We show that the dynamic resource theory of quantum entanglement can be formulated using the superchannel theory. In this formulation, we identify the separable channels and the class of free superchannels that preserve channel separability as free resources, and choose the swap channels as dynamic entanglement golden units. Our first result is that the one-shot dynamic entanglement cost of a bipartite quantum channel under the free superchannels is bounded by the standard log-robustness of channels. The one-shot distillable dynamic entanglement of a bipartite quantum channel under the free superchannels is found to be bounded by a resource monotone that we construct from the hypothesis-testing relative entropy of channels with minimization over separable channels. We also address the one-shot catalytic dynamic entanglement cost of a bipartite quantum channel under a larger class of free superchannels that could generate the dynamic entanglement which is asymptotically negligible; it is bounded by the generalized log-robustness of channels.
We show that the dynamic resource theory of quantum entanglement can be formulated using the superchannel theory. In this formulation, we identify the separable channels and the class of free superchannels that preserve channel separability as free resources, and choose the swap channels as dynamic entanglement golden units. Our first result is that the one-shot dynamic entanglement cost of a bipartite quantum channel under the free superchannels is bounded by the standard log-robustness of channels. The one-shot distillable dynamic entanglement of a bipartite quantum channel under the free superchannels is found to be bounded by a resource monotone that we construct from the hypothesis-testing relative entropy of channels with minimization over separable channels. We also address the one-shot catalytic dynamic entanglement cost of a bipartite quantum channel under a larger class of free superchannels that could generate the dynamic entanglement which is asymptotically negligible; it is bounded by the generalized log-robustness of channels.
We first investigate the one-shot static entanglement cost to simulate a bipartite quantum channel under the set of nonentangling channels. The lower bound on the one-shot static entanglement cost is given by the generalized robustness of the target channel as well as the robust-generating power of the channel, which implies that the cost necessary to generate a bipartite quantum channel might not be retrievable in general. Next, we conceive a set of quantum channels that extends the set of nonentangling channels. We find out that the one-shot static entanglement cost under the extended set of channels is given by the channel's standard log-robustness. This quantity also gives the one-shot dynamic entanglement cost under a set of free superchannels that do not generate a dynamic entanglement resource; the extended set is shown to be equivalent to the set of free superchannels.
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.
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.
Although it is well known that the amount of resources that can be asymptotically distilled from a quantum state or channel does not exceed the resource cost needed to produce it, the corresponding relation in the nonasymptotic regime hitherto has not been well understood.Here, we establish a quantitative relation between the one-shot distillable resource yield and dilution cost in terms of transformation errors involved in these processes.Notably, our bound is applicable to quantum state and channel manipulation with respect to any type of quantum resource and any class of free transformations thereof, encompassing broad types of settings, including entanglement, quantum thermodynamics, and quantum communication.We also show that our techniques provide strong converse bounds relating the distillable resource and the resource dilution cost in the asymptotic regime.Moreover, we introduce a class of channels that generalize the twirling maps encountered in many resource theories, and by directly connecting it with resource quantification, we compute analytically several smoothed resource measures and improve our one-shot yield-cost bound in relevant theories.We use these operational insights to exactly evaluate important measures for various resource states in the resource theory of magic states.
Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.
The thermodynamic resourcefulness of quantum channels primarily depends on their underlying causal structure and their ability to generate quantum correlations. We quantify this interplay within the resource theory of athermality for bipartite quantum channels in the presence of a side channel acting as memory, referred to as the resource theory of conditional athermality. For channels with trivial output Hamiltonians, we characterize the optimal one-shot rates for distilling the identity gate from a given channel, as well as the cost of simulating the channel using the identity gate, under conditional Gibbs-preserving superchannels. We show that these rates have a direct trade-off relation with the conditional channel entropies, attributing operational significance to signaling in quantum processes. Furthermore, we establish an asymptotic equipartition property for the conditional channel min-entropy for classes of channels that are either tele-covariant or no-signaling from the non-conditioning input to the conditioning output. As a consequence, we demonstrate asymptotic reversibility of the resource theory for these channels. The asymptotic conditional athermality capacity of a tele-covariant channel is half the superdense coding capacity of its Choi state. Our work establishes the conditional channel entropy as a primitive information-theoretic concept for quantum processes, elucidating its potential for wider applications in quantum information science.