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arXiv:2103.05665 · Physical Review A

Asymptotically consistent measures of general quantum resources: Discord, non-Markovianity, and non-Gaussianity

Kohdai Kuroiwa, Hayata Yamasaki

Published Aug 2, 2021• 9 citations
arXivHTMLPDF

Abstract

Quantum resource theories provide a unified framework to quantitatively analyze inherent quantum properties as resources for quantum information processing. So as to investigate the best way for quantifying resources, desirable axioms for resource quantification have been extensively studied through axiomatic approaches. However, a conventional way of resource quantification by resource measures with such desired axioms may contradict rates of asymptotic transformation between resourceful quantum states due to an approximation in the transformation. In this paper we establish an alternative axiom, asymptotic consistency of resource measures, and we investigate asymptotically consistent resource measures, which quantify resources without contradicting the rates of the asymptotic resource transformation. We prove that relative entropic measures are consistent with the rates for a broad class of resources, i.e., all convex finite-dimensional resources, e.g., entanglement, coherence, and magic, and even some nonconvex or infinite-dimensional resources such as quantum discord, non-Markovianity, and non-Gaussianity. These results show that consistent resource measures are widely applicable to the quantitative analysis of various inherent quantum-mechanical properties.

DOI
10.1103/physreva.104.l020401

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Quantum resource theories provide a unified framework to quantitatively analyze inherent quantum properties as resources for quantum information processing. So as to investigate the best way for quantifying resources, desirable axioms for resource quantification have been extensively studied through axiomatic approaches. However, a conventional way of resource quantification by resource measures with such desired axioms may contradict rates of asymptotic transformation between resourceful quantum states due to an approximation in the transformation. In this paper we establish an alternative axiom, asymptotic consistency of resource measures, and we investigate asymptotically consistent resource measures, which quantify resources without contradicting the rates of the asymptotic resource transformation. We prove that relative entropic measures are consistent with the rates for a broad class of resources, i.e., all convex finite-dimensional resources, e.g., entanglement, coherence, and magic, and even some nonconvex or infinite-dimensional resources such as quantum discord, non-Markovianity, and non-Gaussianity. These results show that consistent resource measures are widely applicable to the quantitative analysis of various inherent quantum-mechanical properties.
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.
We present a unified axiomatic approach to contextuality and nonlocality based on the fact that both are resource theories. In those theories, the main objects are consistent boxes, which can be transformed by certain operations to achieve certain tasks. The amount of resource is quantified by appropriate measures of the resource. Following a recent paper [J. I. de Vicente, J. Phys. A: Math. Theor. 47, 424017 (2014)], and recent development of abstract approach to resource theories, such as entanglement theory, we propose axioms and welcome properties for operations and measures of resources. As one of the axioms of the measure we propose the asymptotic continuity: the measure should not differ on boxes that are close to each other by more than the distance with a factor depending logarithmically on the dimension of the boxes. We prove that relative entropy of contextuality is asymptotically continuous. Considering another concept from entanglement theory (the convex roof of a measure), we prove that for some nonlocal and contextual polytopes, the relative entropy of a resource is upper bounded up to a constant factor by the cost of the resource. Finally, we prove that providing a measure XXX of resource does not increase under allowed class of operations such as, e.g., wirings, the maximal distillable resource which can be obtained by these operations is bounded from above by the value of XXX up to a constant factor. We show explicitly which axioms are used in the proofs of presented results, so that analogous results may remain true in other resource theories with analogous axioms. We also make use of the known distillation protocol of bipartite nonlocality to show how contextual resources can be distilled.
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess intractable mathematical structure to analyze, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such intractable mathematical structure. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable to mathematically intractable but physically motivated quantum resources in a unified way.
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.
Abstract Robustness measures are increasingly prominent resource quantifiers that have been introduced for quantum resource theories such as entanglement and coherence. Despite the generality of these measures, their usefulness is hindered by the fact that some of their mathematical properties remain unclear, especially when the set of resource-free states is non-convex. In this paper, we investigate continuity properties of different robustness functions. We show that their continuity depends on the shape of the set of free states. In particular, we demonstrate that in many cases, star-convexity is sufficient for Lipschitz-continuity of the robustness, and we provide specific examples of sets leading to non-continuous measures. Finally, we illustrate the applicability of our results by defining a robustness of teleportability and of quantum discord.
We establish the connection between non-Markovianity and negative entropy production rate for various classes of quantum operations. Generalizing the definition of entropy production rate for non-equilibrium case we connect it with the rate of change of free energy of the system and establish complementary relations between non-Markovianity and maximum loss of free energy. We naturally conclude that non-Markovianity, in terms of divisibility breaking is a necessary resource for backflow of other resources like purity or free energy, under corresponding allowed operations. Based on this we propose a resource theory of non-Markovianity by constructing the free operations, free states and a generalized measure of non-Markovianity. The framework satisfies the basic properties of a consistent resource theory. The proposed resource quantifier is lower bounded by the optimization free Rivas-Huelga-Plenio (RHP) measure of non-Markovianity.