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.
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.
Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
Quantum catalysis, the ability to enable previously impossible transformations by using auxiliary systems without degrading them, has emerged as a powerful tool in various resource theories. Although catalytically enabled state transformations have been formally characterized by the monotonic behavior of entropic quantifiers (e.g., the von Neumann entropy or nonequilibrium free energy), such characterizations often rely on unphysical assumptions, namely, the ability of using catalysts of infinitely large dimension. This approach offers very limited insights into the practical significance of using catalysis for quantum information processing. Here, we address this problem across a broad class of quantum resource theories. Leveraging quantum information tools beyond the asymptotic regime, we establish sufficient conditions for the existence of catalytic transformations with finite-size catalysts. We further unveil connections between finite-size catalysis and multicopy transformations. Notably, we discover a phenomenon of catalytic resonance: by carefully tailoring the catalysts's state, one can drastically reduce the required dimension of the catalyst, thus enabling efficient catalytic transformations with minimal resources. Finally, we illustrate our findings with examples from the resource theories of entanglement and thermodynamics, as well in the context of catalytic unitary transformations.
Entanglement quantification aims to assess the value of quantum states for quantum information processing tasks. A closely related problem is state convertibility, asking whether two remote parties can convert a shared quantum state into another one without exchanging quantum particles. Here, we explore this connection for quantum entanglement and for general quantum resource theories. For any quantum resource theory which contains resource-free pure states, we show that there does not exist a finite set of resource monotones which completely determines all state transformations. We discuss how these limitations can be surpassed, if discontinuous or infinite sets of monotones are considered, or by using quantum catalysis. We also discuss the structure of theories which are described by a single resource monotone and show equivalence with totally ordered resource theories. These are theories where a free transformation exists for any pair of quantum states. We show that totally ordered theories allow for free transformations between all pure states. For single-qubit systems, we provide a full characterization of state transformations for any totally ordered resource theory.
Quantum resource theories allow us to quantify a useful quantum phenomenon, to develop new protocols for its detection and determine the exact processes that maximize its use for practical tasks. These theories aim at transforming physical phenomena, such as entanglement and quantum coherence, into useful properties for the execution of concrete tasks related to quantum information. In this thesis, we focus on the resource theories of entanglement, discord-like quantum correlations, and quantum coherence, the most intriguing quantum phenomena exploited so far in quantum information theory. We begin by presenting in detail the theoretical tools of these quantum resources, focusing on the most remarkable techniques and computational problems. In this sense, we discuss several mathematical methods that solve some problems related to their quantifications, and some analytical results for bipartite quantum systems are given. We also examine the intrinsic connections between these quantum resources by extracting the links that unite the corresponding measures. In contrast, the revolution of quantum technology has led to a growing interest in quantum metrology, and quantum entanglement has been employed to overcome the classical limit in several quantum estimation protocols. In this work, we analyze the role of quantum correlations beyond entanglement in improving the accuracy of an unknown parameter. According to our results, correlations can be captured using quantum Fisher information, and quantum discord correlations can be exploited to ensure the accuracy of phase estimation protocols. This thesis includes also the contributions on the dynamics of these quantum resources in various models of open quantum systems. Among our objectives, is to study the effects of the environment on these quantum resources and to obtain techniques to protect them from the effects of intrinsic decoherence.
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 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.