Quantum resource theories provide a structured and elegant framework for quantifying quantum resources. While state-based resource theories have been extensively studied, their measurement-based resource theories remain relatively underexplored. In practical scenarios where a quantum state or a set of measurements is only partially known, conventional resource measures often fall short in capturing the resource content. In such cases, ε-measures offer a robust alternative, making them particularly valuable. In this work, we investigate the quantification of measurement-based resources using distance-based measures, followed by a detailed analysis of the mathematical properties of ε-measures. We also extend our analysis by exploring the connections between ε-measures and some key quantities relevant to resource manipulation tasks. Importantly, the analysis of resources based on sets of measurements are tedious compared to that of single measurements as the former allows more general transformations such as controlled implementation. Yet our framework applies not only to resources associated with individual measurements but also to those arising from sets of measurements. In short, our analysis is applicable to existing resource theories of measurements and has the potential to be useful for all resource theories of measurements that are yet to be developed.
Quantum resource theories provide a structured and elegant framework for quantifying quantum resources. While state-based resource theories have been extensively studied, their measurement-based resource theories remain relatively underexplored. In practical scenarios where a quantum state or a set of measurements is only partially known, conventional resource measures often fall short in capturing the resource content. In such cases, ε-measures offer a robust alternative, making them particularly valuable. In this work, we investigate the quantification of measurement-based resources using distance-based measures, followed by a detailed analysis of the mathematical properties of ε-measures. We also extend our analysis by exploring the connections between ε-measures and some key quantities relevant to resource manipulation tasks. Importantly, the analysis of resources based on sets of measurements are tedious compared to that of single measurements as the former allows more general transformations such as controlled implementation. Yet our framework applies not only to resources associated with individual measurements but also to those arising from sets of measurements. In short, our analysis is applicable to existing resource theories of measurements and has the potential to be useful for all resource theories of measurements that are yet to be developed.
The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.
Chapter 10 delves into the quantification of quantum resources, an essential aspect of quantum resource theories that determines the value of quantum states for specific applications. It begins by defining resource measures and investigating their fundamental properties such as monotonicity under free operations and convexity. The chapter discusses distance-based resource measures, which quantify how far a given quantum state is from the set of free states. Such measures often utilize divergences and metrics explored in earlier chapters. Techniques to compute the relative entropy of a resource are also covered. To refine resource measures, the chapter introduces the concept of smoothing, which considers small deviations from the ideal state to make the measures more robust against perturbations. This approach is crucial in single-shot scenarios where finite resources are available. Furthermore, the chapter examines resource monotones and support functions, offering a comprehensive framework for the theoretical and practical assessment of quantum resources.
The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.
The diverse range of resources which underlie the utility of quantum states in practical tasks motivates the development of universally applicable methods to measure and compare resources of different types. However, many of such approaches were hitherto limited to the finite-dimensional setting or were not connected with operational tasks. We overcome this by introducing a general method of quantifying resources for continuous-variable quantum systems based on the robustness measure, applicable to a plethora of physically relevant resources such as optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. We demonstrate in particular that the measure has a direct operational interpretation as the advantage enabled by a given state in a class of channel discrimination tasks. We show that the robustness constitutes a well-behaved, bona fide resource quantifier in any convex resource theory, contrary to a related negativity-based measure known as the standard robustness. Furthermore, we show the robustness to be directly observable-it can be computed as the expectation value of a single witness operator-and establish general methods for evaluating the measure. Explicitly applying our results to the relevant resources, we demonstrate the exact computability of the robustness for several classes of states.
The quantification of state-based quantum ``resources'' such as entanglement, coherence, and nonstabilizer states lies at the heart of quantum science and technology, providing potential advantages over classical methods. In a realistic scenario, due to the imperfections and uncertainties in physical devices, we are unable to perfectly prepare or detect the true quantum states. Consequently, it is necessary to study the quantification of quantum resources under such circumstances. In this work, by focusing on the state-based quantum resource theory, we introduce a family of resource measures called $\ensuremath{\epsilon}$ measure that relies on a precision parameter to address this issue. This family of resource measures inherits the fundamental properties of the original resource measure, such as weak monotonicity, convexity, monogamy, and so forth. Furthermore, the $\ensuremath{\epsilon}$ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the $\ensuremath{\epsilon}$ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the $\ensuremath{\epsilon}$ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the $\ensuremath{\epsilon}$ measure of the resource rank. Finally, we outline how this work can be extended to channel-based resources among others.
Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. Here we introduce methods for the quantification of resources in general probabilistic theories (GPTs), focusing in particular on the technical issues associated with infinite-dimensional state spaces. We define a universal resource quantifier based on the robustness measure, and show it to admit a direct operational meaning: in any GPT, it quantifies the advantage that a given resource state enables in channel discrimination tasks over all resourceless states. We show that the robustness acts as a faithful and strongly monotonic measure in any resource theory described by a convex and closed set of free states, and can be computed through a convex conic optimization problem. Specializing to continuous-variable quantum mechanics, we obtain additional bounds and relations, allowing an efficient computation of the measure and comparison with other monotones. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. In particular, we establish exact expressions for various classes of states, including Fock states and squeezed states in the resource theory of nonclassicality and general pure states in the resource theory of entanglement, as well as tight bounds applicable in general cases.