Dynamic quantum resource theories study the manipulation of quantum channels by means of a restricted set of free superoperations. In this paper, we formulate general dynamic resource theories using a "top-down" framework, and we provide systematic characterizations for closed and convex resource theories from both information-theoretic and operational perspectives. Our results are summarized as follows. First, we propose and investigate a branch of resource-induced measures of uncertainty, called the free conditional min-entropy (FCME), generalizing the conditional min-entropy and its dynamic extension to scenarios where information processing is subject to variable operational restriction. We provide a complete set of entropic conditions in terms of the FCME for characterizing channel convertibility via free superoperations in any closed and convex resource theory. We also find that the resource global robustness of channels can be equivalently cast as a mutual-information-like quantity derived from the FCME, thereby offering the resource global robustness an information-theoretic interpretation. Apart from the entropic approach, we also study closed and convex resource theories in the contexts of various operational tasks. These tasks are formulated such that each of them induces a complete set of operationally meaningful resource monotones, and therefore they can be used to faithfully test free convertibility between channels. We also systematically study the quantitative relations between the operational advantage of channels in these tasks and the resource robustness measures of channels. In particular, we prove that every well-defined robustness-based measure can be operationally interpreted as some kind of advantage in a task called semiquantum partial preprocessing. Ultimately, our results provide both entropic and operational characterizations for general dynamic quantum resources with a closed and convex structure.
Dynamic quantum resource theories study the manipulation of quantum channels by means of a restricted set of free superoperations. In this paper, we formulate general dynamic resource theories using a "top-down" framework, and we provide systematic characterizations for closed and convex resource theories from both information-theoretic and operational perspectives. Our results are summarized as follows. First, we propose and investigate a branch of resource-induced measures of uncertainty, called the free conditional min-entropy (FCME), generalizing the conditional min-entropy and its dynamic extension to scenarios where information processing is subject to variable operational restriction. We provide a complete set of entropic conditions in terms of the FCME for characterizing channel convertibility via free superoperations in any closed and convex resource theory. We also find that the resource global robustness of channels can be equivalently cast as a mutual-information-like quantity derived from the FCME, thereby offering the resource global robustness an information-theoretic interpretation. Apart from the entropic approach, we also study closed and convex resource theories in the contexts of various operational tasks. These tasks are formulated such that each of them induces a complete set of operationally meaningful resource monotones, and therefore they can be used to faithfully test free convertibility between channels. We also systematically study the quantitative relations between the operational advantage of channels in these tasks and the resource robustness measures of channels. In particular, we prove that every well-defined robustness-based measure can be operationally interpreted as some kind of advantage in a task called semiquantum partial preprocessing. Ultimately, our results provide both entropic and operational characterizations for general dynamic quantum resources with a closed and convex structure.
We offer new methods for characterizing general closed and convex quantum resource theories, including dynamic ones, based on entropic concepts and operational tasks. We propose a resource-theoretic generalization of the quantum conditional min-entropy, termed the free conditional min-entropy (FCME), in the sense that it quantifies an observer’s “subjective” degree of uncertainty about a quantum system given that the observer’s information processing is limited to free operations of the resource theory. Using this generalized concept, we provide a complete set of entropic conditions for free convertibility between quantum states or channels in any closed and convex quantum resource theory. We also derive an information-theoretic interpretation for the resource global robustness of a state or a channel in terms of a mutual-information-like quantity based on the FCME. Apart from this entropic approach, we characterize dynamic resources by also analyzing their performance in operational tasks. We construct operationally meaningful and complete sets of resource monotones with these tasks, which enable faithful tests of free convertibility between quantum channels. Finally, we show that every well-defined robustness-based measure of a channel can be interpreted as an operational advantage of the channel over free channels in a communication task.
Abstract We introduce a comprehensive framework to characterize the weight of resource (WoR) in general closed and convex dynamic quantum resource theories. We show that the WoR precisely captures the maximum operational advantage that a resourceful channel offers over all free channels in entanglement-assisted state exclusion tasks. Specifically, we establish the free convertibility between two quantum channels, and prove that the WoR serves as the universal infimum for the error probability ratio. Furthermore, we introduce the concept of free max relative entropy mutual information and demonstrate that the logarithmic WoR is fundamentally equivalent to the channel’s correlation-generating power. Our results provide a unified ‘triangle of association’ for dynamic resources, bridging the geometric decomposition, operational advantage, and entropic correlation into a single consistent picture.
Quantum channels are quintessential to quantum information, being used in all protocols, and describing how systems evolve in space and time. As such, they play a key role in the manipulation of quantum resources, and they are often resources themselves, called dynamical resources. This forces us to go beyond standard resource theories of quantum states. Here we provide a rigorous foundation for dynamical resource theories, where the resources into play are quantum channels, explaining how to manipulate dynamical resources with free superchannels. In particular, when the set of free superchannels is convex, we present a novel construction of an infinite and complete family of convex resource monotones, giving necessary and sufficient conditions for convertibility under free superchannels. After showing that the conversion problem in convex dynamical resource theories can be solved with conic linear programming, we define various resource-theoretic protocols for dynamical resources. These results serve as the framework for the study of concrete examples of theories of dynamical resources, such as dynamical entanglement theory.
We present a general theory of comparison of quantum channels, concerning with the question of simulability or approximate simulability of a given quantum channel by allowed transformations of another given channel. We introduce a modification of conditional min-entropies, with respect to the set F of allowed transformations, and show that under some conditions on F, these quantities characterize approximate simulability. If F is the set of free superchannels in a quantum resource theory of processes, the modified conditional min-entropies form a complete set of resource monotones. If the transformations in F consist of a preprocessing and a postprocessing of specified forms, approximate simulability is also characterized in terms of success probabilities in certain guessing games, where a preprocessing of a given form can be chosen and the measurements are restricted. These results are applied to several specific cases of simulability of quantum channels, including postprocessings, preprocessings and processing of bipartite channels by LOCC superchannels and by partial superchannels, as well as simulability of sets of quantum measurements. These questions are first studied in a general setting that is an extension of the framework of general probabilistic theories (GPT), suitable for dealing with channels. Here we prove a general theorem that shows that approximate simulability can be characterized by comparing outcome probabilities in certain tests. This result is inspired by the classical Le Cam randomization criterion for statistical experiments and contains its finite dimensional version as a special case.
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.
A central problem in quantum resource theory is to give operational meaning to quantum resources that can provide clear advantages in certain physical tasks compared to the convex set of resource-free states. We propose to extend this basic principle by defining the relative superiority of resources over a specific convex set of resource states, also provide a relative advantage in physical tasks based on this extended principle. This allows the generalized robustness measure to quantify the relative maximal advantage due to a given resource state over a specific convex set of resource states in the subchannel discrimination, thereby showing that the operational interpretation of resource measures also holds in a relative perspective. In addition, we offer a new framework for defining the deficiency of a given state in physical tasks compared to the set of maximum resource states. The geometric measure we provide satisfies the conditions of the framework for quantum coherence and entanglement, and it accurately quantifies the minimal disadvantage due to a given state compared to maximumresourcestates inthe subchannel discrimination in certain situations. These two extensions and new interpretations expand the scope of quantum resource theories and provide a more comprehensive operational interpretation.