This thesis explores the role of modes, states, and symmetries in quantum optics, within the context of quantum information and quantum metrology. It proposes a unified framework to analyze how the modal structure of photonic fields, the statistical nature of states, and their symmetry properties determine the physical resources that can be exploited for quantum information processing and quantum parameter estimation. A first line of investigation develops a description of time-frequency degrees of freedom as continuous quantum variables, highlighting their richness for encoding and manipulating information. A second axis studies entanglement and collective variables, clarifying the link between physical resources and metrological gains, particularly in reaching ultimate precision limits. Interferometric scenarios of the Hong-Ou-Mandel type are then analyzed, and a general formalism centered on the notion of symmetry is developed. This framework enables the analysis of a broad range of situations and leads to several generalizations. Finally, the thesis examines the symmetries imposed by optical superselection rules and their consequences for the structure of quantum states and their operational performance, with the aim of providing a deeper understanding of the foundations of quantum optics.
Comments
508 pages, 56 figures, 6 tables. Please also cite the relevant journal articles when citing the thesis v2: Corrected numerous typographical, language, and mathematical errors, and reworked the proofs of Results 86 and 87 to address gaps in the previous arguments
This thesis explores the role of modes, states, and symmetries in quantum optics, within the context of quantum information and quantum metrology. It proposes a unified framework to analyze how the modal structure of photonic fields, the statistical nature of states, and their symmetry properties determine the physical resources that can be exploited for quantum information processing and quantum parameter estimation. A first line of investigation develops a description of time-frequency degrees of freedom as continuous quantum variables, highlighting their richness for encoding and manipulating information. A second axis studies entanglement and collective variables, clarifying the link between physical resources and metrological gains, particularly in reaching ultimate precision limits. Interferometric scenarios of the Hong-Ou-Mandel type are then analyzed, and a general formalism centered on the notion of symmetry is developed. This framework enables the analysis of a broad range of situations and leads to several generalizations. Finally, the thesis examines the symmetries imposed by optical superselection rules and their consequences for the structure of quantum states and their operational performance, with the aim of providing a deeper understanding of the foundations of quantum optics.
The Hong-Ou-Mandel interferometer is a foundational tool in quantum optics with both fundamental and practical significance. Earlier works identified that input-state symmetry under exchange of the two spatial modes is fundamental in the understanding of the Hong-Ou-Mandel effect. We now show that this notion of symmetry is central to generalizing this effect. In particular, this point of view enables the construction of extensions beyond the standard two single-photon case to arbitrary input states, as well as to configurations with more than two spatial modes via a natural generalization of the beam splitter to a discrete Fourier transform interferometer. Beyond its conceptual significance, this framework offers direct insights into quantum metrology, showing how symmetry properties of input states allow for the computation of explicit precision bounds. By focusing on symmetry, we provide a perspective that simplifies and unifies a range of known results while paving the way for new developments in quantum interference and sensing.
It is impossible to overstate the importance of symmetry in physics and mathematics. Symmetry arguments play a central role in a broad range of problems from simplifying a system of linear equations to a deep role in organizing the fundamental principles of physics. They are used, for instance, in Noether’s theorem to find the consequences of symmetry of a dynamics. For many systems of interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. \n \nIn part I of this thesis we study the problem of finding the consequences of symmetry of a (possibly open) dynamics from an information-theoretic perspective. The study of this problem naturally leads us to the notion of asymmetry of quantum states. The asymmetry of a state relative to some symmetry group specifies how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful to constrain which final states of the system can be reached from a given initial state. Another motivation for the study of asymmetry comes from the field of quantum metrology and relatedly the field of quantum reference frames. It turns out that the degree of success one can achieve in many metrological tasks depends only on the asymmetry properties of the state used for metrology. We show that some ideas and tools developed in the field of quantum information theory are extremely useful to study the notion of asymmetry of states and therefore to find the consequences of symmetry of an open or closed system dynamics. \n \nIn part II of this thesis we present a novel application of symmetry arguments in the field of quantum estimation theory. We consider a family of multi-copy estimation problems wherein one is given n copies of an unknown quantum state according to some prior distribution and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. We introduce a new approach to this problem by considering the symmetries of the prior and the symmetries of the parameters to be estimated. We show that based on these symmetries one can find strong constraints on the amount of entanglement required to implement the optimal measurement. In order to infer properties of the optimal estimation procedure from the symmetries of the parameters and the prior we come up with a generalization of Schur-Weyl duality. Just as Schur-Weyl duality has many applications to quantum information theory and quantum algorithms so too does this generalization. In this thesis we explore some of these applications.
In this thesis, I present experimental and theoretical work which assesses and develops a range of tools that are required for performing quantum information processing, particularly in photonic systems. I investigate the three degrees of freedom of a single photon—its polarisation, spatialmomentum and time-frequency distributions. For polarisation, I show that using wave plates to implement arbitrary, single-qubit rotations is more complicated than is commonly appreciated. In the spatial-momentum and time-frequency domains, I develop ways to perform tomographic analysis of quantum states, and I report the first demonstrations of these techniques. In the time-frequency domain, the tomography technique utilises entanglement in the photon polarisation as a resource to store and provide access to the time-frequency information. In my first two experiments, I use spontaneous parametric down-conversion to produce entanglement between pairs of single photons in all three degrees of freedom. I demonstrate the first characterisation of entanglement in spatial modes and the time-frequency domain, the first quantitative measurement of entangled qutrit states, and the highest quality entangled states yet measured in both polarisation and spatial modes. I also report the first realisation of complete hyperentanglement, and a full, black-box tomography of a 36-dimensional two-photon state—the largest system to be characterised in this way to date. In my final experiment, I model and implement a new architecture for a controlled-Z gate which is much simpler to align than previous implementations. This gate requires only one non-classical interference condition, the visibility of which is the main limitation to its performance. I show that the gate operates effectively as a means of both creating entanglement and discriminating between the four elements in a basis of maximally entangled, Bell-type states. Indeed, its observed performance as a Bell analyser would be sufficient to build a quantum state teleporter which would guarantee that the recipient would be left with a better copy of an unknown input state than any eavesdropper. In a series of numerical simulations, I investigate some of the practicalities that arise when using tomographic reconstruction techniques, including how to estimate errors, which measurements to make and what is actually the optimal reconstruction. In particular, I show that tomographies perform better when based on the results from over-complete sets of measurements. Finally, I discuss the important issue of how to compare two processes, particularly when trying to assess the quality of a measured process by comparing it to some expected ideal. Judging possible candidate measures against a set of experimentally and theoretically motivated
Over the last few decades, quantised excitations of the electromagnetic field have proven to be an ideal system with which to investigate and harness quantum optical phenomena. The techniques developed have enabled fundamental tests of quantum mechanics as well as practical applications in quantum metrology and quantum information processing. Advancing to larger-scale entangled quantum systems will open up new regimes of quantum many-body physics, allowing us to probe the limits of quantum mechanics and enabling truly quantum-enhanced technologies. However, moving towards this goal will require further experimental and theoretical innovations. The work described in this thesis focuses on several different aspects of optical quantum information, but are ultimately all linked by this long-term aim. The first part of this thesis describes a novel method for strain-based active control of quantum optical circuits and a new method for the characterisation of high efficiency detectors. Building on this, I discuss in detail two different fields of quantum optics that stand to benefit from these techniques. I initially consider quantum-enhanced metrology, including work aimed towards demonstrating a truly better-than-classical phase measurement, and a theoretical exploration of multiple-phase estimation. Finally, I focus on linear-optical quantum information processing, exploring in detail the use of time-frequency encodings for quantum computing.
Relativistic quantum information investigates the interplay between quantum mechanics and relativity and is becoming increasingly relevant for future quantum technologies operating over large distances or in gravitational fields. This dissertation studies how relativistic effects influence the storage, transmission, and interference of quantum states of light, with particular emphasis on quantum memories and frequency-entangled photons. A theoretical framework is developed to describe relativistic transformations of photonic quantum states and their interaction with realistic quantum memories. Special attention is given to the influence of gravitational redshift and relativistic motion on quantum coherence, entanglement, and Hong–Ou–Mandel interference. The resulting models provide quantitative predictions for experimentally accessible scenarios and identify conditions under which relativistic effects become relevant for quantum communication and distributed quantum information processing. The presented results contribute to the emerging field of relativistic quantum information by connecting concepts from quantum optics, quantum communication, and general relativity. Beyond their fundamental significance, they provide theoretical tools for the development of future space-based quantum networks and relativistic quantum technologies. Beyond the results presented in this dissertation, the developed framework points toward a broader research program centered on the **Nonlinear Separability Problem**. This problem is proposed as a unifying mathematical perspective for investigating the limits of distinguishability, classification, and information processing in both classical and quantum systems. Its formulation naturally connects to a wide range of contemporary research areas, including computational complexity theory (e.g., P versus NP), quantum information processing, machine learning, optimization, information theory, quantum foundations, and mathematical physics. In particular, the framework suggests new approaches to studying the role of nonlinearity in quantum state discrimination, quantum communication protocols, and the mathematical structure of high-dimensional information spaces. While many of these directions remain to be explored, the present work establishes a conceptual and mathematical basis for future investigations. It is anticipated that further development of the Nonlinear Separability Problem may contribute to a deeper understanding of fundamental questions across mathematics, computer science, and quantum technologies, thereby opening new interdisciplinary avenues of research.
The current frontier of our understanding of the physical universe is dominated by quantum phenomena. Uncovering the prospects and limitations of acquiring and processing information using quantum effects is an outstanding challenge in physical science. This thesis presents an analysis of several new model systems and applications for quantum information processing and metrology.\n\nFirst, we analyze quantum optomechanical systems exhibiting quantum phenomena in both optical and mechanical degrees of freedom. We investigate the strength of non-classical correlations in a model system of two optical and one mechanical mode. We propose and analyze experimental protocols that exploit these correlations for quantum computation.\n\nWe then turn our attention to atom-cavity systems involving strong coupling of atoms with optical photons, and investigate the possibility of using them to store information robustly and as relay nodes. We present a scheme for a robust two-qubit quantum gate with inherent error-detection capabilities. We consider several remote entanglement protocols employing this robust gate, and we use these systems to study the performance of the gate in practical applications.\n\nFinally, we present a new protocol for running multiple, remote atomic clocks in quantum unison. We show that by creating a cascade of independent Greenberger-Horne-Zeilinger states distributed across the network, the scheme asymptotically reaches the Heisenberg limit, the fundamental limit of measurement accuracy. We propose an experimental realization of such a network consisting of neutral atom clocks, and analyze the practical performance of such a system.