Weak islands, individuals, and scope

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The resource theory of steering

We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. To begin with, we characterize the set of steering non-increasing operations (SNIOs) --i.e., those that do not create steering-- on arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device. Next, we introduce the notion of convex steering monotones as the fundamental axiomatic quantifiers of steering. As a convenient example thereof, we present the relative entropy of steering. In addition, we prove that two previously proposed quantifiers, the steerable weight and the robustness of steering, are also convex steering monotones. To end up with, for minimal-dimensional systems, we establish, on the one hand, necessary and sufficient conditions for pure-state steering conversions under stochastic SNIOs and prove, on the other hand, the non-existence of steering bits, i.e., measure-independent maximally steerable states from which all states can be obtained by means of the free operations. Our findings reveal unexpected aspects of steering and lay foundations for further resource-theory approaches, with potential implications in Bell non-locality.Comment: Presentation and structure improve

Some observations about generalized quantifiers in logics of imperfect information

We analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engstrom. comparing them with a more general, higher order definition of team quantifier. We show that Engstrom's definitions (and other quantifiers from the literature) can be identified, by means of appropriate lifts, with special classes of team quantifiers. We point out that the new team quantifiers express a quantitative and a qualitative component, while Engstrom's quantifiers only range over the latter. We further argue that Engstrom's definitions are just embeddings of the first-order generalized quantifiers into team semantics. and fail to capture an adequate notion of team-theoretical generalized quantifier, save for the special cases in which the quantifiers are applied to flat formulas. We also raise several doubts concerning the meaningfulness of the monotone/nonmonotone distinction in this context. In the appendix we develop some proof theory for Engstrom's quantifiers.Peer reviewe

Robustness measures for quantifying nonlocality

We suggest generalized robustness for quantifying nonlocality and investigate its properties by comparing it with white-noise and standard robustness measures. As a result, we show that white-noise robustness does not fulfill monotonicity under local operation and shared randomness, whereas the other measures do. To compare the standard and generalized robustness measures, we introduce the concept of inequivalence, which indicates a reversal in the order relationship depending on the choice of monotones. From an operational perspective, the inequivalence of monotones for resourceful objects implies the absence of free operations that connect them. Applying this concept, we find that standard and generalized robustness measures are inequivalent between even- and odd-dimensional cases up to eight dimensions. This is obtained using randomly performed CGLMP measurement settings in a maximally entangled state. This study contributes to the resource theory of nonlocality and sheds light on comparing monotones by using the concept of inequivalence valid for all resource theories