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arXiv:2211.13349 · PRX Quantum

Witnessing Nonclassicality in a Causal Structure with Three Observable Variables

Pedro Lauand, Davide Poderini, Ranieri Nery, George Moreno, Lucas da Silva Pollyceno, Rafael Rabelo, Rafael Chaves

Published Apr 20, 2023• 8 citations
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Abstract

Seen from the modern lens of causal inference, Bell's theorem is nothing other than the proof that a specific classical causal model cannot explain quantum correlations.It is thus natural to move beyond Bell's paradigmatic scenario and consider different causal structures.For the specific case of three observable variables, it is known that there are three nontrivial causal networks.Two of those are known to give rise to quantum nonclassicality: the instrumental and the triangle scenarios.Here we analyze the third and remaining one, which we name the Evans scenario, akin to the causal structure underlying the entanglement-swapping experiment.We prove a number of results about this elusive scenario and introduce new and efficient computational tools for its analysis that can also be adapted to deal with more general causal structures.We do not solve its main open problem-whether quantum nonclassical correlations can arise from it-but give a significant step in this direction by proving that postquantum correlations, analogous to the paradigmatic Popescu-Rohrlich box, do violate the constraints imposed by a classical description of the Evans causal structure.

DOI
10.1103/prxquantum.4.020311

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Seen from the modern lens of causal inference, Bell's theorem is nothing other than the proof that a specific classical causal model cannot explain quantum correlations.It is thus natural to move beyond Bell's paradigmatic scenario and consider different causal structures.For the specific case of three observable variables, it is known that there are three nontrivial causal networks.Two of those are known to give rise to quantum nonclassicality: the instrumental and the triangle scenarios.Here we analyze the third and remaining one, which we name the Evans scenario, akin to the causal structure underlying the entanglement-swapping experiment.We prove a number of results about this elusive scenario and introduce new and efficient computational tools for its analysis that can also be adapted to deal with more general causal structures.We do not solve its main open problem-whether quantum nonclassical correlations can arise from it-but give a significant step in this direction by proving that postquantum correlations, analogous to the paradigmatic Popescu-Rohrlich box, do violate the constraints imposed by a classical description of the Evans causal structure.
In a Bell experiment, it is natural to seek a causal account of correlations wherein only a common cause acts on the outcomes. For this causal structure, Bell inequality violations can be explained only if causal dependencies are modeled as intrinsically quantum. There also exists a vast landscape of causal structures beyond Bell that can witness nonclassicality, in some cases without even requiring free external inputs. Here, we undertake a photonic experiment realizing one such example: the triangle causal network, consisting of three measurement stations pairwise connected by common causes and no external inputs. To demonstrate the nonclassicality of the data, we adapt and improve three known techniques: (i) a machine-learning-based heuristic test, (ii) a data-seeded inflation technique generating polynomial Bell-type inequalities and (iii) entropic inequalities. The demonstrated experimental and data analysis tools are broadly applicable paving the way for future networks of growing complexity.
Bell's theorem can be understood as establishing the incompatibility of quantum correlations with any classical causal explanation. Causal inference, however, relies not only on passive observations but also on interventions. Here we ask when the observational and interventional data collected on a given causal structure can all be reproduced by a single classical causal model, which is a particular instance of the data fusion problem in causal inference [Bareinboim and Pearl, PNAS 113, 7345 (2016)]. We show that quantum systems give rise to a new form of nonclassicality in this setting, namely: when the data tables associated with each (non)intervention paradigm admit classical explanations, but no single classical model explains the different intervention paradigms when considered jointly. We call this nonclassicality in the synthesis of the data fusion. We assess the prevalence of such fusion by partitioning all marginalized causal structures over three observed variables into exactly two categories: those which can and those which cannot exhibit such nonclassicality. For all those that can, we present explicit quantum violations of the associated causal inequalities.
Bell's theorem is often considered the most stringent notion of nonclassicality. The generalization of Bell's theorem to causal networks offers interesting new perspectives on the phenomenon of quantum nonclassicality and prompts us with a fundamental inquiry: what is the simplest scenario leading to the incompatibility between quantum correlations and the classical theory of causality? Here, we demonstrate that quantum nonclassicality is possible in an entanglement swapping network consisting of only three dichotomic variables, without the need for the locality assumption or external measurement choices. We also show that interventions, a central tool in the field of causal inference, significantly improve the noise robustness of this new kind of nonclassical behavior, making it feasible for experimental tests with current technology.
In a Bell experiment, it is natural to seek a causal account of correlations wherein only a common cause acts on the outcomes. For this causal structure, Bell inequality violations can be explained only if causal dependencies are modelled as intrinsically quantum. There also exists a vast landscape of causal structures beyond Bell that can witness nonclassicality, in some cases without even requiring free external inputs. Here, we undertake a photonic experiment realizing one such example: the triangle causal network, consisting of three measurement stations pairwise connected by common causes and no external inputs. To demonstrate the nonclassicality of the data, we adapt and improve three known techniques: (i) a machine-learning-based heuristic test, (ii) a data-seeded inflation technique generating polynomial Bell-type inequalities and (iii) entropic inequalities. The demonstrated experimental and data analysis tools are broadly applicable paving the way for future networks of growing complexity.
Since Bell's theorem, it is known that local realism fails to explain quantum phenomena. Bell inequality violations manifestly show the incompatibility of quantum theory with classical notions of cause and effect. As recently found, however, the instrumental scenario-a pivotal tool in causal inference-allows for nonclassicality signatures going beyond this paradigm. If we are not limited to observational data and can intervene in our setup, then we can witness quantum violations of classical bounds on the causal influence among the involved variables even when no Bell-like violation is possible. That is, through interventions, the quantum behavior of a system that would seem classical can be demonstrated. Using a photonic setup-faithfully implementing the instrumental causal structure and switching between observation and intervention run by run-we experimentally witness such a nonclassicality. We also test quantum bounds for the causal influence, showing that they provide a reliable tool for quantum causal modeling.
It has long been recognized that certain quantum correlations are incompatible with a particular assumption about classical causal structure. Given a causal structure of unknown classicality, the presence of such correlations certifies the nonclassical nature of the causal structure in a device-independent fashion. In structures where all parties share a common resource, these nonclassical correlations are also known as nonlocal correlations. Any constraint satisfied by all correlations which are classically compatible with a given causal structure defines a causal compatibility criterion. Such criteria were recently derived for the triangle structure (E. Wolfe et al., arXiv:1609.00672) in the form of polynomial inequalities, begging the question of whether any of those inequalities admit violation by quantum correlations. Numerical investigation suggests that they do not, and we further conjecture that the set of correlations admitted by the classical triangle structure is equivalent to the set of correlations admitted by its quantum generalization whenever the three observable variables are binary. Our main contribution in this work, however, is the derivation of causal compatibility inequalities for the triangle structure which do admit quantum violation. This provides a robust-to-noise witness of quantum correlations in the triangle structure. We conclude by considering the possibility of quantum resources potentially qualitatively different from those known previously.