Abstract Resource theories are broad frameworks that capture how useful objects are in performing specific tasks. In this paper we devise a formal resource theory quantum measurements, focusing on the ability of a measurement to acquire information. The objects of the theory are equivalence classes of positive operator-valued measures, and the free transformations are changes to a measurement device that can only deteriorate its ability to report information about a physical system. We show that catalysis and purification, protocols that are possible in other resource theories, are impossible in our resource theory for quantum measurements. Standard measures of information gain are shown to be resource monotones, and the resource theory is applied to the task of quantum state discrimination.
Abstract Resource theories are broad frameworks that capture how useful objects are in performing specific tasks. In this paper we devise a formal resource theory quantum measurements, focusing on the ability of a measurement to acquire information. The objects of the theory are equivalence classes of positive operator-valued measures, and the free transformations are changes to a measurement device that can only deteriorate its ability to report information about a physical system. We show that catalysis and purification, protocols that are possible in other resource theories, are impossible in our resource theory for quantum measurements. Standard measures of information gain are shown to be resource monotones, and the resource theory is applied to the task of quantum state discrimination.
For any resource theory it is essential to identify tasks for which resource objects offer advantage over free objects. We show that this identification can always be accomplished for resource theories of quantum measurements in which free objects form a convex subset of measurements on a given Hilbert space. To this aim we prove that every resourceful measurement offers advantage for some quantum state discrimination task. Moreover, we give an operational interpretation of robustness, which quantifies the minimal amount of noise that must be added to a measurement to make it free. Specifically, we show that this geometric quantity is related to the maximal relative advantage that a resourceful measurement offers in a class of minimal-error state discrimination (MESD) problems. Finally, we apply our results to two classes of free measurements: incoherent measurements (measurements that are diagonal in the fixed basis) and separable measurements (measurements whose effects are separable operators). For both of these scenarios we find, in the asymptotic setting in which the dimension or the number of particles increase to infinity, the maximal relative advantage that resourceful measurements offer for state discrimination tasks.
Resource theories constitute a powerful theoretical framework and a tool that captures, in an abstract structure, pragmatic aspects of the most varied theories and processes.For physical theories, while this framework deals directly with questions about the concrete possibilities of carrying out tasks and processes, resource theories also make it possible to recast these already established theories on a new language, providing not only new perspectives on the potential of physical phenomena as valuable resources for technological development, for example, but they also provide insights into the very foundations of these theories.In this work, we will investigate some properties of a resource theory for quantum contextuality, an essential characteristic of quantum phenomena that ensures the impossibility of interpreting the results of quantum measurements as revealing properties that are independent of the set of measurements being made.We will present the resource theory to be studied and investigate certain global properties of this theory using tools and methods that, although already developed and studied by the community in other resource theories, had not yet been used to characterize resource theories of contextuality.In particular, we will use the so called cost and yield monotones, making use of their power in the study of resource theories for nonlocality, in an attempt to extend these results to this more general class of phenomena, contextuality.
Coherence is a cornerstone of quantum theory and a prerequisite for the advantage of quantum technologies. In recent work, the notion of coherence with respect to a general quantum measurement, i.e., positive operator-valued measure (POVM), was introduced and embedded into a resource-theoretic framework that generalizes the standard resource theory of coherence. In particular, POVM-incoherent (free) states and operations were established. In this work, we explore features of this framework which arise due to the rich structure of POVMs compared to projective measurements. Moreover, we introduce a rigorous, probabilisitic framework for POVM-based coherence measures and free operations. This leads to the introduction of strongly monotonic resource measures that neatly generalize well-known standard coherence measures. Finally, we show that the relative entropy of POVM coherence is equal to the cryptographic randomness gain, providing an important operational meaning to the concept of coherence with respect to a general measurement.
Resource theories constitute a powerful theoretical framework and a tool that captures, in an abstract structure, pragmatic aspects of the most varied theories and processes. For physical theories, while this framework deals directly with questions about the concrete possibilities of carrying out tasks and processes, resource theories also make it possible to recast these already established theories on a new language, providing not only new perspectives on the potential of physical phenomena as valuable resources for technological development, for example, but they also provide insights into the very foundations of these theories. In this work, we will investigate some properties of a resource theory for quantum contextuality, an essential characteristic of quantum phenomena that ensures the impossibility of interpreting the results of quantum measurements as revealing properties that are independent of the set of measurements being made. We will present the resource theory to be studied and investigate certain global properties of this theory using tools and methods that, although already developed and studied by the community in other resource theories, had not yet been used to characterize resource theories of contextuality. In particular, we will use the so called cost and yield monotones, making use of their power in the study of resource theories for nonlocality, in an attempt to extend these results to this more general class of phenomena, contextuality.
Quantum resources are certain features of the quantum world that provide advantages in certain information-theoretic, thermodynamic, or other useful operational tasks that are outside the realm of what classical theories can achieve. Quantum resource theories provide us with an elegant framework for studying these resources quantitatively and rigorously. While numerous state-based quantum resource theories have already been investigated, and to some extent, measurement-based resource theories have also been explored, instrument-based resource theories remain largely unexplored, with only a few notable exceptions. As quantum instruments are devices that provide both the classical outcomes of induced measurements and the post-measurement quantum states, they are quite important, especially for scenarios where multiple parties sequentially act on a quantum system. In this work, we study several instrument-based resource theories, namely (1) the resource theory of information preservability, (2) the resource theory of (strong) entanglement preservability, (3) the resource theory of (strong) incompatibility preservability, (4) the resource theory of traditional incompatibility, and (5) the resource theory of parallel incompatibility. Furthermore, we outline the hierarchies of these instrument-based resources and provide measures to quantify them. We then also established a relationship between our resource measure and the advantage in an information-theoretic task. In short, we provide a detailed framework for a wide variety of instrument-based quantum resource theories.
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.