Quantum physics currently provides our deepest description of nature. To understand precisely how it differs from classical physics is not only of foundational interest, but also important to advance technology: the advent of quantum information has led to the insight that certain quantum phenomena, e.g., entanglement and coherence, are resources which enable us to solve practical tasks that are unsolvable within the realm of classical physics. But how can we quantify quantum resources and how do we decide which resource is relevant for a specific operational advantage in quantum cryptography, quantum computation, or quantum metrology? How can we employ these resources optimally? Addressing these questions is the motivation for the development of quantum resource theories, a mathematically rigorous framework that allows us to investigate a specific quantum phenomenon as a resource. Formally, such theories emerge from restrictions which are imposed in addition to the laws of quantum mechanics. A prominent example is the restriction to local operations and classical communication, which singles out entanglement as a resource. Typically, such theories describe the value of quantum states, i.e., static resources. An important question in such theories concerns their structure as determined by the restriction under consideration, which is analyzed for coherence theory on qubits in this thesis. Only recently, so-called dynamical resource theories that quantify the value of operations were proposed, which are advantageous for several reasons: they include resource theories of states as special cases, allow us to quantify additional quantum properties such as the detection or usage of static resources, which will be a main topic of this thesis, and are more natural from an operational perspective. But are different types of dynamical resources present in quantum operations related? For the case of dynamical coherence and entanglement, it will be shown that there is not only a quantitative, but also a qualitative connection.
Quantum physics currently provides our deepest description of nature. To understand precisely how it differs from classical physics is not only of foundational interest, but also important to advance technology: the advent of quantum information has led to the insight that certain quantum phenomena, e.g., entanglement and coherence, are resources which enable us to solve practical tasks that are unsolvable within the realm of classical physics. But how can we quantify quantum resources and how do we decide which resource is relevant for a specific operational advantage in quantum cryptography, quantum computation, or quantum metrology? How can we employ these resources optimally? Addressing these questions is the motivation for the development of quantum resource theories, a mathematically rigorous framework that allows us to investigate a specific quantum phenomenon as a resource. Formally, such theories emerge from restrictions which are imposed in addition to the laws of quantum mechanics. A prominent example is the restriction to local operations and classical communication, which singles out entanglement as a resource. Typically, such theories describe the value of quantum states, i.e., static resources. An important question in such theories concerns their structure as determined by the restriction under consideration, which is analyzed for coherence theory on qubits in this thesis. Only recently, so-called dynamical resource theories that quantify the value of operations were proposed, which are advantageous for several reasons: they include resource theories of states as special cases, allow us to quantify additional quantum properties such as the detection or usage of static resources, which will be a main topic of this thesis, and are more natural from an operational perspective. But are different types of dynamical resources present in quantum operations related? For the case of dynamical coherence and entanglement, it will be shown that there is not only a quantitative, but also a qualitative connection.
It was recently realized that many quantum phenomena, such as entanglement or non-locality, can be expressed in the framework of quantum resource theories (QRTs) as resources that can be expended to perform tasks. This approach has revolutionized our understanding of these phenomena and has provided a framework to rigorously characterize them, quantify them, and study the question of convertibility of resources within them. Moreover, this framework has enabled the discovery of similar structures common to many QRTs, and thus offered a better and deeper understanding of the phenomena they describe. Most of the research that has been done so far in the area of QRTs has been concerned with resource theories of states, that is, where the objects in the QRT are static resources that are divided among free and resource states. Free operations, defined as quantum channels that do not generate resources, are used to transform these states. However, this concept can be generalized to include dynamical resources, or non-free operations. We call the resource theories that arise from this, QRTs of quantum processes. This thesis introduces different QRTs of quantum processes that we call QRTs of non-projectiveness. More precisely, these are QRTs where projective measurements are considered free. There are different resource theories that arise based on exactly which additional operations are allowed and whether quantum measurements are described with a classical output, a quantum output, or both. This research looks into four such QRTs of non-projectiveness.
In the works leading to this thesis the aim was to give a clear operational interpretation of different aspects of non-classicality. Doing so, enables to see the different properties as different aspects of the same underlying question, namely whether there is a classical explanation of the measurements at hand. If one incorporates the additional assumptions in the setting (such as locality by the separation of the two parties in entanglement theory), the question of whether some system is classical, boils down to ask whether one can simulate everything one can possibly observe by classical variables governed by a stochastic process. This new perspective allows to characterise dynamic properties of non-classicality, and also to connect them to more static types of non-classicality such as entanglement. Indeed we unify entanglement, discord and coherence from one single perspective. We do this by restricting our setting such that the operations have a clear classical interpretation and hence incorporating natural restrictions of classical systems, if the initial state of the system is not entangled does not have discord or does not have coherence. If the initial state of the system is different, we see the magic of quantum mechanics happening again. For instance we show for one specific quantum algorithm, which is thought to be better than any classical algorithm for its task, that the precision of the algorithm scales with the coherence present at its start. In other words, it is these resources (coherence, discord and entanglement) that make the magic possible and with our perspective we can interpret the role of the different resources and analyse how they are linked to each other. This does not give an explanation of the magic of quantum mechanics, but it does make the mechanisms more transparent allowing for new explanations and there are some examples already in this thesis. Finally we also look at the dual of when states are non-classical, asking, and answering, in a specific setting, what makes measurements non-classical.
This book delves into the burgeoning field of quantum resource theories, a novel and vibrant area of research within quantum information science that seeks to unify diverse quantum phenomena under a single framework. By recognizing various attributes of physical systems as "resources," this approach offers a fresh perspective on quantum phenomena, transforming our understanding and application of concepts such as quantum entanglement, coherence, and more. With a focus on the pedagogical, the book aims to equip readers with the advanced mathematical tools and physical principles needed to navigate and contribute to this rapidly evolving field. It covers a wide range of topics, from the foundational aspects of quantum mechanics and quantum information to detailed explorations of specific resource theories, including entanglement, asymmetry, and thermodynamics. Through rigorous mathematical exposition and a unique axiomatic approach, the book provides deep insights into the operational and conceptual frameworks that underpin quantum resource theories, making it an invaluable resource for graduate students, early-career researchers, and anyone interested in the cutting-edge developments in quantum information science.
Quantum theory provides us with a mathematical framework to understand nature at the microscopic level.It explains phenomena that classical physics could not account for.A deeper understanding of these non-classical phenomena (e.g., entanglement and quantum coherence) is not only of a foundational interest, but also paves the way towards potential technological advancements.Developments in the field of quantum information showed that such quantum features help us to perform operational tasks which would be impossible in the framework of classical physics.Therefore, these quantum phenomena can be viewed as resources which help us to go beyond the restrictions imposed by classical physics.Inspired by this, the mathematical framework of quantum resource theories was developed.Each resource theory singles out a particular quantum resource by setting the states which contain no resource as free states.Furthermore, the set of free operations are defined such that they do not generate resource states out of free states.An important problem in any resource theory is to understand how different resources transform into each other under the action of free operations.This problem of state transformation will be the main focus of this thesis.We will start by studying the notion of deterministic transformations, where one aims to achieve a target state without a chance of failure.We will explore the connection between deterministic transformations and the problem of quantification of resources.We will then go further and concentrate on transformations which allow for a probability of failure, and study the fundamental restrictions imposed on the achievable error and the probability of success.Finally we will explore the idea of catalytic transformations, where we allow for an ancillary quantum system which increases the transformation power while remaining invariant in the process.For the resource theory of entanglement, we will give a full solution for this problem when the target state is pure and the initial state is distillable.Finally, we discuss potential applications of catalysis to the problem of sending quantum information through noisy channels.
A central problem in quantum resource theory is to give operational meaning to quantum resources that can provide clear advantages in certain physical tasks compared to the convex set of resource-free states. We propose to extend this basic principle by defining the relative superiority of resources over a specific convex set of resource states, also provide a relative advantage in physical tasks based on this extended principle. This allows the generalized robustness measure to quantify the relative maximal advantage due to a given resource state over a specific convex set of resource states in the subchannel discrimination, thereby showing that the operational interpretation of resource measures also holds in a relative perspective. In addition, we offer a new framework for defining the deficiency of a given state in physical tasks compared to the set of maximum resource states. The geometric measure we provide satisfies the conditions of the framework for quantum coherence and entanglement, and it accurately quantifies the minimal disadvantage due to a given state compared to maximumresourcestates inthe subchannel discrimination in certain situations. These two extensions and new interpretations expand the scope of quantum resource theories and provide a more comprehensive operational interpretation.
Quantum resource theories provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing. State conversion can be implemented by free operations. The partial trace and the incoherent operation, similarly to the local operation and classical communication, are all free operations in the corresponding quantum resource theory. This work mainly studies the limitations of these two types of free operations: (1) Masking information is a protocol that encodes quantum information into a bipartite correlated state while the information is completely unknown to local systems. While the no-masking theorem turns out that it is not possible to determinately mask arbitrary quantum states, quantum information in certain restricted sets of states can be masked. We consider the new regime of probabilistic and approximate masking quantum information. Here, the goal is to mask a set of quantum states with maximal probability, allowing for a small error in the masking. And we provide limits on both the fidelity and the probability of masking, and derive limitations on the masking of quantum information by providing bounds on the maximal masking fidelity. (2) From quantum sensors to quantum computers, many technologies require quantum resources. The properties of such states are coherent or correlated in a way that would not be possible in classical physics. Ideally, for technology applications, these states should be in the so-called maximally resource state, one in which all possible measures of quantum resource are maximized. In a quantum system with noise, an important question is whether there exists maximal resource states. This work answered this important quantum question with a negative answer, i.e., there are no maximal resource states in general. In order to do so, this work considers the case of rank-2 states and shows that for particular values of the eigenvalues, there exists no state that can be transformed to all other states under free operations. This in particular implies that in these cases the state that maximizes a given resource measure among all states depends on the choice of resource measure, i.e., it cannot be that the aforementioned family of states maximizes all resource measures.