We investigate resource nongenerating operations in both static and dynamical quantum resource theories. For the static scenarios, we derive a sufficient condition for state transformations under resource nongenerating operations. Then we construct a dynamical resource theory where resource nongenerating operations constitute the set of free operations, and we propose an axiomatic approach to quantify the dynamical resource. We further analyze the erasure of the dynamical resources. As applications, we establish bounds on the rate of state conversions under resource nongenerating operations in a generic convex resource theory and obtain capacity bounds for classical communication tasks assisted by dynamical coherence. Our results clarify the key roles of resource nongenerating operations in quantum information processing tasks.
We investigate resource nongenerating operations in both static and dynamical quantum resource theories. For the static scenarios, we derive a sufficient condition for state transformations under resource nongenerating operations. Then we construct a dynamical resource theory where resource nongenerating operations constitute the set of free operations, and we propose an axiomatic approach to quantify the dynamical resource. We further analyze the erasure of the dynamical resources. As applications, we establish bounds on the rate of state conversions under resource nongenerating operations in a generic convex resource theory and obtain capacity bounds for classical communication tasks assisted by dynamical coherence. Our results clarify the key roles of resource nongenerating operations in quantum information processing tasks.
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 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.
We introduce a resource theory of channels relevant to communication via quantum channels, in which the set of constant channels-useless channels for communication tasks-are considered as free resources. We find that our theory with such a simple structure is useful to address central problems in quantum Shannon theory-in particular, we provide a converse bound for the one-shot nonsignaling assisted classical capacity that naturally leads to its strong converse property, as well as obtain the one-shot channel simulation cost with nonsignaling assistance. We clarify an intimate connection between the nonsignaling assistance and our formalism by identifying the nonsignaling assisted channel coding with the channel transformation under the maximal set of resource nongenerating superchannels, providing a physical characterization of the latter. Our results provide new perspectives and concise arguments to those problems, connecting the recently developed fields of resource theories to "classic" settings in quantum information theory and shedding light on the validity of resource theories of channels as effective tools to address practical problems.
One of the main goals of any resource theory such as entanglement, quantum thermodynamics, quantum coherence, and asymmetry, is to find necessary and sufficient conditions that determine whether one resource can be converted to another by the set of free operations. Here we find such conditions for a large class of quantum resource theories which we call affine resource theories. Affine resource theories include the resource theories of athermality, asymmetry, and coherence, but not entanglement. Remarkably, the necessary and sufficient conditions can be expressed as a family of inequalities between resource monotones (quantifiers) that are given in terms of the conditional min-entropy. The set of free operations is taken to be (1) the maximal set (i.e., consists of all resource nongenerating quantum channels) or (2) the self-dual set of free operations (i.e., consists of all resource nongenerating maps for which the dual map is also resource nongenerating). As an example, we apply our results to quantum thermodynamics with Gibbs preserving operations, and several other affine resource theories. Finally, we discuss the applications of these results to resource theories that are not affine and, along the way, provide the necessary and sufficient conditions that a quantum resource theory consists of a resource destroying map.
Resource theories in quantum information science are helpful for the study and quantification of the performance of information-processing tasks that involve quantum systems. These resource theories also find applications in other areas of study; e.g., the resource theories of entanglement and coherence have found use and implications in the study of quantum thermodynamics and memory effects in quantum dynamics. In this paper, we introduce the resource theory of unextendibility, which is associated with the inability of extending quantum entanglement in a given quantum state to multiple parties. The free states in this resource theory are the k-extendible states, and the free channels are k-extendible channels, which preserve the class of k-extendible states. We make use of this resource theory to derive nonasymptotic, upper bounds on the rate at which quantum communication or entanglement preservation is possible by utilizing an arbitrary quantum channel a finite number of times, along with the assistance of k-extendible channels at no cost. We then show that the bounds obtained are significantly tighter than previously known bounds for quantum communication over both the depolarizing and erasure channels.
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.