General Resource Theories Research Articles

Every Quantum Helps: Operational Advantage of Quantum Resources beyond Convexity.

Identifying what quantum-mechanical properties are useful to untap a superior performance in quantum technologies is a pivotal question. Quantum resource theories provide a unified framework to analyze and understand such properties, as successfully demonstrated for entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a nonconvex set of free states and their interpretation has so far remained elusive. Here we address the fundamental question of the usefulness of quantum resources without convexity assumption, by providing two operational interpretations of the generalized robustness measure in general resource theories. First, we characterize the generalized robustness in terms of a nonlinear resource witness and reveal that any state is more advantageous than a free one in some multicopy channel discrimination task. Next, we consider a scenario where a theory is characterized by multiple constraints and show that the generalized robustness coincides with the worst-case advantage in a single-copy channel discrimination setting. Based on these characterizations, we conclude that every quantum resource state shows a qualitative and quantitative advantage in discrimination problems in a general resource theory even without any specification on the structure of the free states.

Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence.

Quantum resource manipulation may include an ancillary state called a catalyst, which aids the transformation while restoring its original form at the end, and characterizing the enhancement enabled by catalysts is essential to reveal the ultimate manipulability of the precious resource quantity of interest. Here, we show that allowing correlation among multiple catalysts can offer arbitrary power in the manipulation of quantum coherence. We prove that any state transformation can be accomplished with an arbitrarily small error by covariant operations with catalysts that may create a correlation within them while keeping their marginal states intact. This presents a new type of embezzlement-like phenomenon, in which the resource embezzlement is attributed to the correlation generated among multiple catalysts. We extend our analysis to general resource theories and provide conditions for feasible transformations assisted by catalysts that involve correlation, putting a severe restriction on other quantum resources for showing this anomalous enhancement, as well as characterizing achievable transformations in relation to their asymptotic state transformations. Our results provide not only a general overview of the power of correlation in catalysts but also a step toward the complete characterization of the resource transformability in quantum thermodynamics with correlated catalysts.

One-Shot Manipulation of Dynamical Quantum Resources.

We develop a unified framework to characterize one-shot transformations of dynamical quantum resources in terms of resource quantifiers, establishing universal conditions for exact and approximate transformations in general resource theories. Our framework encompasses all dynamical resources represented as quantum channels, including those with a specific structure-such as boxes, assemblages, and measurements-thus immediately applying in a vast range of physical settings. For the particularly important manipulation tasks of distillation and dilution, we show that our conditions become necessary and sufficient for broad classes of important theories, enabling an exact characterization of these tasks and establishing a precise connection between operational problems and resource monotones based on entropic divergences. We exemplify our results by considering explicit applications to quantum communication, where we obtain exact expressions for one-shot quantum capacity and simulation cost assisted by no-signaling, separability-preserving, and positive partial transpose-preserving codes; as well as to nonlocality, contextuality, and measurement incompatibility, where we present operational applications of a number of relevant resource measures.

Simple bounds for one-shot pure-state distillation in general resource theories

We present bounds for distilling many copies of a pure state from an arbitrary initial state in a general quantum resource theory. Our bounds apply to operations that are able to generate no more than a $\ensuremath{\delta}$ amount of resource, where $\ensuremath{\delta}\ensuremath{\ge}0$ is a given parameter. To maximize applicability of our upper bound, we assume little structure on the set of free states under consideration besides a weak form of superadditivity of the function ${G}_{\mathrm{min}}(\ensuremath{\rho})$, which measures the overlap between $\ensuremath{\rho}$ and the set of free states. Our bounds are given in terms of this function and the robustness of resource. Known results in coherence and entanglement theory are reproduced in this more general framework.

General Theory of Resources

This article focuses on presenting the modalities of appeals under the NCPC/15, showing some points of the formalities required for its interposition, which will be differentiated in the course of the article. It will also be addressed, in addition to the aspects, the verification of each of them in case of possible occurrence in the case itself, with a view to sending to the proper Court, dealing with the notes that are possible of its effects when occurred and admitted aspects.

Universal limitations on implementing resourceful unitary evolutions

We derive a trade-off relation between the accuracy of implementing a desired unitary evolution using a restricted set of free unitaries and the size of the assisting system, in terms of the resource generating/losing capacity of the target unitary. In particular, this relation implies that, for any theory equipped with a resource measure satisfying lenient conditions, any resource changing unitary cannot be perfectly implemented by a free unitary applied to a system and an environment if the environment has finite dimensions. Our results are applicable to a wide class of resources including energy, asymmetry, coherence, entanglement, and magic, imposing ultimate limitations inherent in such important physical settings, as well as providing insights into operational restrictions in general resource theories.

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

We establish an operational characterization of general convex resource theories -- describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -- in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as well as establishing that such discrimination problems can fully characterize the allowed transformations within the given resource theory. Specifically, we introduce quantifiers of resourcefulness of states, measurements, and channels in any GPT based on the generalized robustness, and show that they exactly characterize the maximum advantage that a given resource provides over all free states, measurements, or channels in a class of state or channel discrimination tasks. In quantum mechanics, we show that the robustness of measurement can be alternatively understood as the maximal increase in one-shot accessible information when compared to free measurements. We furthermore endow the standard robustness of a state with an operational meaning as the quantifier of the maximum advantage in binary channel discrimination tasks. Finally, we show that several classes of channel and state discrimination tasks can form complete families of monotones fully characterizing the transformations of states and measurements under any chosen class of free operations. Our results establish a fundamental connection between operational tasks of discrimination and core concepts of resource theories, valid for all physical theories with no additional assumptions about the structure of the GPT required.

Hypothesis testing and entropies of quantum channels

Entropic quantifiers of states lie at the cornerstone of the quantum information theory. While a quantum state can be abstracted as a device that only has outputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend the entropic quantifiers of states to the ones of channels. In the one-shot and asymptotic scenarios, we propose relative entropies of channels under the task of hypothesis testing. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarising channel. We also study properties of relative entropies of channels and the interplay with entanglement. Finally, based on relative entropies of channels, we propose general resource theories of channels and discuss the coherence of general channels and measurements, and the entanglement of channels.

Amortized entanglement of a quantum channel and approximately teleportation-simulable channels

This paper defines the amortized entanglement of a quantum channel as the largest difference in entanglement between the output and the input of the channel, where entanglement is quantified by an arbitrary entanglement measure. We prove that the amortized entanglement of a channel obeys several desirable properties, and we also consider special cases such as the amortized relative entropy of entanglement and the amortized Rains relative entropy. These latter quantities are shown to be single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of a quantum channel, respectively. Of especial interest is a uniform continuity bound for these latter two special cases of amortized entanglement, in which the deviation between the amortized entanglement of two channels is bounded from above by a simple function of the diamond norm of their difference and the output dimension of the channels. We then define approximately teleportation- and positive-partial-transpose-simulable (PPT-simulable) channels as those that are close in diamond norm to a channel which is either exactly teleportation- or PPT-simulable, respectively. These results then lead to single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of channels that are approximately teleportation- or PPT-simulable, respectively. Finally, we generalize many of the concepts in the paper to the setting of general resource theories, defining the amortized resourcefulness of a channel and the notion of ν-freely-simulable channels, connecting these concepts in an operational way as well.

Application of the Eco-field and General Theory of Resources to Bark Beetles: Beyond the Niche Construction Theory

A new approach to landscape ecology involves the application of the eco-field hypothesis and the General Theory of Resources. In this study, we describe the putative eco-field of bark beetles as a spatial configuration with a specific meaning-carrier for every organism-resource interaction. Bark beetles are insects with key roles in matter and energy cycles in coniferous forests, which cause significant changes to forestry landscapes when outbreaks occur. Bark beetles are guided towards host trees by the recognition of semiotic signals using a specific eco-field. These signals mainly comprise a group of scents, which are called the odourtope. Their interactions with other organisms (fungi, bacteria, nematodes, predators, etc.) occur by sharing relevant information from the eco-field networks (representamen networks) in the forest ecosystem. The eco-field networks modulate the expansion of the realized semiotic niche of the bark beetle towards the potential semiotic niche. Moreover, the niche construction process can be initiated by interchanging signals among species living in the same place, where these signals allow the exploitation of the required resources. If different organisms are interdependent on signals in eco-field networks, then this process may result in the establishment of mutualistic relationships. This is an example of how evolutionary processes are initiated by the recognition of signals in a network of eco-fields.

A Biosemiotic Perspective of the Resource Criterion: Toward a General Theory of Resources

Describing resources and their relationships with organisms seems to be a useful approach to a ‘unified ecology’, contributing to fill the gap between natural and human oriented processes, and opening new perspectives in dealing with biological complexity. This Resource Criterion defines the main properties of resources, describes the mechanisms that link them to individual species, and gives a particular emphasis to the biosemiotic approach that allows resources to be identified inside a heterogeneous ecological medium adopting the eco-field model. In particular, this Criterion allows to couple matter, structured energy and information composing the ecological systems to the biosemiotic and cognitive mechanisms adopted by individual species to track resources, transforming neutral surroundings into meaningful species-specific Umwelten. The expansion of the human semiotic niche that is a relevant evolutionary process of the present time, assigns the role of powerful and efficient agency to the Resources Criterion to evaluate the effect of human intrusion into the natural systems with habits of key stone species, under the challenge of a growing use of alloctonous, immaterial and symbolic resources of the actual globalized societal models. The Resource Criterion interprets the ecological dynamics contributing to complete the epistemology of the ecology, to open a bridge toward economy and other societal sciences, and to contribute to formulate a General Theory of Resources.

Landscape Ecology and the General Theory of Resources: Comparing Two Paradigms

This paper is an attempt to connect the General Theory of Resources (GTR) with the principles that guide the Landscape Ecology Principles (LEP). The recent GTR is based on the assumption that resources are the common requirement for individual species, populations, communities and ecosystems. We therefore describe the main characteristics of resources, while the biosemiotic mechanisms with which to track them are also discussed.

Rethinking the Landscape: New Theoretical Perspectives for a Powerful Agency

An ecological description of a landscape transcends its geographical definition to characterize it in terms of a complex agency composed of a spatial mosaic, structured energy, information and meaning. Because the dimensions of the landscape encompasses both natural and human processes, it requires a more robust set of theories that incorporate the material components and their perceptual meaning. A biosemiotic approach defines the landscape as the sum of its organisms’ eco-fields, which are spatial configurations that carry meanings connected to specific needs and related functions. In this perspective, the new postulated General Theory of Resources, offers a substantial contribution to complete the paradigmatic framework of a “private landscape” species-, individual- and function-specific. From this theory resources are considered every material (f.i. food) or immaterial elements (f.i. safety) necessary to fullfill individual needs. The habitat becomes the “private landscape” and the well-being versus the ill-being are the emergent conditions of each individual under the environmental constraints. Definitively space is the entity on which material and immaterial resources are distributed and the dimension on which individuals species are interacting by using biosemiotic mechanisms.

Entanglement theory and the second law of thermodynamics

Entanglement is central both to the foundations of quantum theory and, as a novel resource, to quantum information science. The theory of entanglement establishes basic laws that govern its manipulation, in particular the non-increase of entanglement under local operations on the constituent particles. Such laws aim to draw from them formal analogies to the second law of thermodynamics; however, whereas in the second law the entropy uniquely determines whether a state is adiabatically accessible from another, the manipulation of entanglement under local operations exhibits a fundamental irreversibility, which prevents the existence of such an order. Here, we show that a reversible theory of entanglement and a rigorous relationship with thermodynamics may be established when considering all non-entangling transformations. The role of the entropy in the second law is taken by the asymptotic relative entropy of entanglement in the basic law of entanglement. We show the usefulness of this approach to general resource theories and to quantum information theory.