Bell’s 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell’s theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. The main concepts and tools which have been developed to describe and study the nonlocality of quantum theory and which have raised this topic to the status of a full subfield of quantum information science are reviewed

Einstein-Podolsky-Rosen (EPR) steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It allows for entanglement certification when the measurements performed by one of the parties are not characterised (or are untrusted) and has applications in quantum key distribution. Despite its foundational and applied importance, EPR steering lacks a quantitative assessment. Here we propose a way of quantifying this phenomenon and use it to study the steerability of several quantum states. In particular we show that every pure entangled state is maximally steerable, the projector onto the anti-symmetric subspace is maximally steerable for all dimensions, we provide a new example of one-way steering, and give strong support that states with positive-partial-transposition are not steerable.Introduction.-Quantum systems display correlations that do not have a counterpart in classical physics. In the early days of quantum theory E. Schrodinger noticed a consequence of these stronger-than-classical correlations and named it EPR steering [1]. EPR steering refers to the following phenomenon: two parties, Alice and Bob, share an entangled state |ψ AB . By measuring her subsystem, Alice can remotely change (i.e. steer) the state of Bob's subsystem in such a way that would be impossible if their systems were only classically correlated. The simplest example of steering is given by the maximally entangled state of two qubits |φ + = (|00 +|11 )/ √ 2. Alice can project Bob's system into the basis {|a , |a ⊥ } by making a measurement of her subsystem in the conjugate basis {|a * , |a ⊥ * }. As such, she can remotely prepare any state on Bob's subsystem, a feature that is impossible if they share only separable states.EPR steering was recently given an operational interpretation as the distribution of entanglement by an untrusted party [2]: Alice wants to convince Bob, who does not trust her, that they share an entangled state. Bob, in order to be convinced, asks Alice to remotely prepare a collection of states of his subsystems. Alice performs her measurements (which are unknown to Bob) and communicates the results to him. By looking at the conditional states prepared by Alice, Bob is able to certify if they must have come from measurements on an entangled state. Interestingly, EPR steering is a form of quantum correlation that lies in between entanglement [3] and Bell nonlocality [4] since, on the one hand not every entangled state is steerable, and on the other hand some steerable states do not violate a Bell inequality [2]. Furthermore, like nonlocality, steering can be demonstrated in simple 'tests', for example it is sufficient to consider only two measurements with two outcomes for Alice, preparing a collection of four states for Bob. As such, steering can be certified experimentally through the violation of steering inequalities, the analogue of Bell inequalities [5]. In fact several steering tests have been reported [6,7], including a recent loophole-free experiment [8]...

Quantum discord quantifies non-classical correlations in a quantum system including those not captured by entanglement. Thus, only states with zero discord exhibit strictly classical correlations. We prove that these states are negligible in the whole Hilbert space: typically a state picked out at random has positive discord; and, given a state with zero discord, a generic arbitrarily small perturbation drives it to a positive-discord state. These results hold for any Hilbert-space dimension, and have direct implications on quantum computation and on the foundations of the theory of open systems. In addition, we provide a simple necessary criterion for zero quantum discord. Finally, we show that, for almost all positive-discord states, an arbitrary Markovian evolution cannot lead to a sudden, permanent vanishing of discord.PACS numbers: 03.67. Ac, 03.65.Yz, 03.67.Lx The emergence of quantum information science motivated a major effort towards the characterization of entangled states, generally believed to be an essential resource for quantum information tasks that outperform their classical counterparts. In particular, the geometry of the sets of entangled/non-entangled states received much attention [1] -starting from the fundamental result that the set of separable (non-entangled) states has non-zero volume in a finite dimensional Hilbert space [2]. In other words, separable states are not at all negligible, which has direct implications on some implementations of quantum computing [3] and on the definition of entanglement quantifiers [4].Apart from entanglement, quantum states display other correlations [5][6][7] not present in classical systems (meaning, here, systems where all observables commute). Aiming at capturing such correlations, Ollivier and Zurek introduced the quantum discord [5]. They showed that only in the absence of discord there exists a measurement protocol that enables distant observers to extract all the information about a bipartite system without perturbing it. This completeness of local measurements is featured by any classical state, but not by quantum states, even some separable ones. Thus, zero discord is a necessary condition for only-classical correlations.Very recently, quantum discord has received increasing attention [8][9][10][11][12][13][14][15]. A prevailing observation in all results obtained so far is that the absence or presence of discord is directly associated to non-trivial properties of states. Thus, it is natural to question how typical are positivediscord states?. Here we prove that a particular subset of states that contains the set of zero-discord states, has measure zero and is nowhere dense. That is, it is topologically negligible: typically, every state picked out at random has positive discord; and given a state with zero discord, a generic (arbitrarily small) perturbation will take it to a state of strictly positive discord. Remarkably, these results hold true for any Hilbert space dimension and are thus in contrast with expectations based on the structure ...