Fundamental limitations on distillation of quantum channel resources

Arbitrary Amplification of Quantum Coherence in Asymptotic and Catalytic Transformation

Quantum coherence is one of the fundamental aspects distinguishing classical and quantum theories. Coherence between different energy eigenstates is particularly important, as it serves as a valuable resource under the law of energy conservation. A fundamental question in this setting is how well one can prepare good coherent states from low coherent states and whether a given coherent state is convertible to another one. Here, we show that any low coherent state is convertible to any high coherent state arbitrarily well in two operational settings: asymptotic and catalytic transformations. For a variant of asymptotic coherence manipulation where one aims to prepare desired states in local subsystems, the rate of transformation becomes unbounded regardless of how weak the initial coherence is. In a non-asymptotic transformation with a catalyst, a helper state that locally remains in the original form after the transformation, we show that an arbitrary state can be obtained from any low coherent state. Applying this to the standard asymptotic setting, we find that a catalyst can increase the coherence distillation rate significantly-from zero to infinite rate. We also prove that such anomalous transformation requires small but nonzero coherence in relevant modes, establishing the condition under which a sharp transition of the operational capability occurs. Our results provide a general characterization of the coherence transformability in these operational settings and showcase their peculiar properties compared to other common resource theories such as entanglement and quantum thermodynamics.

Experimental Virtual Distillation of Entanglement and Coherence

Noise is, in general, inevitable and detrimental to practical and useful quantum communication and computation. Under the resource theory framework, resource distillation serves as a generic tool to overcome the effect of noise. Yet, conventional resource distillation protocols generally require operations on multiple copies of resource states, and strong limitations exist that restrict their practical utilities. Recently, by relaxing the setting of resource distillation to only approximating the measurement statistics instead of the quantum state, a resource-frugal protocol, "virtual resource distillation," is proposed, which allows more effective distillation of noisy resources. Here, we report its experimental implementation on a photonic quantum system for the distillation of quantum coherence (up to dimension four) and bipartite entanglement. We show the virtual distillation of the maximal superposed state of dimension four from the state of dimension two, an impossible task in conventional coherence distillation. Furthermore, we demonstrate the virtual distillation of entanglement with operations acting only on a single copy of the noisy Einstein-Podolsky-Rosen (EPR) pair and showcase the quantum teleportation task using the virtually distilled EPR pair with a significantly improved fidelity of the teleported state. These results illustrate the feasibility of the virtual resource distillation method and pave the way for accurate manipulation of quantum resources with noisy quantum hardware.

Every Quantum Helps: Operational Advantage of Quantum Resources beyond Convexity

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.

Virtual Quantum Resource Distillation

Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted "free" operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call "virtual resource distillation," provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources.

Probabilistic Transformations of Quantum Resources

The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over deterministic ones, significantly strengthening previous findings and extending recent no-go theorems. We apply our results to obtain a substantial improvement in bounds for the errors and overheads of probabilistic distillation protocols, directly applicable to tasks such as entanglement or magic state distillation, and computable through convex optimization. In broad classes of resources, we strengthen our results to show that the monotone completely governs probabilistic transformations-it serves as a necessary and sufficient condition for state convertibility. This endows the monotone with a direct operational interpretation, as it can exactly quantify the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol.

One-Shot Manipulation of Dynamical Quantum Resources

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.