One-Shot Manipulation of Dynamical Quantum Resources

Abstract

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 tighten the bounds further and show that our conditions become necessary and sufficient for broad classes of important theories. This enables an exact characterization of these fundamental tasks and establishes 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-signalling, separability-preserving, and positive partial transpose-preserving codes; as well as to nonlocality, contextuality, and measurement incompatibility, where we present an operational application of a number of relevant resource measures.

Convertibility and Guessing Games in Dynamic Resource Theories

“…The generalized robustness of resource reflects how robust an object is in terms of remaining being resourceful against generic noise [42,50,57,62]. Its definition on a state ρ A1 in a static QRT is given by…”

“…In a general QRT, whether or not a set of resource monotones is complete depends on the free operations of the theory. It turns out that the problem is simplified considerably if one considers the "maximal" set of free operations, which is the full set of maps that act invariantly on the set of free objects [19,41,42]. Unfortunately, in almost all physically-motivated QRTs, the natural choice for free operations does not coincide with the maximal set.…”

“…We arrive at these results by establishing a general framework for studying dynamic QRTs based on an entropic quantity which we call the free min-entropy. Most approaches to unifying different QRTs focus on the resource robustness measure [42,50,57,62]. Our use of the free minentropy to characterize general QRTs is slightly different, and it has the advantage that the complete convertibility conditions emerge naturally within the framework.…”

Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

“…Our approach is based on the notion of robustness measures [22] -inspired by recent applications of such quantities in the study of general resource theories of channels [23][24][25][26][27][28][29][30][31][32][33], we use them to quantify the amount of noise needed to turn a given map into a quantum channel. Such measures allow for several different generalisations to the setting of linear maps, motivating us to study and compare these definitions.…”

Incompatibility as a resource for programmable quantum instruments

“…The incompatibility in one family of POVMs/channels/instruments can then be quantitatively compared to another. Resource theories also provide tools for detecting or "witnessing" the incompatibility present in general measurement devices [27,[29][30][31][32]. This certification can also be done in a semi-device-independent way [12,18,19,23,[33][34][35].…”