We develop a general framework for higher-order quantum resources that treats the resourcefulness of quantum states, channels, and supermaps (also called processes) in a unified manner. By generalizing well-established concepts from resource theories of states and channels, we introduce and study supermap resource monotones and supermap distillation and dilution. As free supermaps, we consider free networks (supermaps that can be implemented with free channels), compatible supermaps (sets of supermaps closed under arbitrary composition), and maximally free superchannels. Specializing to approximate channel-to-channel conversions, we show that under mild assumptions, the conversion error with respect to the diamond distance and the worst-case infidelity coincide. We then apply the general framework to resource theories of dynamical coherence, where we establish the existence of largest sets of compatible supermaps. By characterizing these sets with semidefinite constraints, we show that they are strictly larger than the corresponding free networks but do not contain all maximally free superchannels. We finish by investigating the operational differences of these sets in one-shot channel distillation and dilution, providing solutions for both tasks in the parallel setting and for dilution in the adaptive setting.
We develop a general framework for higher-order quantum resources that treats the resourcefulness of quantum states, channels, and supermaps (also called processes) in a unified manner. By generalizing well-established concepts from resource theories of states and channels, we introduce and study supermap resource monotones and supermap distillation and dilution. As free supermaps, we consider free networks (supermaps that can be implemented with free channels), compatible supermaps (sets of supermaps closed under arbitrary composition), and maximally free superchannels. Specializing to approximate channel-to-channel conversions, we show that under mild assumptions, the conversion error with respect to the diamond distance and the worst-case infidelity coincide. We then apply the general framework to resource theories of dynamical coherence, where we establish the existence of largest sets of compatible supermaps. By characterizing these sets with semidefinite constraints, we show that they are strictly larger than the corresponding free networks but do not contain all maximally free superchannels. We finish by investigating the operational differences of these sets in one-shot channel distillation and dilution, providing solutions for both tasks in the parallel setting and for dilution in the adaptive setting.
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.
Higher-order quantum transformations allow multiple channel uses to be combined through different causal architectures, from parallel and fixed-order sequential networks to general higher-order processes. Whether this causal freedom improves channel transformation when the higher-order operation is also constrained by a resource theory remains largely unexplored. We study this question in the dynamical resource theory of coherence using a unified semidefinite-programming framework. For two qubit amplitude-damping channels and the identity target, we prove a strict causal hierarchy at every nontrivial damping strength under both maximally incoherent superchannels (MISC) and dephasing-covariant incoherent superchannels (DISC). In contrast, mixed-Pauli channels admit a common teleportation simulation that transfers the channel dependence to Bell-diagonal program states prepared in parallel. The remaining processing can then be absorbed into a single quantum operational, so parallel, fixed-order sequential, and general higher-order strategies achieve the same optimal error for any target. These results identify free program-state parallelisation as a structural obstruction to causal enhancement.
A fundamental problem in quantum information is to understand the operational significance of quantum resources. Quantum resource theories (QRTs) provide a powerful theoretical framework that aids in the analysis and comprehension of the operational meaning of these resources. Early resource theories primarily focused on analyzing static quantum resources. Recently, interest in the study of dynamic quantum resources has been growing. In this paper, we utilize superchannel theory to describe the dynamic resource theory of quantum coherence. In this dynamic resource theory, we treat classical channels as free channels and consider two classes of free superchannels that preserve channel incoherence [maximally incoherent superchannels (MISCs) and dephasing-covariant incoherent superchannels (DISCs)] as free resources. We regard the quantum Fourier transform as the golden unit of dynamic coherence resources. We first establish the one-shot theory of dynamic coherence cost and dynamic coherence distillation, which involves converting the quantum Fourier transform into an arbitrary quantum channel using MISCs and DISCs. Next, we introduce a class of free superchannels known as $\ensuremath{\delta}$-MISCs, which asymptotically generate negligible dynamic coherence. Finally, we provide upper and lower bounds for the one-shot catalytic dynamic coherence cost of quantum channels under the action of these $\ensuremath{\delta}$-MISC superchannels.
Quantum information processing relies on a variety of resources, including entanglement, coherence, non-Gaussianity, and magic. In realistic settings, protocols run on networks of parties with heterogeneous local resource constraints, so different resources coexist and interact. Yet, resource theories have mostly treated each resource in isolation, and a general theory for manipulation in such distributed settings has been lacking. We develop a unified framework for composite quantum resource theories that describes distributed networks of locally constrained parties. We formulate natural axioms a composite theory should satisfy to respect the local structure, and from these axioms derive fundamental bounds on resource manipulation that hold universally, independent of the particular network characteristics. We apply our results to central operational tasks, including resource conversion and assisted distillation, and introduce new methods to construct new resource monotones from this setup. Our framework further reveals previously unexplored phenomena in the remote certification of quantum resources. Together, these results establish foundational laws for distributed quantum resource manipulation across diverse physical platforms.
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.
A fundamental problem in resource theory is to study the manipulation of the resource. Focusing on a general dynamical resource theory of quantum channels, here we consider tasks of one-shot resource distillation and dilution with a single copy of the resource. For any target of unitary channel or pure state preparation channel, we establish a universal strategy to determine upper and lower bounds on rates that convert between any given resource and the target. We show that the rates are related to resource measures based on the channel robustness and the channel hypothesis testing entropy, with regularization factors of the target resource measures. The strategy becomes optimal with converged bounds when the channel robustness is finite and measures of the target resource collapse to the same value. The single-shot result also applies to asymptotic parallel manipulation of channels to obtain asymptotic resource conversion rates. We give several examples of dynamical resources, including the purity, classical capacity, quantum capacity, non-uniformity, coherence, and entanglement of quantum channels. Our results are applicable to general dynamical resource theories with potential applications in quantum communication, fault-tolerant quantum computing, and quantum thermodynamics.