Abstract
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.
Highlights
Bell’s theorem [1, 2] highlights a precise sense in which quantum theory requires a departure from a classical worldview
We argue that the natural set of free operations on such processes are those that are achieved by embedding the process in a circuit for which the only connection between the wings is a classical common cause, and we demonstrate that this is equivalent to the set of local operations and shared randomness, as the latter is formalized in Refs. [16, 17]
As noted in the introduction and as will be demonstrated in Section 3.2, in the special case of Bell scenarios—the focus of this article—the natural set of free operations within our causal modelling paradigm is equivalent to one of the proposals for the set of free operations made in earlier works within the strictly operational paradigm, namely, local operations and shared randomness (LOSR), as the latter is defined in Refs. [16, 17]
Summary
Bell’s theorem [1, 2] highlights a precise sense in which quantum theory requires a departure from a classical worldview. Violations of Bell inequalities provide a means for certifying the nonclassicality of nature, independently of the correctness of quantum theory. We take a resource-theoretic approach to quantifying the nonclassicality of a given correlation in a Bell scenario, grounded in a new perspective on Bell’s theorem. This is the perspective of causal modelling, which differs from the traditional operational approaches both conceptually and in practice. Our causal perspective on quantifying Bell nonclassicality generalizes naturally to a framework for quantifying the nonclassicality of correlations in more general causal scenarios We discuss this generalization, but leave its development to future work
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