arXiv subjects

Ana Belén Sainz

Publications and source records attributed to Ana Belén Sainz.

44 records · Page 3Linked to original sources

arXiv2017

Multipartite Composition of Contextuality Scenarios

Contextuality is a particular quantum phenomenon that has no analogue in classical probability theory. Given two independent systems, a natural question is how to represent such a situation as a single test space. In other words, how separate contextuality scenarios combine into a joint scenario. Under the premise that the the allowed probabilistic models satisfy the No Signalling principle, Foulis and Randall defined the unique possible way to compose two contextuality scenarios. When composing strictly-more than two test spaces, however, a variety of possible composition methods have been conceived. Nevertheless, all these formally-distinct composition methods appear to give rise to observationally equivalent scenarios, in the sense that the different compositions all allow precisely the same sets of classical and quantum probabilistic models. This raises the question of whether this property of invariance-under-composition-method is special to classical and quantum probabilistic models, or if it generalizes to other probabilistic models as well, our particular focus being $\mathcal{Q}_1$ models. $\mathcal{Q}_1$ models are physically important since, when applied to scenarios constructed by a particular composition rule, coincide with the well-defined "Almost Quantum Correlations". In this work we see that some composition rules, however, give rise to scenarios with inequivalent allowed sets of $\mathcal{Q}_1$ models. We find that the non-trivial dependence of $\mathcal{Q}_1$ models on the choice of composition method is apparently an artifact of failure of those composition rules to capture the orthogonality relations given by the Local Orthogonality principle. We prove that $\mathcal{Q}_1$ models satisfy invariance-under-composition-method for all the constructive compositions protocols which do capture this notion of Local Orthogonality.

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arXiv2016

On the tightness of correlation inequalities with no quantum violation

We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question whether every proper (tight) Bell inequality for two parties, other than the trivial ones from positivity, normalization and no-signaling can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games, illustrating it for the case of three outcomes. We then show as a main result that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the non-local computation games do not constitute tight Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest CHSH Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the non-contextual polytope of single-party correlation inequalities with the cut polytope $CUT(\nabla G)$, where $G$ denotes the compatibility graph of observables in the contextuality scenario and $\nabla G$ denotes the suspension graph of $G$. We observe that there exist tight non-contextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the Consistent Exclusivity principle.

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arXiv2016

Adjusting inequalities for detection-loophole-free steering experiments

We study the problem of certifying quantum steering in a detection-loophole-free manner in experimental situations that require post-selection. We present a method to find the modified local-hidden-state bound of steering inequalities in such a post-selected scenario. We then present a construction of linear steering inequalities in arbitrary finite dimension and show that they certify steering in a loophole-free manner as long as the detection efficiencies are above the known bound below which steering can never be demonstrated. We also show how our method extends to the scenarios of multipartite steering and Bell nonlocality, in the general case where there can be correlations between the losses of the different parties. In both cases we present examples to demonstrate the techniques developed.

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arXiv2016

Classical communication cost of quantum steering

Quantum steering is observed when performing appropriate local measurements on an entangled state. Here we discuss the possibility of simulating classically this effect, using classical communication instead of entanglement. We show that infinite communication is necessary for exactly simulating steering for any pure entangled state, as well as for a class of mixed entangled states. Moreover, we discuss the communication cost of steering for general entangled states, as well as approximate simulation. Our findings reveal striking differences between Bell nonlocality and steering, and provide a natural way of measuring the strength of the latter.

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arXiv2016

A new property of the Lovász number and duality relations between graph parameters

We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $α(G) \leq \vartheta(G)$ between the independence number and the Lovász number of $G$ can be made arbitrarily tight: Precisely, the inequality \[ α(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H) \] becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that \[ α(G \times H) \leq α^*(G)\,α(H), \] with the fractional packing number $α^*(G)$, and for every $G$ there exists $H$ that makes the above an equality; conversely, for every graph $H$ there is a $G$ that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which $α$ and $α^*$ are dual to each other, and the Lovász number $\vartheta$ is self-dual. We also show duality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$ of the Lovász number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.

math.CO↗

arXiv2015

Macroscopic non-contextuality as a principle for Almost Quantum Correlations

Quantum mechanics allows only certain sets of experimental results (or "probabilistic models") for Bell-type quantum non-locality experiments. A derivation of this set from simple physical or information theoretic principles would represent an important step forward in our understanding of quantum mechanics, and this problem has been intensely investigated in recent years. "Macroscopic locality," which requires the recovery of locality in the limit of large numbers of trials, is one of several principles discussed in the literature that place a bound on the set of quantum probabilistic models. A similar question can also be asked about probabilistic models for the more general class of quantum contextuality experiments. Here, we extend the Macroscopic Locality principle to this more general setting, using the hypergraph approach of Acín, Fritz, Leverrier and Sainz [Comm. Math. Phys. 334(2), 533-628 (2015)], which provides a framework to study both phenomena of nonlocality and contextuality in a unified manner. We find that the set of probabilistic models allowed by our Macroscopic Non-Contextuality principle is equivalent to an important and previously studied set in this formalism, which is slightly larger than the quantum set. In the particular case of Bell Scenarios, this set is equivalent to the set of "Almost Quantum" models, which is of particular interest since the latter was recently shown to satisfy all but one of the principles that have been proposed to bound quantum probabilistic models, without being implied by any of them (or even their conjunction). Our condition is the first characterisation of the almost quantum set from a simple physical principle.

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arXiv2015

A Combinatorial Approach to Nonlocality and Contextuality

So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from a certain product of contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to several graph invariants. The recently proposed Local Orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is strictly dominated by a low level of the Navascués-Pironio-Acín hierarchy of semidefinite programs, which we also apply to contextuality scenarios. We derive a wealth of results in our framework, many of these relating to quantum and supraquantum contextuality and nonlocality, and state numerous open problems. For example, we show that the set of quantum models on a contextuality scenario can in general not be characterized in terms of a graph invariant. In terms of graph theory, our main result is this: there exist two graphs $G_1$ and $G_2$ with the properties \begin{align*} α(G_1) &= Θ(G_1), & α(G_2) &= \vartheta(G_2), \\[6pt] Θ(G_1\boxtimes G_2) & > Θ(G_1)\cdot Θ(G_2),& Θ(G_1 + G_2) & > Θ(G_1) + Θ(G_2). \end{align*}

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arXiv2014

Probabilistic models on contextuality scenarios

We introduce a framework to describe probabilistic models in Bell experiments, and more generally in contextuality scenarios. Such a scenario is a hypergraph whose vertices represent elementary events and hyperedges correspond to measurements. A probabilistic model on such a scenario associates to each event a probability, in such a way that events in a given measurement have a total probability equal to one. We discuss the advantages of this framework, like the unification of the notions of contexuality and nonlocality, and give a short overview of results obtained elsewhere.

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