Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Epsilon measures of state-based quantum resource theory

Yu Luo, Fanxu Meng, Youle Wang

Published May 8, 2024• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.109.052413

Related papers

Framework for resource quantification in infinite-dimensional general probabilistic theories

Ludovico Lami, Bartosz Regula, Ryuji Takagi +1 more

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.

arXiv:2009.11313•Physical Review A•Mar 18, 2021•34 citations

Distance-based resource quantification for sets of quantum measurements

Lucas Tendick, Martin Kliesch, Hermann Kampermann +1 more

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.

Quantum•May 15, 2023•7 citations

Distance-based resource quantification for sets of quantum measurements

Lucas Tendick, Martin Kliesch, Hermann Kampermann +1 more

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.

arXiv:2205.08546•May 17, 2022•1 citation

Quantification of Quantum Resources

Gilad Gour

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.

Cambridge University Press eBooks•Apr 10, 2025•0 citations

Convex geometry of quantum resource quantification

Bartosz Regula

Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.

arXiv:1707.06298•Journal of Physics A Mathematical and Theoretical•Oct 4, 2017•100 citations

Epsilon-measure of coherence

Zhengjun Xi, Shanshan Yuwen

To estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in actually prepared state, we discuss the smooth version of coherence quantifiers which depend on a precision parameter $\epsilon$. In this paper, we call it the $\epsilon$-measure of coherence, it can be interpreted as the minimal coherence guaranteed to present in an $\epsilon$ ball around given quantum state. We find that the $\epsilon$-measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the $\epsilon$-measure of coherence is continuous even if the original coherence quantifier is not. In particular, we study some properties of the $\epsilon$-measure of distance-based coherence quantifier.

•Oct 9, 2018•0 citations
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ 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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ 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.
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 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.
Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.
To estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in actually prepared state, we discuss the smooth version of coherence quantifiers which depend on a precision parameter ϵ\epsilonϵ. In this paper, we call it the ϵ\epsilonϵ-measure of coherence, it can be interpreted as the minimal coherence guaranteed to present in an ϵ\epsilonϵ ball around given quantum state. We find that the ϵ\epsilonϵ-measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the ϵ\epsilonϵ-measure of coherence is continuous even if the original coherence quantifier is not. In particular, we study some properties of the ϵ\epsilonϵ-measure of distance-based coherence quantifier.