In this thesis we examine a variety of question regarding several quantum information-theoretic concepts as they manifest in the context of relativistic quantum fields. In particular, we study the qualitative and quantitative structure of entanglement (as well as more generalized quantum correlations) in field theory. Primarily we study the nature of vacuum entanglement, but consider spatial correlations in more general field states as well. We also discuss several novel aspects of particle creation phenomena, which generally go hand in hand with the presence of field entanglement. Such investigations are important from two perspectives. First, we gain enlightening new insight into the nature of quantum fields and, therefore, into the fundamental constituents of the matter contained in our universe. Second, we are able to consider the possibility of utilizing the relativistic nature of quantum fields as a resource for quantum technological tasks such as entanglement distribution, information processing, and metrology. \n \nAs a general guiding principle, here we approach such questions from an operational perspective. That is, we focus less on the mathematical facts derivable from a given theory and instead attempt to answer in what concretely physical ways such facts manifest themselves. For example, in what ways can vacuum entanglement actually be experimentally detected and measured, at least in principle? Such an approach allows us to circumvent otherwise problematic interpretational issues. It also lends itself more naturally to the proposal of real world experiments regarding, and utilizations of, the physics being investigated. In this thesis we utilize two models of detection as applied to field theory. \n \nThe first is one that we will develop from scratch and use to great effect in a number of studies; this is the non-perturbative oscillator detector. In this context, a ``detector" is simply some quantum system that is allowed to interact in some way with a quantum field. The response of this system then gives us information about the field properties. Typically such studies are done perturbatively, which presents several severe limitations on what may be investigated. We develop an alternative model, in which the detector is identified as a quantum harmonic oscillator, that allows non-perturbative solutions. We go on to apply this model to a variety of studies. We start by the examining the Unruh effect in a cavity setting. Not only are we able to examine the thermalization behavior of such a detector (which is stronger than what can be achieved perturbatively), we also demonstrate that the Unruh response is largely independent of the cavity boundary conditions, indicating a surprising amount of generality and universality to the phenomenon. We also consider the spacelike extraction of spatial entanglement by a set of such detectors. This includes the harvesting of genuine tripartite entanglement out of the vacuum, as well as the observation that the extractable quantum discord (a generalized measure of quantum correlations) can be greatly amplified by thermal fluctuations in the field, in opposite behavior to entanglement. This calls into question the nature of quantum discord, but also suggests at the possibility of actually using environmental noise as a quantum computational resource. Next, we use the oscillator detector model to great effect by proposing a method of reliably and sustainably generating and distributing entanglement via a cavity field. This protocol is inherently non-perturbative, and we refer to it as entanglement farming. Lastly, we modify the farming scenario to discover a very sensitive resonance effect that generates a large difference in the protocol output due to a minute change in system parameters, including the length of cavity being used. We put this forth as a potential method of quantum seismic metrology. \n \nThe second model that we will consider simply involves performing projective measurements onto a given set of field modes. We will find, however, that such an approach allows us to solve previously intractable problems. In particular, we apply this procedure to the physically sensible choice of localized, positive frequency field modes. Using this model we perform two different studies. In the first, we consider the response of an observer accelerating not through the vacuum state but rather a thermal state. Such a setting has not been (properly) studied previously, due to the computational challenges involved. By applying the localized projective measurement model we are able determine that such an observer sees both an Unruh response from acceleration and a response from the field thermality and that, critically, these responses are distinguishable. We then go on to examine the degradation of quantum correlations between two observers due to the acceleration of one of the observers. This is a prototypical scenario of interest and, by applying realistic measurement assumptions, we obtain results qualitatively different from many previous works on the issue. \n \nIn addition to these models, we question the nature of vacuum entanglement and particle creation by considering an operationally straightforward procedure that greatly demystifies these phenomena and puts them on to solid, physically concrete grounds. This operation is that of very quickly introducing a boundary condition onto one's field, for example by creating a mirror. The real particles generated in this process will then be quantum entangled with each other, even if they are created at spatially separate locations, and this entanglement derives exactly from the previously present vacuum entanglement. Not only does this realization provide an operationally satisfying interpretation of vacuum entanglement, it also suggests a straightforward procedure for its experimental verification. \n \nIn all of this, the computational framework that we will rely upon is that of Gaussian quantum mechanics. This is a restriction of continuous variable quantum mechanics that, while limited in its applicability, provides extremely powerful computational tools in the cases that it may be used. The scenarios of interest in this thesis fall within this category and lend themselves naturally to a Gaussian approach. With this we are able to perform, near trivially, many otherwise intractable studies.
In this thesis we examine a variety of question regarding several quantum information-theoretic concepts as they manifest in the context of relativistic quantum fields. In particular, we study the qualitative and quantitative structure of entanglement (as well as more generalized quantum correlations) in field theory. Primarily we study the nature of vacuum entanglement, but consider spatial correlations in more general field states as well. We also discuss several novel aspects of particle creation phenomena, which generally go hand in hand with the presence of field entanglement. Such investigations are important from two perspectives. First, we gain enlightening new insight into the nature of quantum fields and, therefore, into the fundamental constituents of the matter contained in our universe. Second, we are able to consider the possibility of utilizing the relativistic nature of quantum fields as a resource for quantum technological tasks such as entanglement distribution, information processing, and metrology. \n \nAs a general guiding principle, here we approach such questions from an operational perspective. That is, we focus less on the mathematical facts derivable from a given theory and instead attempt to answer in what concretely physical ways such facts manifest themselves. For example, in what ways can vacuum entanglement actually be experimentally detected and measured, at least in principle? Such an approach allows us to circumvent otherwise problematic interpretational issues. It also lends itself more naturally to the proposal of real world experiments regarding, and utilizations of, the physics being investigated. In this thesis we utilize two models of detection as applied to field theory. \n \nThe first is one that we will develop from scratch and use to great effect in a number of studies; this is the non-perturbative oscillator detector. In this context, a ``detector" is simply some quantum system that is allowed to interact in some way with a quantum field. The response of this system then gives us information about the field properties. Typically such studies are done perturbatively, which presents several severe limitations on what may be investigated. We develop an alternative model, in which the detector is identified as a quantum harmonic oscillator, that allows non-perturbative solutions. We go on to apply this model to a variety of studies. We start by the examining the Unruh effect in a cavity setting. Not only are we able to examine the thermalization behavior of such a detector (which is stronger than what can be achieved perturbatively), we also demonstrate that the Unruh response is largely independent of the cavity boundary conditions, indicating a surprising amount of generality and universality to the phenomenon. We also consider the spacelike extraction of spatial entanglement by a set of such detectors. This includes the harvesting of genuine tripartite entanglement out of the vacuum, as well as the observation that the extractable quantum discord (a generalized measure of quantum correlations) can be greatly amplified by thermal fluctuations in the field, in opposite behavior to entanglement. This calls into question the nature of quantum discord, but also suggests at the possibility of actually using environmental noise as a quantum computational resource. Next, we use the oscillator detector model to great effect by proposing a method of reliably and sustainably generating and distributing entanglement via a cavity field. This protocol is inherently non-perturbative, and we refer to it as entanglement farming. Lastly, we modify the farming scenario to discover a very sensitive resonance effect that generates a large difference in the protocol output due to a minute change in system parameters, including the length of cavity being used. We put this forth as a potential method of quantum seismic metrology. \n \nThe second model that we will consider simply involves performing projective measurements onto a given set of field modes. We will find, however, that such an approach allows us to solve previously intractable problems. In particular, we apply this procedure to the physically sensible choice of localized, positive frequency field modes. Using this model we perform two different studies. In the first, we consider the response of an observer accelerating not through the vacuum state but rather a thermal state. Such a setting has not been (properly) studied previously, due to the computational challenges involved. By applying the localized projective measurement model we are able determine that such an observer sees both an Unruh response from acceleration and a response from the field thermality and that, critically, these responses are distinguishable. We then go on to examine the degradation of quantum correlations between two observers due to the acceleration of one of the observers. This is a prototypical scenario of interest and, by applying realistic measurement assumptions, we obtain results qualitatively different from many previous works on the issue. \n \nIn addition to these models, we question the nature of vacuum entanglement and particle creation by considering an operationally straightforward procedure that greatly demystifies these phenomena and puts them on to solid, physically concrete grounds. This operation is that of very quickly introducing a boundary condition onto one's field, for example by creating a mirror. The real particles generated in this process will then be quantum entangled with each other, even if they are created at spatially separate locations, and this entanglement derives exactly from the previously present vacuum entanglement. Not only does this realization provide an operationally satisfying interpretation of vacuum entanglement, it also suggests a straightforward procedure for its experimental verification. \n \nIn all of this, the computational framework that we will rely upon is that of Gaussian quantum mechanics. This is a restriction of continuous variable quantum mechanics that, while limited in its applicability, provides extremely powerful computational tools in the cases that it may be used. The scenarios of interest in this thesis fall within this category and lend themselves naturally to a Gaussian approach. With this we are able to perform, near trivially, many otherwise intractable studies.
Quantum effects in non-inertial frames and curved space-time have been studied for decades. One of the most intriguing discoveries is the existence of virtual particles in the quantum vacuum. One example in which such particles can be observed is through the Unruh effect, which predicts that a uniformly accelerated observer would see a thermal bath of (virtual) particles in the quantum vacuum. Initial doubts on the existence of these virtual particles were allayed by the Unruh De-Witt (UDW) detector, which promoted these virtual particles to real excitations of the detector. Ever since, semiclassical approaches to relativistic quantum field theory (QFT) in flat and curved space-time have been predominantly analysed through a two-level Unruh De-Witt detector utilizing perturbation theory. It is noted, however, that the change of relativistic frame preserves Gaussian properties (e.g. Gaussian distribution of the Wigner function) of the system. In this thesis, we specifically explore the effects of relativistic (non-inertial) reference frames on QFT by considering relativistic QFT in flat-space time. These systems are analysed from the perspective of Gaussian quantum information, via homodyne detection and UDW detectors with a harmonic oscillator degree of freedom.Our project starts by reviewing and developing bipartite/multi-partite entanglement measures (specifically, entanglement of formation), in the hope that we can utilize these measures to understand the entanglement properties of the quantum vacuum. In general, these measure have infinite degrees of freedom and are incomputable. In this project the multi-partite entanglement measure is applied to the Gaussian regime, which reduces the number of degrees of freedom to a finite one, hence making it a computable measure.In our next project, we attempt to understand how relativistic reference frames affect the communication between different observers. To obtain an intuition of its effect, a passive (time-delay) signal sent from a uniformly accelerated observer to a stationary observer was analysed. When we naively implemented the self-homodyne detection scheme, the signal appeared noisy. This effect was referred to as apparent decoherence, and its effect could be traced back to our naive assumption that all of the signal’s information would be stored in the time-delayed mode; the vacuum entanglement that pre-existed before the (time-delay) signal was created must be accounted for. We follow by analysing the effects that emerge due to communication between different relativistic reference frames. We then develop a communication technique referred to as ideal-homodyne which is a homodyne detection scheme robust to these effects. We used this detection scheme to analyse the correlation/entanglement properties of an accelerated signal (created via accelerated mirror, squeezer and phase shifter) sent to an inertial observer.In our last project, we utilize the knowledge that was gained through the prior projects to analyse (subcycle) electro-optic sampling. In the literature, there were doubts as to whether electro-optic sampling were truly detecting virtual particles in the quantum vacuum, or particle excitations that were created as a by-product of the detection mechanism. We were able to pin-point some similarities between electro-optic sampling and subcycle probe via the UDW detector. The regime in which the electro-optic sampling directly maps virtual particles from the vacuum into real excitations of the probe field was identified by establishing an equivalence to the UDW detector in certain parameter regimes.
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 fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras. In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data. Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization. In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
Quantum effects in accelerated frames and gravitational fields have been studied for decades. One of the most influential outcomes is the discovery of thermal radiation from a black hole by Hawking in 1975. Other important discoveries include the Unruh effect, dynamical Casimir effect etc.. Although these discoveries are very exciting, experimental verification of them is extremely challenging. Even in the theoretical aspect, not all the issues have been resolved, e.g., the well known black hole information paradox. Quantum information science was developed rapidly during the last thirty years. The well established concepts and tools in quantum information science have been used to explore the quantum effects in gravitational fields and relativistic frames, giving birth to a new research field named relativistic quantum information. This thesis studies quantum effects in accelerated frames and gravitational fields by exploiting the concepts and techniques in quantum information science. The Unruh effect implies that the state of the fields confined within part of the Minkowski spacetime can appear thermal, and entanglement exists between different spacetime regions. We show that the particle number distribution of the field modes confined within a finite diamond region is also thermal in the Minkowski vacuum, an analogue to the Unruh effect; and there exists entanglement between different diamonds. The vacuum entanglement can be extracted and utilized for some quantum information protocols, e.g., quantum key distribution. Furthermore, we show that the presence of a horizon and the Unruh thermal noise has important consequences to the quantum communication protocols where one of the parties is a uniformly accelerated observer. Interactions between uniformly accelerated objects and quantum fields are traditionally studied using perturbation theory. The quantum circuit model, a crucial tool in quantum communication and computation, can be exploited to calculate radiations from the uniformly accelerated objects non-perturbatively. By further combining field detection scheme in quantum optics, e.g., homodyne detection, the output field from the uniformly accelerated objects can be fully studied. These techniques help to study decoherence effect in non-inertial frames, which may provide important insights for the black hole information paradox. Dynamical spacetimes generally create quantum particles. Gravitational perturbations around a black hole oscillate and decay, due to the emission of gravitational waves to spatial infinity and into the black hole. We show that they play the role as a multimode squeezer, squeezing the state of the quantum fields and creating particles.
In recent decades, substantial theoretical and experimental progress was achieved in understanding the quantum nature of physical phenomena that serves as the foundation of present and future quantum technologies. Quantum correlations like the entanglement of the states of composite systems, the phenomenon of quantum discord, which captures other aspects of quantum correlations, quantum contextuality and, connected with these phenomena, uncertainty relations for conjugate variables and entropies, like Shannon and Rényi entropies, and the inequalities for spin states, like Bell inequalities, reflect the recently understood quantum properties of micro and macro systems. The mathematical methods needed to describe all quantum phenomena mentioned above were also the subject of intense studies in the end of the last, and beginning of the new, century. In this section of CAMOP 'Mathematical Methods of Studying Physical Phenomena' new results and new trends in the rapidly developing domain of quantum (and classical) physics are presented. Among the particular topics under discussion there are some reviews on the problems of dynamical invariants and their relations with symmetries of the physical systems. In fact, this is a very old problem of both classical and quantum systems, e.g. the systems of parametric oscillators with time-dependent parameters, like Ermakov systems, which have specific constants of motion depending linearly or quadratically on the oscillator positions and momenta. Such dynamical invariants play an important role in studying the dynamical Casimir effect, the essence of the effect being the creation of photons from the vacuum in a cavity with moving boundaries due to the presence of purely quantum fluctuations of the electromagnetic field in the vacuum. It is remarkable that this effect was recently observed experimentally. The other new direction in developing the mathematical approach in physics is quantum tomography that provides a new vision of quantum states. In the tomographic picture of quantum mechanics, the states are identified with fair conditional probability distributions, which contain the same information on the states as the wave function or the density matrix. The mathematical methods of the tomographic approach are based on studying the star-product (associative product) quantization scheme. The tomographic star-product technique provides an additional understanding of the associative product, which is connected with the existence of specific pairs of operators called quantizers and dequantizers. These operators code information on the kernels of all the star-product schemes, including the traditional phase-space Weyl–Wigner–Moyal picture describing the quantum-system evolution. The new equation to find quantizers, if the kernel of the star product of functions is given, is presented in this CAMOP section. For studying classical systems, the mathematical methods developed in quantum mechanics can also be used. The case of paraxial-radiation beams propagating in waveguides is a known example of describing a purely classical phenomenon by means of quantum-like equations. Thus, some quantum phenomenon like the entanglement can be mimicked by the properties of classical beams, for example, Gaussian modes. The mathematical structures and relations to the symplectic symmetry group are analogous for both classical and quantum phenomena. Such analogies of the mathematical classical and quantum methods used in research on quantum-like communication channels provide new tools for constructing a theoretical basis of the new information–transmission technologies. The conventional quantum mechanics and its relation to classical mechanics contain mathematical recipes of the correspondence principle and quantization rules. Attempts to find rules for deriving the quantum-mechanical formalism starting from the classical field theory, taking into account the influence of classical fluctuations of the field, is considered in these papers. The methods to solve quantum equations and formulate the boundary conditions in the problems with singular potentials are connected with the mathematical problems of self-adjointness of the Hamiltonians. The progress and some new results in this direction are reflected in this CAMOP section. The Gaussian states of the photons play an important role in quantum optics. The multimode electromagnetic field and quantum correlations in the Gaussian states are considered in this section. The new results in the statistical properties of the laser radiation discussed here are based on applications of mathematical methods in this traditional domain of physics. It is worth stressing that the universality of the mathematical procedures permitted to consider the physical phenomena in the ocean is on the same footing as the phenomena in the microworld. In this CAMOP section, there are also papers devoted to traditional problems of solving the Schrödinger equation for interesting quantum systems. Recently obtained results related to different domains of theoretical physics are united by applying mathematical methods and tools, that provide new possibilities to better understand the theoretical foundations needed to develop new quantum technologies like quantum computing and quantum communications. The papers are arranged alphabetically by the name of the first author. We are grateful to all authors who accepted our invitation to contribute to this CAMOP section.