This thesis explores a broad range of topics in the foundations of relativity and quantum information theory. The first and main topic of this thesis is on relativistic quantum information theory. Here we construct a reformulation of quantum information, which is consistent with relativity theory. We will see that by providing a rigorous formulation starting with the field equations for a massive fermion and a photon we can construct a theory for relativistic quantum information. In particular we provide a measurement formalism, a transport equation which describes the unitary evolution of a state through spacetime as well as how to extend this to multipartite systems. The second topic concerns the nature of time, duration and clocks in current physical theories and in particular for Newtonian mechanics. We analyse the relationship between the readings of clocks in Newtonian mechanics with absolute time. We will see that in order to answer this question we must provide not only a model for a clock but also solve what is referred to as Newton’s Scholium problem. We then compare this with other dynamical theories in particular quantum mechanics and general relativity where the treatment of time is quite different from Newtonian mechanics. The final topic is rather different from the first two. In this chapter we investigate a range of methods to perform tomography in a solid-state qubit device, for which a priori initialization and measurement of the qubit is restricted to a single basis of the Bloch sphere. We explore and compare several methods to acquire precise descriptions of additional states and measurements, quantifying both stochastic and systematic errors, ultimately leading to a tomographically-complete set that can be subsequently used in process tomography. We focus in detail on the example of a spin qubit formed by the singlet-triplet subspace of two electron spins in a GaAs double quantum dot, although our approach is quite general.
This thesis explores a broad range of topics in the foundations of relativity and quantum information theory. The first and main topic of this thesis is on relativistic quantum information theory. Here we construct a reformulation of quantum information, which is consistent with relativity theory. We will see that by providing a rigorous formulation starting with the field equations for a massive fermion and a photon we can construct a theory for relativistic quantum information. In particular we provide a measurement formalism, a transport equation which describes the unitary evolution of a state through spacetime as well as how to extend this to multipartite systems. The second topic concerns the nature of time, duration and clocks in current physical theories and in particular for Newtonian mechanics. We analyse the relationship between the readings of clocks in Newtonian mechanics with absolute time. We will see that in order to answer this question we must provide not only a model for a clock but also solve what is referred to as Newton’s Scholium problem. We then compare this with other dynamical theories in particular quantum mechanics and general relativity where the treatment of time is quite different from Newtonian mechanics. The final topic is rather different from the first two. In this chapter we investigate a range of methods to perform tomography in a solid-state qubit device, for which a priori initialization and measurement of the qubit is restricted to a single basis of the Bloch sphere. We explore and compare several methods to acquire precise descriptions of additional states and measurements, quantifying both stochastic and systematic errors, ultimately leading to a tomographically-complete set that can be subsequently used in process tomography. We focus in detail on the example of a spin qubit formed by the singlet-triplet subspace of two electron spins in a GaAs double quantum dot, although our approach is quite general.
This thesis is a compilation of research in relativistic quantum information theory, and research in quantum reference frames. The research in the former category provides a fundamental construction of quantum information theory of localised qubits in curved spacetimes. For example, this concerns quantum experiments on free-space photons and electrons in the vicinity of the Earth. From field theory a description of localised qubits that traverse classical trajectories in curved spacetimes is obtained, for photons and massive spin-1/2 fermions. The equations governing the evolution of the two-dimensional quantum state and its absolute phase are determined. Quantum information theory of these qubits is then developed. The Stern-Gerlach measurement formalism for massive spin-1/2 fermions is also derived from field theory. In the latter category of research, the process of changing reference frames is considered for the case where the reference frames are quantum systems. As part of this process, it is shown that decoherence is necessarily induced on any quantum system described relative to these frames. This process is explored using examples involving quantum reference frames for phase and orientation. Quantifying the effect of changing quantum reference frames provides a theoretical description for this process in quantum experiments, and serves as a first step in developing a relativity principle for theories in which all objects including reference frames are necessarily quantum.
This thesis is a collection of works focusing on interrelations between the fields of special relativity, open quantum systems and complex networks. As each of the aforementioned fields encompasses a huge variety of topics, the thesis contains a selection of particular connections. The first half of the thesis considers an open quantum systems approach to the description of relativistic phenomena. As the aforementioned phenomena often involve dynamically evolving counterparts, some non-Markovian aspects of open quantum systems are initially explored. Conditions for complete positivity are derived for a type of phase-covariant time-local master equation and then applied to a non-Markovian master equation with heuristically-derived decay rates. This is relevant in cases when the physicality of the master equation cannot be postulated. The first relativistic system considered is a superposition of coherent states in a gravitational gradient. It is shown to exhibit decoherence induced by gravitational time dilation through interaction between inner degrees of freedom and the centre of mass of the superposition. The decoherence is quantified using various, widely-used non-classicality measures. The decoherence rates are then compared to decoherence induced by classical noise to roughly evaluate the experimental precision necessary to detect gravitationally induced decoherence. In the second relativistic system, the Unruh effect is modeled as an open system where an Unruh-deWitt detector is interacting with bosonic fields. A master equation with time-dependent decay rates for the open system is derived by assuming a non-conventional spacetime path profile for the detector. A particular parameter governing the physical evolution is identified. The system is shown to exhibit non-Markovianity within the completely positive domain of the master equation, the latter ensured by the conditions derived before. The second half of the thesis starts with a definition of complex networks, which are yet not a commonly used tool in quantum physics. The concept of pairwise tomography networks is then introduced as a way to represent many-body quantum states. As the number of pairwise connections grows quadratically with the number of nodes, an efficiently scaling measurement scheme to recover the pairwise tomography networks is presented. The scheme and the concept of pairwise tomography networks are demonstrated to be useful in various applications and the results of a proof-of-principle experiment are shown. One application explored is a paradigmatic spin chain model known as the XX model. A pairwise entanglement network representation of the XX model is shown to suggest new phenomena such as gradual establishment of quasi-long range order accompanied by a symmetry regarding single-spin concurrence distributions. The existence of structural classes and a cyclic self-similarity in the state are revealed.
This thesis is split into two parts, which are united in the sense that they involve applying ideas from quantum information to fundamental physics. The first part is focused on examining discrete-time models in quantum computation (discrete-time quantum walks and quantum cellular automata) as discretized models of relativistic systems. One of the results here is a theorem demonstrating that a large class of discrete-time quantum walks have relativistic dynamics in the continuum limit. Additionally, the problem of fermion doubling for these models is investigated, and it is seen that the problem can be circumvented in two dimensional space. This was already known for one dimensional systems. Another result involves taking the limits of causal free field theories in discrete spacetime to recover continuum field theories, something that is not straightforward because of the nontrivial nature of the vacuum in quantum field theory. Additionally, it is shown that general systems of fermions evolving causally in discrete spacetime can be represented by quantum cellular automata, which makes them efficiently simulable by quantum computers. A related result is that quantum cellular automata composed of fermions are equivalent to regular quantum cellular automata. In the second part of this thesis, the focus is on the foundations of statistical physics. The main result of this part is a general bound on the time it takes a quantum system to effectively reach equilibrium. The discussion also includes a practical definition of equilibration that takes our measurement capabilities into account. Finally, the nature of the equilibrium state is also discussed, with a focus on initial state independence, which relates to the important question of when the equilibrium state is a Gibbs state.
Quantum mechanics, which describes the behaviour of matter and energy on very small scales, is one of the most successful theories in the history of science. Einstein's theory of special relativity, which describes the relationship between space and time, is likewise a highly successful and widely accepted theory. And yet there is a well-documented tension between the two theories, to the extent that it is still not clear that the two can ever be reconciled. This thesis is concerned with furthering the current understanding of the relationship between quantum mechanics and special relativity. In the first part of the thesis we study the behaviour of quantum information in relativistic spacetime. The field of quantum information arose from the realisation that quantum information has a number of crucial properties that distinguish it from classical information, such as the no-cloning property, quantum contextuality, and quantum discord. More recently, it has been realised that placing quantum information under relativistic constraints leads to the emergence of further unique features which are not exhibited by either non-relativistic quantum information or relativistic classical information; as part of this ongoing research programme we develop a new relativistic quantum `paradox' which puts pressure on conventional views about the spatiotemporal persistence of quantum states over time. We then study a new set of relativistic quantum protocols which involve the distribution of entangled states over spacetime, defining one task involving the distribution of the two halves of a known entangled state, and another task involving the distribution of the two halves of an unknown entangled state. The second part of the thesis deals with relativistic quantum cryptography, a field which first began attracting serious attention when it was realised that a cryptographic task known as `bit commitment,' can be implemented with perfect security under relativistic constraints. This result was highly significant, since it is provably impossible to implement bit commitment with perfect security in a purely classical or purely quantum context, and hence bit commitment is an ideal starting point for probing the power of relativistic quantum cryptography. In this thesis we propose several new relativistic quantum bit commitment protocols which have notable advantages over previously known protocols. We then move to a related task, a generalization of zero-knowledge proving which we refer to as knowledge-concealing evidencing of knowledge of a quantum state; we prove no-go theorems concerning the possibility of implementing this task with perfect security, and then set out a relativistic protocol for the task which is asymptotically secure as the dimension of the state in question becomes large. These results have interesting foundational significance above and beyond their applications in the field of cryptography, providing a new perspective on the connections between knowledge, realism and quantum states.
Measurements play a crucial role in quantum mechanics because they serve two fundamental purposes. First, they enable the extraction of information about a quantum system; second, they give rise to new physical effects that do not appear in classical physics, one important example being the measurement-induced state update, where the act of measurement alters the system’s state. For these reasons, the mathematical description of measurements plays a primary role in the foundational postulates of the theory. However, the essence of what happens during a measurement remains poorly understood, even within the relatively easy realm of non-relativistic quantum theory. The situation becomes even less transparent in the relativistic setting, where the requirement to preserve causality and prevent faster-than-light signalling introduces both formal and interpretational challenges, complicating the very mathematical formulation of a measurement theory. As for quantum gravity, the idea of applying quantum measurements within or onto a quantum spacetime remains virtually unexplored, presenting a vast and uncharted frontier for theoretical inquiry. Building on recent advances across several areas at the intersection of quantum foundations, quantum information theory, relativistic quantum theory, and quantum gravity, the last few years saw a surge of interest in formulating a consistent framework for describing local measurements in a relativistic context. The main results of this endeavour are two distinct proposals for a relativistic measurement postulate for quantum field theory in generic spacetimes. While these two proposals are still being understood, and a growing research community is working to clarify their implications and potential interconnections, the field of relativistic quantum measurements has now reached a level of maturity that allows us to ask both formal and interpretive questions on a solid footing. All this only applies to the regime in which quantum field theory in curved spacetime remains valid: nothing can yet be said about measurements at the Planck energy, where quantum gravitational effects are expected to become important. This thesis presents an introduction to the theory of measurements in non-relativistic and relativistic quantum mechanics, along with four original research articles that explore aspects of measurements in both fields, as well as the potential extension of measurement theory to the domain of quantum gravity. The first article examines the limitations of realistic measurement apparatuses in non-relativistic quantum theory, establishing reasonable bounds for the precision that can be achieved with a given apparatus composed of many quantum components. The second article explores how to extract information from a newly introduced class of quantum systems, defined as those that cannot be replicated and thus fall outside the usual scope of quantum mechanics. Then, it applies the methodology to quantum field theory, demonstrating that classical results, such as the Unruh effect, can still be observed in this setting. The third article discusses detector-based measurements in quantum field theory. This is a recently proposed method that extends the non-relativistic measurement postulate to the relativistic regime by means of Unruh-de Witt detectors. The article highlights some conceptual and mathematical issues with the framework and proposes a solution using algebraic quantum field theory. The outcome is a self-consistent scheme for introducing local measurements in quantum field theory via measuring detectors. Finally, the fourth article uses the measurement scheme above to study how what we learned in the past years about local measurements can be extended to quantum gravity. This is achieved through the holographic principle realised in terms of the AdS-CFT correspondence. In conclusion, this thesis contributes to the development of both non-relativistic and relativistic measurement theories and introduces ideas and questions related to the role of measurement in quantum gravity. Though preliminary, this work opens a pathway toward studying measuring observers in quantum gravity.
Time in relativity theory has a status different from that adopted by standard quantum mechanics, where time is considered as a parameter measured with reference to an external absolute Newtonian frame. This status strongly restricts its role in the dynamics of systems and hinders any formulation to merge quantum mechanics with general relativity, specifically when considering quantum gravity. To overcome those limitations, several authors tried to construct an operator which is conjugate to the Hamiltonian of quantum systems implementing some essential features of the relativistic time. These formulations use the concept of internal or intrinsic time instead of the universal coordinate time used in textbooks. Furthermore, recently it is remarked that the consideration of time with relativistic features could enhance the analysis techniques in quantum information processing and have an impact on its status in causal orders and causal structures of quantum information. The role of clocks, their accuracy and stability has become an important issue in quantum information processing. This article present a substantiative review of recent works which reflect the possibility of utilizing quantum time, measured by quantum clock devised according to Page-Wootters scheme, to stand as a resource for quantum information processing.