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Laser Physics Letters

Unified robustness of general quantum resources of a quantum system

Min Chen, Bo Zhou, Hongjian Wen, Huaixin CAO

Published Feb 1, 2026• 0 citations
Publisher

Abstract

Abstract Treating robustness as a key measure for various quantum resources, this work proposes a general quantum resource theory (GQRT) for a quantum system (GQRT4QS), which was formally a pair consisting of a free-state set and a free-operation set. Here, the free-state set whose elements are called free states is defined as any compact convex set of quantum states of a quantum system that includes the maximally random state, while the free-operation set whose elements are called free operations is a semigroup consisting of quantum operations of the system that map free states to free states. Within this GQRT4QS framework, we first introduce the unified robustness of quantum resources and rigorously prove its validity as a quantum resource measure. Subsequently, we define the unified geometric robustness of quantum resources and further demonstrate that it also satisfies the criteria for a valid quantum resource measure. For practical applications, we derive a series of resource-specific robustness measures, encompassing the robustness and geometric robustness of entanglement, Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering, and quantum coherence.

DOI
10.1088/1612-202x/ae4191

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Abstract Treating robustness as a key measure for various quantum resources, this work proposes a general quantum resource theory (GQRT) for a quantum system (GQRT4QS), which was formally a pair consisting of a free-state set and a free-operation set. Here, the free-state set whose elements are called free states is defined as any compact convex set of quantum states of a quantum system that includes the maximally random state, while the free-operation set whose elements are called free operations is a semigroup consisting of quantum operations of the system that map free states to free states. Within this GQRT4QS framework, we first introduce the unified robustness of quantum resources and rigorously prove its validity as a quantum resource measure. Subsequently, we define the unified geometric robustness of quantum resources and further demonstrate that it also satisfies the criteria for a valid quantum resource measure. For practical applications, we derive a series of resource-specific robustness measures, encompassing the robustness and geometric robustness of entanglement, Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering, and quantum coherence.
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.
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.
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.
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.
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.
Abstract Recently, the projective robustness of quantum states was introduced in (Regula 2022 Phys. Rev. Lett. 128 110505). This demonstrates that the projective robustness is a useful resource monotone and can comprehensively characterize the capabilities and limitations of probabilistic protocols that manipulate quantum resources deterministically. In this paper, we will extend the projective robustness to any convex resource theory of quantum channels and measurements. First, we introduce the projective robustness of quantum channels and prove that it satisfies some good properties, especially sub-multiplicativity under any free quantum process. Moreover, we show that the projective robustness of channels quantifies the maximal advantage that a given channel outperforms all free channels in the simultaneous discrimination and exclusion of a fixed-state ensemble. Second, we define the projective robustness of quantum measurements and prove that it exactly quantifies the maximal advantage that a given measurement outperforms all free measurements in the simultaneous discrimination and exclusion of two fixed-state ensembles. Finally, within a specific channel resource setting based on measurement incompatibility, we show that the projective robustness of quantum channels coincides with the projective robustness of measurement incompatibility.