We address the problem of testing the dimensionality of classical and quantum systems in a "black-box" scenario. We develop a general formalism for tackling this problem. This allows us to derive lower bounds on the classical dimension necessary to reproduce given measurement data. Furthermore, we generalize the concept of quantum dimension witnesses to arbitrary quantum systems, allowing one to place a lower bound on the Hilbert space dimension necessary to reproduce certain data. Illustrating these ideas, we provide simple examples of classical and quantum dimension witnesses.

We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. For arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device, we show that local operations assisted by one-way classical communication (1W-LOCCs) from the quantum part to the black box cannot create steering. Based on this, we build a resource theory of steering with 1W-LOCCs as the free operations. 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 1W-LOCCs 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 research, with potential implications in Bell non-locality. arXiv:1409.5804v3 [quant-ph] 12 Nov 2015

Due to the importance of entanglement for quantum information purposes, a framework has been developed for its characterization and quantification as a resource based on the following operational principle: entanglement among N parties cannot be created by local operations and classical communication, even when N − 1 parties collaborate. More recently, nonlocality has been identified as another resource, alternative to entanglement and necessary for device-independent quantum information protocols. We introduce an operational framework for nonlocality based on a similar principle: nonlocality among N parties cannot be created by local operations and allowed classical communication even when N − 1 parties collaborate. We then show that the standard definition of multipartite nonlocality, due to Svetlichny, is inconsistent with this operational approach: according to it, genuine tripartite nonlocality could be created by two collaborating parties. We finally discuss alternative definitions for which consistency is recovered.Introduction. The fundamental importance of entanglement in Quantum Information Science has driven a strong theoretical effort devoted to its characterization, detection and quantification. The resulting entanglement theory [1] has produced new mathematical tools, such as entanglement witnesses or entanglement measures, which find application also beyond the quantum information scenario for which they were initially derived, e.g. in Condensed Matter Physics [2], Quantum Thermodynamics [3] or Biology [4].The first step when deriving this theoretical formalism consists in identifying the relevant objects and set of operations, see Table I. The relevant objects in the entanglement scenario are quantum states in systems composed by N observers, labeled by A i with i = 1, . . . , N . The relevant set of operations is the set of local operations and classical communication (LOCC). The whole formalism then relies on the following principle, which has a clear operational motivation: entanglement of a quantum state is a resource that cannot be created by LOCC. This implies that those states that can be created by LOCC are not entangled. These states are called separable and can be written as [1]

Do completely unpredictable events exist? Classical physics excludes fundamental randomness. Although quantum theory makes probabilistic predictions, this does not imply that nature is random, as randomness should be certified without relying on the complete structure of the theory being used. Bell tests approach the question from this perspective. However, they require prior perfect randomness, falling into a circular reasoning. A Bell test that generates perfect random bits from bits possessing high-but less than perfectrandomness has recently been obtained. Yet, the main question remained open: does any initial randomness suffice to certify perfect randomness? Here we show that this is indeed the case. We provide a Bell test that uses arbitrarily imperfect random bits to produce bits that are, under the non-signalling principle assumption, perfectly random. This provides the first protocol attaining full randomness amplification. Our results have strong implications onto the debate of whether there exist events that are fully random.