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Laser Physics Letters

Operational and entropic characterization of weight-based dynamic resource measures

Zhanyuan Fu, Yiran Wang, Li Li

Published Apr 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1612-202x/ae53c7

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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.
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.
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 introduce the resource quantifier of weight of resource for convex quantum resource theories of states and measurements with arbitrary resources. We show that it captures the advantage that a resourceful state (measurement) offers over all possible free states (measurements) in the operational task of exclusion of subchannels (states). Furthermore, we introduce information-theoretic quantities related to exclusion for quantum channels and find a connection between the weight of resource of a measurement and the exclusion-type information of quantum-to-classical channels. Our results apply to the resource theory of entanglement in which the weight of resource is known as the best-separable approximation or Lewenstein-Sanpera decomposition introduced in 1998. Consequently, the results found here provide an operational interpretation to this 21-year-old entanglement quantifier.
Quantum resource theories (QRTs) provide a unified framework to analyze quantum properties as resources for achieving advantages in quantum information processing. The generalized robustness and the weight of resource have been gaining increasing attention as useful resource quantifiers. However, the existing analyses of these measures were restricted to the cases where convexity of the set of free states is assumed, and physically motivated resources do not necessarily satisfy this restriction. In this paper, we give characterizations of robustness- and weight-based measures in general QRTs without convexity restriction through two different yet related approaches. On the one hand, we characterize the generalized robustness and the weight of resource by introducing a nonlinear witness. We show a general construction of new witness observables that detect the resourcefulness of a given state from multiple copies of the state and, using these witnesses, we provide operational interpretations of the above resource measures even without any convexity assumption. On the other hand, we find that the generalized robustness and the weight of resource can also be interpreted as the worst-case maximum advantage in variants of channel-discrimination and channel-exclusion tasks, respectively, where the set of free states consists of several convex subsets corresponding to multiple restrictions. We further extend these results to QRTs for quantum channels and quantum instruments. These characterizations show that every quantum resource exhibits an advantage for the corresponding tasks, even in general QRTs without convexity assumption. Thus, we establish the usefulness of robustness-based and weight-based techniques beyond the conventional scope of convex QRTs, leading to a better understanding of the general structure of QRTs.
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.
A key ingredient in quantum resource theories is a notion of measure. Such as a measure should have a number of fundamental properties, and desirably also a clear operational meaning. Here we show that a natural measure known as the convex weight, which quantifies the resource cost of a quantum device, has all the desired properties. In particular, the convex weight of any quantum resource corresponds exactly to the relative advantage it offers in an exclusion (or antidistinguishability) task. After presenting the general result, we show how the construction works for state assemblages, sets of measurements, and sets of transformations. Moreover, in order to bound the convex weight analytically, we give a complete characterization of the convex components and corresponding weights of such devices.