Quantifying Operations with an Application to Coherence

Abstract

To describe certain facets of non-classicality, it is necessary to quantify properties of operations instead of states. This is the case if one wants to quantify how well an operation detects non-classicality, which is a necessary prerequisite for its use in quantum technologies. To do so rigorously, we build resource theories on the level of operations, exploiting the concept of resource destroying maps. We discuss the two basic ingredients of these resource theories, the free operations and the free super-operations, which are sequential and parallel concatenations with free operations. This leads to defining properties of functionals that are well suited to quantify the resources of operations. We introduce these concepts at the example of coherence. In particular, we present two measures quantifying the ability of an operation to detect, i.e. to use, coherence, one of them with an operational interpretation, and provide methods to evaluate them. arXiv:1806.07332v2 [quant-ph]

Application of the Resource Theory of Channels to Communication Scenarios

“…which proves (32). In addition, when A ∈ A and F is the set of constant channels, the denominator of the l.h.s.…”

“…By definition of R F (N ), there exists a free channel F ∈ F and some channel L such that N = (1 + R F (N ))Ξ − R F (N )L. Let {M i } be the optimal measurement for the numerator of the l.h.s in (32). Then, for any ensemble…”

“…We introduce resource theory for communication, a resource theory of channels relevant to communication via quantum channels, in which we choose the set of constant channels, the useless channels for communication, as free resources. Unlike many of the resource theories of channels with underlying state theories [15,27,29,[31][32][33][34][35], our setting is not equipped with any theory of state, making the consideration of the resource theories of channels crucial. We introduce generalized robustness for communication and another equivalent quantity, max-relative entropy for communication, as resource quantifiers, and discuss resource transformations under free operations, for which we consider the maximal set of superchannels that map constant channels to constant channels.…”

Operational resource theory of quantum channels

“…As some channels may generate more resource from input states than others, we mainly focus on the resource generation ability of channels. One can also consider general manipulation abilities of channels, such as resource detection [60], which we leave as future works.Interplay with state resource theories.-Consider a state resource theory S = (Ω, Φ, µ) with a tensor product structure, i.e., φ ⊗ id ∈ Φ, ∀φ ∈ Φ. To characterize the state resource generating power, we construct a corresponding channel resource theory C = (F, O, R).…”

“…Now we construct a channel resource theory of coherence to characterize the coherence generating power. This has been partially done in several works [39,40,60,[68][69][70][71], whereas they did not treat channel coherence as an operational resource. Following our resource framework, we define free channels as resource non-generating 5 channels, i.e., MIO.…”