Einstein-Podolsky-Rosen steering is a form of inseparability in quantum theory commonly acknowledged to be intermediate between entanglement and Bell nonlocality. However, this statement has so far only been proven for a restricted class of measurements, namely, projective measurements. Here we prove that entanglement, one-way steering, two-way steering, and nonlocality are genuinely different considering general measurements, i.e., single round positive-operator-valued measures. Finally, we show that the use of sequences of measurements is relevant for steering tests, as they can be used to reveal "hidden steering."

We construct a single observable measurement of which mean value on four copies of an unknown two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so-called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight upper and lower bounds for negativity and concurrence. Elementary quantum computing device estimating unknown two-qubit entanglement is designed.PACS numbers: 03.65.-wIntroduction .-One of the main challenges of both theoretical and experimental Quantum Information Theory is a determination of entanglement properties of a given state. There is an extensive literature covering the problem of deciding entanglement of a state [1,2,3,4,5,6,7]. As one knows from the seminal paper of Peres and Wootters [8] collective measurement on several copies of a system in a given quantum state may provide better results than measurements performed on each copy separately. This fact was reflected in the method of entanglement detection with collective measurements. The method initiated for pure states [9, 10], then developed for mixed states with help of quantum networks [11,12,13,14,15,16,17] and the concept of collective entanglement witnesses [18], has found its first experimental demonstration in coalescence-anti coalescence coincidence experiment [19]. In particular, somewhat surprisingly, it was shown how to estimate and/or even measure amount of entanglement (concurrence) without prior state reconstruction [11,12,13]. Recently the method got the new twist thanks to application of such collective measurements [20,21,22,23] that are directly related to quantum concurrence (see [24]) including photon polarization-momentum experimental demonstration for pure states in distant laboratories paradigm [20]. Recently collective entanglement witnesses were also shown to lead to easily measurable lower bounds on entanglement [21]. The idea of collective entanglement witnesses was also implemented in continuous variables setup [22].We show that a single observable if measured on four copies of a unknown two-qubit state is sufficient for discrimination between entanglement and separability of it. Moreover it can serve for limited quantitative purposes. To this aim we explore the two-qubit separability test (equivalent to the PPT one [2,25]) stating that a state is separable iff the determinant of its partially transposed density matrix is nonnegative [26,27]. The result, known for a few years, was barely mentioned in the literature in that form (see e.g. [28]) and up to our knowledge this is the first time an operative physical meaning is assigned

Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled, but are genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.

While entanglement and violation of Bell inequalities were initially thought to be equivalent quantum phenomena, we now have different examples of entangled states whose correlations can be described by local hidden-variable models and, therefore, do not violate any Bell inequality. We provide an up to date overview of the existing local hidden-variable models for entangled quantum states, both in the bipartite and multipartite case, and discuss some of the most relevant open questions in this context. Our review covers twenty five years of this line of research since the seminal work by Werner [R. F. Werner, Phys. Rev. A 40, 8 (1989)] providing the first example of an entangled state with a local model, which in turn appeared twenty five years after the seminal work by Bell [J. S. Bell, Physics 1, 195 (1964)], about the impossibility of recovering the predictions of quantum mechanics using a local hidden-variables theory. arXiv:1405.7321v1 [quant-ph]

Understanding the relation between nonlocality and entanglement is one of the fundamental problems in quantum physics. In the bipartite case, it is known that these two phenomena are inequivalent, as there exist entangled states of two parties that do not violate any Bell inequality. However, except for a single example of an entangled three-qubit state that has a local model, almost nothing is known about such a relation in multipartite systems. We provide a general construction of genuinely multipartite entangled states that do not display genuinely multipartite nonlocality, thus proving that entanglement and nonlocality are inequivalent for any number of parties.