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We investigate entanglement detection in quantum materials through criteria based on the simultaneous suppression of collective matter excitations. Unlike other detection schemes, these criteria can be applied to continuous and unbounded variables. By considering a system of interacting dipoles on a lattice, we show the detection of collective entanglement arising from two different physical mechanisms, namely, the ferroelectric ordering and the dressing of matter degrees of freedom by light. In the latter case, the detection shows the formation of a collective entangled phase not directly related to spontaneous symmetry breaking. These results open a new perspective for the entanglement characterization of competing orders in quantum materials, and have direct application to quantum paraelectrics with large polariton splittings., Comment: 5 pages, 3 figures

Entanglement-breaking channels (equivalently, measure-and-prepare channels) are an important class of quantum operations noted for their ability to destroy multipartite spatial quantum correlations. Inspired by this property, they have also been employed in defining notions of "classical memory", under the assumption that such channels effectively act as a classical resource. We show that, in a single-system multi-time scenario, entanglement-breaking channels are still a quantum memory resource: a qudit going through an entanglement-breaking channel cannot be simulated by a classical system of same dimension. We provide explicit examples of memory-based output generation tasks where entanglement-breaking channels outperform classical memories of the same size. Our results imply that entanglement-breaking channels cannot be generally employed to characterize classical memory effects in temporal scenarios without additional assumptions., Comment: 7 pages, 3 figures

We study what can or cannot be certified in communication scenarios where the assumption of independence and identical distribution (iid) between experimental rounds fails. In this respect, we prove that membership tests for non-convex sets of correlations cannot be formulated in the non-iid regime. Similarly, it is impossible to self-test non-extreme quantum operations, such as mixed states, or noisy quantum measurements, unless one allows more than a single use thereof within the same experimental round. One consequence of our results is that non-classicality in causal networks without inputs cannot be experimentally demonstrated. By analyzing optimal non-iid strategies in the triangle scenario, we raise the need to take into account the prior communication required to set up a causal network.

Steering resources, central for quantum advantages in one-sided device-independent quantum information tasks, can be enhanced via local filters. Recently, reversible steering conversion under local filters has been fully characterised. Here, we solve the problem in the irreversible scenario, which leads to a complete understanding of stochastic steering distillation. This result also provides an operational interpretation of the max-relative entropy as the optimal filter success probability. We further show that all steering measures can be used to quantify measurement incompatibility in certain stochastic steering distillation scenarios. Finally, for a broad class of steering robustness measures, we show that their maximally achievable values in stochastic steering distillation are always upper bounded by different types of incompatibility robustness measures. Hence, measurement incompatibility sets the fundamental limitations for stochastic steering distillation., Comment: 5+4 pages; 2 figures

We show that the incompatibility of a set of measurements cannot be increased by subjecting them to a filter, namely, by combining them with a device that post-selects the incoming states on a fixed outcome of a stochastic transformation. This result holds for several measures of incompatibility, such as those based on robustness and convex weight. Expanding these ideas to Einstein-Podolsky-Rosen steering experiments, we are able to solve the problem of the maximum steerability obtained with respect to the most general local filters in a way that allows for an explicit calculation of the filter operation. Moreover, our results generalize to nonphysical maps, i.e., positive but not completely positive linear maps., Comment: 12 pages, 1 figure, comments welcome

We introduce a framework to compute upper bounds for temporal correlations achievable in open quantum system dynamics, obtained by repeated measurements on the system. As these correlations arise by virtue of the environment acting as a memory resource, such bounds are witnesses for the minimal dimension of an effective environment compatible with the observed statistics. These witnesses are derived from a hierarchy of semidefinite programs with guaranteed asymptotic convergence. We compute non-trivial bounds for various sequences involving a qubit system and a qubit environment, and compare the results to the best known quantum strategies producing the same outcome sequences. Our results provide a numerically tractable method to determine bounds on multi-time probability distributions in open quantum system dynamics and allow for the witnessing of effective environment dimensions through probing of the system alone., Comment: 24 pages, 7 figures

Many problems in quantum information theory can be formulated as optimizations over the sequential outcomes of dynamical systems subject to unpredictable external influences. Such problems include many-body entanglement detection through adaptive measurements, computing the maximum average score of a preparation game over a continuous set of target states and limiting the behavior of a (quantum) finite-state automaton. In this work, we introduce tractable relaxations of this class of optimization problems. To illustrate their performance, we use them to: (a) compute the probability that a finite-state automaton outputs a given sequence of bits; (b) develop a new many-body entanglement detection protocol; (c) let the computer invent an adaptive protocol for magic state detection. As we further show, the maximum score of a sequential problem in the limit of infinitely many time steps is in general incomputable. Nonetheless, we provide general heuristics to bound this quantity and show that they provide useful estimates in relevant scenarios., Comment: Close to published version

Leggett and Garg formulated macrorealist models encoding our intuition on classical systems, i.e., physical quantities have a definite value that can be measured with minimal disturbance, and with the goal of testing macroscopic quantum coherence effects. The associated inequalities, involving the statistics of sequential measurements on the system, are violated by quantum mechanical predictions and experimental observations. Such tests, however, are subject to loopholes: a classical explanation can be recovered assuming specific models of measurement disturbance. We review recent theoretical and experimental progress in characterizing macrorealist and quantum temporal correlations, and in closing loopholes associated with Leggett-Garg tests. Finally, we review recent definitions of nonclassical temporal correlations, which go beyond macrorealist models by relaxing the assumption on the measurement disturbance, and their applications in sequential quantum information processing., Comment: 22 pages, 7 figures. Final version

Quantum steering is the central resource for one-sided device-independent quantum information. It is manipulated via one-way local operations and classical communication, such as local filtering on the trusted party. Here, we provide a necessary and sufficient condition for a steering assemblage to be transformable into another one via local filtering. We characterize the equivalence classes with respect to filters in terms of the steering equivalent observables measurement assemblage (SEO), first proposed to connect the problem of steerability with measurement incompatibility. We provide an efficient method to compute the maximal extractable steerability via local filters and show that it coincides with the incompatibility of the SEO. Moreover, we show that there always exists a bipartite state that provides an assemblage with steerability equal to the incompatibility of the measurements on the untrusted party. Finally, we investigate the optimal success probability and rates for transformation protocols (distillation and dilution) in the single-shot scenario together with examples., Comment: 13 pages, 2 figures, comments welcome!

We investigate temporal correlations in the simplest measurement scenario, i.e., that of a physical system on which the same measurement is performed at different times, producing a sequence of dichotomic outcomes. The resource for generating such sequences is the internal dimension, or memory, of the system. We characterize the minimum memory requirements for sequences to be obtained deterministically, and numerically investigate the probabilistic behavior below this memory threshold, in both classical and quantum scenarios. A particular class of sequences is found to offer an upper-bound for all other sequences, which suggests a nontrivial universal upper-bound of $1/e$ for the classical probability of realization of any sequence below this memory threshold. We further present evidence that no such nontrivial bound exists in the quantum case., Comment: 21 pages, 20 figures

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed., Comment: 68 pages, 20 figures. Final version

Incompatible measurements, i.e., measurements that cannot be simultaneously performed, are necessary to observe nonlocal correlations. It is natural to ask, e.g., how incompatible the measurements have to be to achieve a certain violation of a Bell inequality. In this work, we provide the direct link between Bell nonlocality and the quantification of measurement incompatibility. This includes quantifiers for both incompatible and genuine-multipartite incompatible measurements. Our method straightforwardly generalizes to include constraints on the system's dimension (semi-device-independent approach) and on projective measurements, providing improved bounds on incompatibility quantifiers, and to include the prepare-and-measure scenario., Comment: 15 pages, 4 figures. Comments welcome! v2: Accepted version

In the context of epidemiology, policies for disease control are often devised through a mixture of intuition and brute-force, whereby the set of logically conceivable policies is narrowed down to a small family described by a few parameters, following which linearization or grid search is used to identify the optimal policy within the set. This scheme runs the risk of leaving out more complex (and perhaps counter-intuitive) policies for disease control that could tackle the disease more efficiently. In this article, we use techniques from convex optimization theory and machine learning to conduct optimizations over disease policies described by hundreds of parameters. In contrast to past approaches for policy optimization based on control theory, our framework can deal with arbitrary uncertainties on the initial conditions and model parameters controlling the spread of the disease, and stochastic models. In addition, our methods allow for optimization over policies which remain constant over weekly periods, specified by either continuous or discrete (e.g.: lockdown on/off) government measures. We illustrate our approach by minimizing the total time required to eradicate COVID-19 within the Susceptible-Exposed-Infected-Recovered (SEIR) model proposed by Kissler \emph{et al.} (March, 2020)., Comment: New material: effect of vaccination campaigns on the minimum time under lockdown, use of optimization constraints to control the complexity of the generated policies for disease control, methods to optimize over weekly adaptive lockdown policies. The current pre-print is close to the published version

We investigate the role of nonclassical temporal correlations in enhancing the performance of ticking clocks in a discrete-time scenario. We show that the problem of optimal models for ticking clocks is related to the violation of Leggett-Garg-type temporal inequalities formulated in terms of, possibly invasive, sequential measurements, but on a system with a bounded memory capacity. Ticking clocks inspire the derivation of a family of temporal inequalities showing a gap between classical and quantum correlations, despite involving no input. We show that quantum ticking-clock models achieving accuracy beyond the classical bound are also those violating Leggett-Garg-type temporal inequalities for finite sequences and we investigate their continuous-time limit. Interestingly, we show that optimal classical clock models in the discrete-time scenario do not have a well-defined continuous-time limit, a feature that is absent in quantum models., Comment: 8+12 pages, 5 figures

The correlations arising from sequential measurements on a single quantum system form a polytope. This is defined by the arrow-of-time (AoT) constraints, meaning that future choices of measurement settings cannot influence past outcomes. We discuss the resources needed to simulate the extreme points of the AoT polytope, where resources are quantified in terms of the minimal dimension, or "internal memory" of the physical system. First, we analyze the equivalence classes of the extreme points under symmetries. Second, we characterize the minimal dimension necessary to obtain a given extreme point of the AoT polytope, including a lower scaling bound in the asymptotic limit of long sequences. Finally, we present a general method to derive dimension-sensitive temporal inequalities for longer sequences, based on inequalities for shorter ones, and investigate their robustness to imperfections., Comment: 20 pages (including appendix), 8 figures

Quantum field theory is completely characterized by the field correlations between spacetime points. In turn, some of these can be accessed by locally coupling to the field simple quantum systems, a.k.a. particle detectors. In this work, we consider what happens when a quantum-controlled superposition of detectors at different space-time points is used to probe the correlations of the field. We show that, due to quantum interference effects, two detectors can gain information on field correlations which would not be otherwise accessible. This has relevant consequences for information theoretic quantities, like entanglement and mutual information harvested from the field. In particular, the quantum control allows for extraction of entanglement in scenarios where this is otherwise provably impossible., Comment: 5+6 pages, 5 figures. This version matches the one accepted for publication in Phys.Rev.Lett

A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general., Comment: 15 pages, 1 figure

In contrast with classical physics, in quantum physics some sets of measurements are incompatible in the sense that they can not be performed simultaneously. Among other applications, incompatibility allows for contextuality and Bell nonlocality. This makes of crucial importance developing tools for certifying whether a set of measurements posses a certain structure of incompatibility. Here we show that, for quantum or nonsignaling models, if the measurements employed in a Bell test satisfy a given type of compatibility, then the amount of violation of some specific Bell inequalities become limited. Then, we show that correlations arising from local measurements on two-qubit states violate these limits, which rules out in a device-independent way such structures of incompatibility. In particular, we prove that quantum correlations allow for a device-independent demonstration of genuine triplewise incompatibility. Finally, we translate these results into a semi-device-independent Einstein-Podolsky-Rosen-steering scenario., Comment: Substantial improvements, several new results added, new author added. 18 pages, 4 figures

We investigate the relation between a refined version of Leggett and Garg conditions for macrorealism, namely the no-signaling-in-time (NSIT) conditions, and the quantum mechanical notion of nondisturbance between measurements. We show that all NSIT conditions are satisfied for any state preparation if and only if certain compatibility criteria on the state-update rules relative to the measurements, i.e. quantum instruments, are met. The systematic treatment of NSIT conditions supported by structural results on nondisturbance provides a unified framework for the discussion of the the clumsiness loophole. This extends previous approaches and allows for a tightening of the loophole via a hierarchy of tests able to disprove a larger class of macrorealist theories. Finally, we discuss perspectives for a resource theory of quantum mechanical disturbance related to violations of macrorealism., Comment: Final version

We propose a funding scheme for theoretical research that does not rely on project proposals, but on recent past scientific productivity. Given a quantitative figure of merit on the latter and the total research budget, we introduce a number of policies to decide the allocation of funds in each grant call. Under some assumptions on scientific productivity, some of such policies are shown to converge, in the limit of many grant calls, to a funding configuration that is close to the maximum total productivity of the whole scientific community. We present numerical simulations showing evidence that these schemes would also perform well in the presence of statistical noise in the scientific productivity and/or its evaluation. Finally, we prove that one of our policies cannot be cheated by individual research units. Our work must be understood as a first step towards a mathematical theory of the research activity., Comment: Some edits to the published version

In a recent work [Phys. Rev. Lett. 116, 240401 (2016)], a framework known by the name of "assemblage moment matrices" (AMMs) has been introduced for the device-independent quantification of quantum steerability and measurement incompatibility. In other words, even with no assumption made on the preparation device nor the measurement devices, one can make use of this framework to certify, directly from the observed data, the aforementioned quantum features. Here, we further explore the framework of AMM and provide improved device-independent bounds on the generalized robustness of entanglement, the incompatibility robustness and the incompatibility weight. We compare the tightness of our device-independent bounds against those obtained from other approaches. Along the way, we also provide an analytic form for the generalized robustness of entanglement for an arbitrary two-qudit isotropic state. When considering a Bell-type experiment in a tri- or more-partite scenario, we further show that the framework of AMM provides a natural way to characterize a superset to the set of quantum correlations, namely, one which also allows post-quantum steering., Comment: 17 pages, 6 figures. Comments welcome!

A quantum process encodes the causal structure that relates quantum operations performed in local laboratories. The process matrix formalism includes as special cases quantum mechanics on a fixed background space-time, but also allows for more general causal structures. Motivated by the interpretation of processes as a resource for quantum information processing shared by two (or more) parties, with advantages recently demonstrated both for computation and communication tasks, we investigate the notion of composition of processes. We show that under very basic assumptions such a composition rule does not exist. While the availability of multiple independent copies of a resource, e.g. quantum states or channels, is the starting point for defining information-theoretic notions such as entropy (both in classical and quantum Shannon theory), our no-go result means that a Shannon theory of general quantum processes will not possess a natural rule for the composition of resources., Comment: 9 pages, 3 figures. Published version

In quantum mechanics, spatial correlations arising from measurements at separated particles are well studied. This is not the case, however, for the temporal correlations arising from a single quantum system subjected to a sequence of generalized measurements. We first characterize the polytope of temporal quantum correlations coming from the most general measurements. We then show that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identify the correlations in the simplest scenario that can not be reached by two-dimensional systems. This leads to a temporal inequality for a dimension test, and we discuss a possible implementation using nitrogen-vacancy centers in diamond., Comment: 13 pages (including appendix), 1 figure

We propose a measure of quantum steerability, namely a convex steering monotone, based on the trace distance between a given assemblage and its corresponding closest assemblage admitting a local-hidden-state (LHS) model. We provide methods to estimate such a quantity, via lower and upper bounds, based on semidefinite programming. One of these upper bounds has a clear geometrical interpretation as a linear function of rescaled Euclidean distances in the Bloch sphere between the normalized quantum states of: (i) a given assemblage and (ii) an LHS assemblage. For a qubit-qubit quantum state, the above ideas also allow us to visualize various steerability properties of the state in the Bloch sphere via the so-called LHS surface. In particular, some steerability properties can be obtained by comparing such an LHS surface with a corresponding quantum steering ellipsoid. Thus, we propose a witness of steerability corresponding to the difference of the volumes enclosed by these two surfaces. This witness (which reveals the steerability of a quantum state) enables finding an optimal measurement basis, which can then be used to determine the proposed steering monotone (which describes the steerability of an assemblage) optimized over all mutually-unbiased bases.

The existence of a global causal order between events places constraints on the correlations that parties may share. Such "causal correlations" have been the focus of recent attention, driven by the realization that some extensions of quantum mechanics may violate so-called causal inequalities. In this paper we study causal correlations from an entropic perspective, and we show how to use this framework to derive entropic causal inequalities. We consider two different ways to derive such inequalities. Firstly, we consider a method based on the causal Bayesian networks describing the causal relations between the parties. In contrast to the Bell-nonlocality scenario, where this method has previously been shown to be ineffective, we show that it leads to several interesting entropic causal inequalities. Secondly, we consider an alternative method based on counterfactual variables that has previously been used to derive entropic Bell inequalities. We compare the inequalities obtained via these two methods and discuss their violation by noncausal correlations. As an application of our approach, we derive bounds on the quantity of information - which is more naturally expressed in the entropic framework - that parties can communicate when operating in a definite causal order., Comment: 21 pages, 3 figures; updated to published version

The term Einstein-Podolsky-Rosen steering refers to a quantum correlation intermediate between entanglement and Bell nonlocality, which has been connected to another fundamental quantum property: measurement incompatibility. In the finite-dimensional case, efficient computational methods to quantify steerability have been developed. In the infinite-dimensional case, however, less theoretical tools are available. Here, we approach the problem of steerability in the continuous variable case via a notion of state-channel correspondence, which generalizes the well-known Choi-Jamio\l{}kowski correspondence. Via our approach we are able to generalize the connection between steering and incompatibility to the continuous variable case and to connect the steerability of a state with the incompatibility breaking property of a quantum channel, e.g., noisy NOON states and amplitude damping channels. Moreover, we apply our methods to the Gaussian steering setting, proving, among other things, that canonical quadratures are sufficient for steering Gaussian states., Comment: 13 pages, 2 figures

We investigate the possibility of distinguishing among different causal relations starting from a limited set of marginals. Our main tool is the notion of adhesivity, that is, the extension of probability or entropies defined only on subsets of variables, which provides additional independence constraints among them. Our results provide a criterion for recognizing which causal structures are indistinguishable when only limited marginal information is accessible. Furthermore, the existence of such extensions greatly simplifies the characterization of a marginal scenario, a result that facilitates the derivation of Bell inequalities both in the probabilistic and entropic frameworks, and the identification of marginal scenarios where classical, quantum, and postquantum probabilities coincide., Comment: 16 pages, 9 figures

We introduce the concept of assemblage moment matrices, i.e., a collection of matrices of expectation values, each associated with a conditional quantum state obtained in a steering experiment. We demonstrate how it can be used for quantum states and measurements characterization in a device-independent manner, i.e., without invoking any assumption about the measurement or the preparation device. Specifically, we show how the method can be used to lower bound the steerability of an underlying quantum state directly from the observed correlation between measurement outcomes. Combining such device-independent quantifications with earlier results established by Piani and Watrous [Phys. Rev. Lett. 114, 060404 (2015)], our approach immediately provides a device-independent lower bound on the generalized robustness of entanglement, as well as the usefulness of the underlying quantum state for a type of subchannel discrimination problem. In addition, by proving a quantitative relationship between steering robustness and the recently introduced incompatibility robustness, our approach also allows for a device-independent quantification of the incompatibility between various measurements performed in a Bell-type experiment. Explicit examples where such bounds provide a kind of self-testing of the performed measurements are provided., Comment: The core of these results were already presented at the Workshop on Quantum Nonlocality, Causal Structure and Device-independent Quantum Information on 14/12/2016; v2: closely approximates journal version; v3: title is updated as journal version

We introduce the concept of entropic nonsignaling correlations, i.e., entropies arising from probabilistic theories that are compatible with the fact that we cannot transmit information instantaneously. We characterize and show the relevance of these entropic correlations in a variety of different scenarios, ranging from typical Bell experiments to more refined descriptions such as bilocality and information causality. In particular, we apply the framework to derive the first entropic inequality testing genuine tripartite nonlocality in quantum systems of arbitrary dimension and also prove the first known monogamy relation for entropic Bell inequalities. Further, within the context of complex Bell networks, we show that entropic nonlocal correlations can be activated., Comment: 14 pages (including appendix), 2 figures, final version

Quantum steering refers to the possibility for Alice to remotely steer Bob's state by performing local measurements on her half of a bipartite system. Two necessary ingredients for steering are entanglement and incompatibility of Alice's measurements. In particular, it has been recently proven that for the case of pure states of maximal Schmidt rank the problem of steerability for Bob's assemblage is equivalent to the problem of joint measurability for Alice's observables. We show that such an equivalence holds in general, namely, the steerability of any assemblage can always be formulated as a joint measurability problem, and vice versa. We use this connection to introduce steering inequalities from joint measurability criteria and develop quantifiers for the incompatibility of measurements., Comment: Definition of the SDP in Eq. (31) corrected, thanks to a comment by Marco Piani and Benjamin A. Ross

Hypergraph states form a family of multiparticle quantum states that generalizes the well-known concept of Greenberger-Horne-Zeilinger states, cluster states, and more broadly graph states. We study the nonlocal properties of quantum hypergraph states. We demonstrate that the correlations in hypergraph states can be used to derive various types of nonlocality proofs, including Hardy-type arguments and Bell inequalities for genuine multiparticle nonlocality. Moreover, we show that hypergraph states allow for an exponentially increasing violation of local realism which is robust against loss of particles. Our results suggest that certain classes of hypergraph states are novel resources for quantum metrology and measurement-based quantum computation., Comment: 31 pages, 1 figure, v3: final version

We show how a test of macroscopic realism based on Leggett-Garg inequalities (LGIs) can be performed in a macroscopic system. Using a continuous-variable approach, we consider quantum non-demolition (QND) measurements applied to atomic ensembles undergoing magnetically-driven coherent oscillation. We identify measurement schemes requiring only Gaussian states as inputs and giving a significant LGI violation with realistic experimental parameters and imperfections. The predicted violation is shown to be due to true quantum effects rather than to a classical invasivity of the measurement. Using QND measurements to tighten the "clumsiness loophole" forces the stubborn macrorealist to re-create quantum back action in his or her account of measurement.

We present a general framework for contextuality tests in phase space using displacement operators. First, we derive a general condition that a single-mode displacement operator should fulfill in order to construct Peres-Mermin square and similar scenarios. This approach offers a straightforward scheme for experimental implementations of the tests via modular variable measurements. In addition to the continuous variable case, our condition can also be applied to finite-dimensional systems in discrete phase space, using Heisenberg-Weyl operators. This approach, therefore, offers a unified picture of contextuality with a geometric flavor., Comment: 5+3 pages, 4 figures. Missing appendix added

We solve the problem of whether a set of quantum tests reveals state-independent contextuality and use this result to identify the simplest set of the minimal dimension. We also show that identifying state-independent contextuality graphs [R. Ramanathan and P. Horodecki, Phys. Rev. Lett. 112, 040404 (2014)] is not sufficient for revealing state-independent contextuality., Comment: 5 pages, 3 graphs

We show that the quantum bound for temporal correlations in a Leggett-Garg test, analogous to the Tsirelson bound for spatial correlations in a Bell test, strongly depends on the number of levels $N$ that can be accessed by the measurement apparatus via projective measurements. We provide exact bounds for small $N$, that exceed the known bound for the Leggett-Garg inequality, and show that in the limit $N\rightarrow \infty$ the Leggett-Garg inequality can be violated up to its algebraic maximum., Comment: 6 pages, 2 figures

Sequential measurements on a single particle play an important role in fundamental tests of quantum mechanics. We provide a general method to analyze temporal quantum correlations, which allows us to compute the maximal correlations for sequential measurements in quantum mechanics. As an application, we present the full characterization of temporal correlations in the simplest Leggett-Garg scenario and in the sequential measurement scenario associated with the most fundamental proof of the Kochen-Specker theorem., Comment: 8 pages, 2 figures

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. The resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness., Comment: 12 pages, no figures

The problem of separating classical from quantum correlations is in general intractable and has been solved explicitly only in few cases. In particular, known methods cannot provide general solutions for an arbitrary number of settings. We provide the complete characterization of the classical correlations and the corresponding maximal quantum violations for the case of n >= 4 observables X_0, ...,X_{n-1}, where each consecutive pair {X_i,X_{i+1}}, sum modulo n, is jointly measurable. This generalizes both the Clauser-Horne-Shimony-Holt and the Klyachko-Can-Binicioglu-Shumovsky scenarios, which are the simplest ones for, respectively, locality and noncontextuality. In addition, we provide explicit quantum states and settings with maximal quantum violation and minimal quantum dimension., Comment: 5+3 pages, 4 figures, 11 countries. v3: Published version

Contextuality is a natural generalization of nonlocality which does not need composite systems or spacelike separation and offers a wider spectrum of interesting phenomena. Most notably, in quantum mechanics there exist scenarios where the contextual behavior is independent of the quantum state. We show that the quest for an optimal inequality separating quantum from classical noncontextual correlations in an state-independent manner admits an exact solution, as it can be formulated as a linear program. We introduce the noncontextuality polytope as a generalization of the locality polytope, and apply our method to identify two different tight optimal inequalities for the most fundamental quantum scenario with state-independent contextuality., Comment: REVTeX4.1, 5 pages, 1 figure; v2: improved presentation and significantly extended results

Tight Bell inequalities are facets of Pitowsky's correlation polytope and are usually obtained from its extreme points by solving the hull problem. Here we present an alternative method based on a combination of algebraic results on extensions of measures and variable elimination methods, e.g., the Fourier-Motzkin method. Our method is shown to overcome some of the computational difficulties associated with the hull problem in some non-trivial cases. Moreover, it provides an explanation for the arising of only a finite number of families of Bell inequalities in measurement scenarios where one experimenter can choose between an arbitrary number of different measurements.

The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations < A_i >, < B >, < A_i B >, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation < A_1 A_2 > associated to different choices for B. The same analysis applies to the Hardy model and to the "perfect correlations" discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality., Comment: Some modifications have been done in order to improve clarity of presentation and comparison with other approaches

Alternative partial Boolean structures, implicit in the discussion of classical representability of sets of quantum mechanical predictions, are characterized, with definite general conclusions on the equivalence of the approaches going back to Bell and Kochen-Specker. An algebraic approach is presented, allowing for a discussion of partial classical extension, amounting to reduction of the number of contexts, classical representability arising as a special case. As a result, known techniques are generalized and some of the associated computational difficulties overcome. The implications on the discussion of Boole-Bell inequalities are indicated., Comment: A number of misprints have been corrected and some terminology changed in order to avoid possible ambiguities