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arXiv:2407.02120 · Physical Review A

Convexity of noncontextual wirings and how they order the set of correlations

Tiago Felipe de Abreu Santos, Rafael Wagner, Bárbara Amaral

Published Sep 19, 2024• 0 citations
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Abstract

The resource theory of contextuality considers resourceful objects to be probabilistic data tables, known as correlations or behaviors, that fail to have an explanation in terms of Kochen-Specker noncontextual models. In this work, we advance this resource theory, considering free operations to be noncontextual wirings (NCWs). We show that all such wirings form a convex set. When restricted to Bell scenarios, we show that such wirings are not equivalent to local operations assisted by a common source of classical shared randomness (LOSR). The set of all NCW operations contains LOSR, but is strictly larger. We also prove several elementary facts about how different resources can be converted via NCW. As a concrete example, we show that there are pairs of behaviors that cannot be converted one into the other using NCW. Since resource conversion mathematically induces a pre-order over the set of all behaviors, our results reveal the intricate ordering induced by NCW in scenarios beyond Bell scenarios.

DOI
10.1103/physreva.110.032217

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The resource theory of contextuality considers resourceful objects to be probabilistic data tables, known as correlations or behaviors, that fail to have an explanation in terms of Kochen-Specker noncontextual models. In this work, we advance this resource theory, considering free operations to be noncontextual wirings (NCWs). We show that all such wirings form a convex set. When restricted to Bell scenarios, we show that such wirings are not equivalent to local operations assisted by a common source of classical shared randomness (LOSR). The set of all NCW operations contains LOSR, but is strictly larger. We also prove several elementary facts about how different resources can be converted via NCW. As a concrete example, we show that there are pairs of behaviors that cannot be converted one into the other using NCW. Since resource conversion mathematically induces a pre-order over the set of all behaviors, our results reveal the intricate ordering induced by NCW in scenarios beyond Bell scenarios.
Bell nonlocality can be formulated in terms of a resource theory with local-hidden variable models as resourceless objects. Two such theories are known, one built upon local operations assisted by shared randomness (LOSRs) and the other one allowing, in addition, for prior-to-input classical communication. We show that prior communication, although unable to create nonlocality, leads to wirings not only beyond LOSRs but also not contained in a much broader class of (nonlocality-generating) global wirings. Technically, this is shown by proving that it can improve the statistical distinguishability between Bell correlations optimized over all fixed measurement choices. This has implications in nonlocality quantification, and leads us to a natural universal definition of Bell nonlocality measures. To end up with, we also consider the statistical strength of nonlocality proofs. We point out some issues of its standard definition in the resource-theoretic operational framework, and suggest simple fixes for them. Our findings reveal nontrivial features of the geometry of the set of wirings and may have implications in the operational distinguishability of nonlocal behaviors.
We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables having a particular causal structure, namely, one wherein the wings are only connected by a common cause. We term them common-cause boxes. We define the distinction between classical and nonclassical resources in terms of whether or not a classical causal model can explain the correlations. One can then quantify the relative nonclassicality of resources by considering their interconvertibility relative to the set of operations that can be implemented using a classical common cause (which correspond to local operations and shared randomness). We prove that the set of free operations forms a polytope, which in turn allows us to derive an efficient algorithm for deciding whether one resource can be converted to another. We moreover define two distinct monotones with simple closed-form expressions in the two-party binary-setting binary-outcome scenario, and use these to reveal various properties of the pre-order of resources, including a lower bound on the cardinality of any complete set of monotones. In particular, we show that the information contained in the degrees of violation of facet-defining Bell inequalities is not sufficient for quantifying nonclassicality, even though it is sufficient for witnessing nonclassicality. Finally, we show that the continuous set of convexly extremal quantumly realizable correlations are all at the top of the pre-order of quantumly realizable correlations. In addition to providing new insights on Bell nonclassicality, our work also sets the stage for quantifying nonclassicality in more general causal networks.
We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables having a particular causal structure, namely, one wherein the wings are only connected by a common cause. We term them "common-cause boxes". We define the distinction between classical and nonclassical resources in terms of whether or not a classical causal model can explain the correlations. One can then quantify the relative nonclassicality of resources by considering their interconvertibility relative to the set of operations that can be implemented using a classical common cause (which correspond to local operations and shared randomness). We prove that the set of free operations forms a polytope, which in turn allows us to derive an efficient algorithm for deciding whether one resource can be converted to another. We moreover define two distinct monotones with simple closed-form expressions in the two-party binary-setting binary-outcome scenario, and use these to reveal various properties of the pre-order of resources, including a lower bound on the cardinality of any complete set of monotones. In particular, we show that the information contained in the degrees of violation of facet-defining Bell inequalities is not sufficient for quantifying nonclassicality, even though it is sufficient for witnessing nonclassicality. Finally, we show that the continuous set of convexly extremal quantumly realizable correlations are all at the top of the pre-order of quantumly realizable correlations. In addition to providing new insights on Bell nonclassicality, our work also sets the stage for quantifying nonclassicality in more general causal networks.
Contextuality is a fundamental feature of quantum theory necessary for certain models of quantum computation and communication. Serious steps have therefore been taken towards a formal framework for contextuality as an operational resource. However, the main ingredient of a resource theory-a concrete, explicit form of free operations of contextuality-was still missing. Here we provide such a component by introducing noncontextual wirings: a class of contextuality-free operations with a clear operational interpretation and a friendly parametrization. We characterize them completely for general black-box measurement devices with arbitrarily many inputs and outputs. As applications, we show that the relative entropy of contextuality is a contextuality monotone and that maximally contextual boxes that serve as contextuality bits exist for a broad class of scenarios. Our results complete a unified resource-theoretic framework for contextuality and Bell nonlocality.
A standard approach to quantifying resources is to determine which operations on the resources are freely available, and to deduce the partial order over resources that is induced by the relation of convertibility under the free operations. If the resource of interest is the nonclassicality of the correlations embodied in a quantum state, i.e., e n t a n g l e m e n t , then the common assumption is that the appropriate choice of free operations is Local Operations and Classical Communication (LOCC). We here advocate for the study of a different choice of free operations, namely, Local Operations and Shared Randomness (LOSR), and demonstrate its utility in understanding the interplay between the entanglement of states and the nonlocality of the correlations in Bell experiments. Specifically, we show that the LOSR paradigm (i) provides a resolution of the anomalies of nonlocality , wherein partially entangled states exhibit more nonlocality than maximally entangled states, (ii) entails new notions of genuine multipartite entanglement and nonlocality that are free of the pathological features of the conventional notions, and (iii) makes possible a resource-theoretic account of the self-testing of entangled states which generalizes and simplifies prior results. Along the way, we derive some fundamental results concerning the necessary and sufficient conditions for convertibility between pure entangled states under LOSR and highlight some of their consequences, such as the impossibility of catalysis for bipartite pure states. The resource-theoretic perspective also clarifies why it is neither surprising nor problematic that there are mixed entangled states which do not violate any Bell inequality. Our results motivate the study of LOSR-entanglement as a new branch of entanglement theory.
Kochen--Specker contextuality is a fundamental feature of quantum mechanics and a crucial resource for quantum computational advantage and reduction of communication complexity. Its presence is witnessed in empirical data by the violation of noncontextuality inequalities. However, all known noncontextuality inequalities corresponding to facets of noncontextual polytopes are either Bell inequalities or refer to cyclic or state-independent contextuality scenarios. We introduce a general method for lifting noncontextuality inequalities, deriving facets of noncontextual polytopes for more complex scenarios from known facets of simpler subscenarios. Concretely, starting from an arbitrary scenario, the addition of a new measurement or a new outcome preserves the facet-defining nature of any noncontextuality inequality. This extends the results of Pironio [J. Math. Phys. 46, 062112 (2005)] from Bell nonlocality scenarios to contextuality scenarios, unifying liftings of Bell and noncontextuality inequalities. Our method produces facet-defining noncontextuality inequalities in all scenarios with contextual correlations, and we present examples of facet-defining noncontextuality inequalities for scenarios where no examples were known. Our results shed light on the structure of noncontextuality polytopes and the relationship between such polytopes across different scenarios.