantum field theory (QFT), (II) a description of local probes in QFT, (III) a discussion of entanglement in QFT and how to probe it, (IV) a description of the regimes where QFT interactions can be approximated by direct interactions, and (V) a discussion the information about the geometry of spacetime contained in quantum fields. The partial goal of this thesis is to serve as a guide for students aiming to tackle these different research programs. If the reader is interested in pursuing one or more research projects detailed here, they are encouraged to contact me for collaboration in these topics.
Recently, there has been increased interest in understanding entanglement and quantum communication in black hole spacetimes and in using quantum information techniques to address questions in gravity. Studies on relativistic entanglement show the emergence of conceptually important qualitative differences to a non-relativistic treatment. For instance, entanglement was found to be an observer-dependent property that changes from the perspective of accelerated observers moving in flat spacetime. Relativisitic quantum information theory uses well-known tools coming from quantum information and quantum optics to study quantum effects provoked by gravity to learn information about the spacetime. We can take advantage of our knowledge about quantum correlations and effects produced by the gravitational interaction to set the basis for experimental proposals ultimately aiming at finding corrections due to quantum gravity effects, too mild to be directly observed. This doctoral thesis dissertation summarises most of the research carried out during my PhD on this topic.
This thesis aims to establish the study of a somewhat novel form of entanglement in field-theoretical quantum systems: that between degrees of freedom in momentum space. Although scattered results can be found in the literature, with some even going back decades, this work, based on the papers published during the PhD, provides a systematic investigation of the matter and arrive at non-perturbative results, shedding light on unexplored structures present in Quantum Field Theories (QFTs) and their ground states. This line of research is motivated by the inherent connection between the notion of momentum modes of fields and the Wilsonian formulation of the Renormalization Group (RG), a cornerstone of the modern understanding of field theories and phases of matter which is naturally defined by the partition of the degrees of freedom into “fast' and “slow' modes, according to their momenta. Furthermore, by glancing at the achievements of techniques such as the Density Matrix Renormalization Group and Tensor Networks, which are based on ideas from renormalization in real space, there is reason to believe that similarly efficient methods may be derived by connecting the fundaments of the RG with quantum information in momentum-space. With this in mind, the thesis begins by introducing the main notions of information theory and renormalization used throughout, with the inclusion of a brief discussion on the use of von Neumann algebras in quantum physics, a formalism which is not directly applied here, but nevertheless provides a solid basis for many intuitions on the nature of entanglement. Afterwards, the idea of degrees of freedom in momentum-space is defined and its different subtleties clarified. A key insight, that any field theory possesses multiple definitions of partitioning its Hilbert space into momentum modes, based on a choice of a complete set of local and commuting observables or field variables in a path integral, is then obtained. Given this solid grasp on the fundaments of quantum information in momentum-space, two directions of further investigation are pursued. First, a path-integral method is developed for the calculation of the associated entanglement entropies in ground states and applied perturbatively to several models, which also allows for the quantitative confirmation of certain predictions. Secondly, the Wilsonian renormalization procedure is translated to this formalism and is shown to be equivalent to a so-called quantum channel which acts on momentum-space. While an interesting result on its own right, this also leads directly to a powerful constraint on the structure of ground states of renormalization group fixed points: they must be separable, and thus uncorrelated, when the degrees of freedom are partitioned based on their momentum scale. This opens the way for further analytical and numerical studies of these special QFTs beyond the scope of this thesis. Finally, the concept of fidelity of quantum states is invoked in order to better understand the information aspects of the renormalization group flow for theories in the critical manifold of a given fixed point. It is used to quantify the idea that such QFTs are indistinguishable from the fixed point itself when long-distance observables are measured and new “critical exponents' are introduced in this context, providing tight bounds on the differences of expectation values of operators in the critical and fixed-point theories. The results obtained in this work mark only the beginning of the in-depth research on different structures present in the momentum-space of Quantum Field Theories and may, hopefully, serve as a basis for further developments.:Contents Acknowledgements iii 1 Introduction 1 1.1 Notions of quantum information 5 1.2 Comments on von Neumann algebras 10 1.3 Basics of Wilsonian renormalization 14 2 Defining momentum-space entanglement 18 2.1 Momentum modes from the path integral 19 2.2 Non-uniqueness of partitions 24 3 The replica method for non-local actions 33 3.1 Construction of the replica trick 34 3.2 Entanglement between momentum scales in perturbation theory 38 3.2.1 Perturbation theory and the replica trick 39 3.2.2 Coupled Oscillators 40 3.2.3 Perturbative interacting QFT 42 3.2.4 Sine-Gordon and Thirring models 48 4 Wilsonian RG as a quantum channel 53 4.1 Quantum operations from the RG steps 54 4.2 Separability of ground states at fixed points 59 4.3 Further implications 63 4.3.1 The RG is a quantum dynamical semigroup 64 4.3.2 First-order phase transitions 67 5 Critical theories and operator estimations 74 5.1 Quantum state distinguishability 75 5.2 Fidelity in translation-invariant systems 77 5.3 Hyperscaling in the critical manifold 80 6 Conclusions and further directions 86 Bibliography 88
This thesis consists of four research papers and an introduction covering the most important concepts appearing in the papers. The papers deal with applications of gauge/gravity dualities in the study of various physical quantities and systems. Gauge/gravity dualities are equivalences between certain quantum field theories and classical theories of gravity. These dualities can be used as computational tools in a wide range of applications across different fields of physics, and as such they have garnered much attention in the last two decades. The great promise of these new tools is the ability to tackle difficult problems in strongly interacting quantum field theories by translating them to problems in classical gravity, where progress is much easier to make. Quantum information theory studies the information contained in quantum systems. Entanglement is the fundamental property of quantum mechanics that sets it apart from classical theories of physics. Entanglement is commonly quantified by entanglement entropy, a quantity which is difficult to compute in interacting quantum field theories. Gauge/gravity dualities provide a practical way for computing the entanglement entropy via the Ryu-Takayanagi formula. The primary focus of this thesis is to use this formula for computing various entanglement measures in strongly interacting quantum field theories via their gravity duals. The purpose of this thesis is to introduce quantum information theory concepts that have been important in our research. When applicable, quantities of interest are first introduced in the classical setting in order to build intuition about their behaviour. Quantum properties of entanglement measures are discussed in detail, along with their holographic counterparts, and remarks are made concerning their applications.
This thesis (Habilitation à diriger des recherches) presents some of my research contributions since my Ph.D defense in 2008. I have had the chance to participate in the development of quantum programming languages since their early developments: the presentation aims to present my point of view on the evolution of the subject, my contributions, and the current research trends in the community. The target audience is a graduate student interested in pointers to the field of quantum programming languages.
This thesis collects my own and collaborative work I have been involved with finding localised systems in quantum field theory that are useful for quantum information. It draws from many well established physical theories such as quantum field theory in curved spacetimes, quantum optics and Gaussian state quantum information. The results are split between three chapters. For the first results, we set-up the basic framework for working with quantum fields confined to cavities. By considering the real Klein-Gordon field, we describe how to model the non-uniform motion of a rigid cavity through spacetime. We employ the use of Bogoliubov transformations to describe the effects of changing acceleration. The second set of results investigate how the Dirac field can be confined to a cavity for quantum information purposes. By again considering Bogoliubov transformations, we thoroughly investigate how the entanglement shared between two cavities is affected by non-uniform motion. It is shown that different types of Dirac field states are more robust against motion than others. The final results look at using our second notion of localisation, Unruh- DeWitt detectors. We investigate how allowing for a "non-point-like" spatial profile of the Unruh-DeWitt detector affects how it interacts with a quantum field around it. By engineering suitable detector-field interactions, we use techniques from symplectic geometry to compute the dynamics of a quantum state beyond commonly used perturbation theory. There is also a conclusions chapter at the end of the thesis which summarises the results presented and suggestions of possible new directions of research in relativistic quantum information are made.
This thesis will be focused on the classical capacity of quantum channels, one of the first areas treated by quantum information theorists. The problem is fairly solved since some years. Nevertheless, this work will give me a reason to introduce a consistent formalism of the quantum theory, as well as to review fundamental facts about quantum non-locality and how it can be used to enhance communication. Moreover, this reflects my dwelling in the spirit of classical information theory, and it is intended to be a starting point towards a thorough study of how quantum technologies can help to shape the future of telecommunications. Whenever it was possible, heuristic reasonings were introduced instead of rigorous mathematical proofs. This finds an explanation in that I am a self-taught neophyte in the field, and just about every time I came across a new concept, physical arguments were always more compelling to me than just maths. The technical content of the thesis is twofold. On one hand, a quadratic classification based on optimization programs that I devised for distinguishing entangled states is presented in Chapter 4. In second place, a less difficult yet I hope equally interesting technical part consists of versions of some proofs throughout the text.