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arXiv:2308.02380 · Physical Review Research

Classifying causal structures: Ascertaining when classical correlations are constrained by inequalities

Shashaank Khanna, Marina Maciel Ansanelli, Matthew F. Pusey, Elie Wolfe

Published Apr 9, 2024• 4 citations
arXivHTMLPDF

Abstract

The classical causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independencies) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d separation, e separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal, and Pusey (HLP). Considering all scenarios with up to four observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.023038

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The classical causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independencies) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d separation, e separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal, and Pusey (HLP). Considering all scenarios with up to four observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive. Published by the American Physical Society 2024
Abstract For two causal structures with the same set of visible variables, one is said to observationally dominate the other if the set of distributions over the visible variables realizable by the first contains the set of distributions over the visible variables realizable by the second. Knowing such dominance relations is useful for adjudicating between these structures given observational data. Here, we consider the problem of determining the partial order of equivalence classes of causal structures with latent variables relative to observational dominance. We provide a complete characterization of the dominance order in the case of three visible variables, and a partial characterization in the case of four visible variables. Our techniques also help to identify which observational equivalence classes have a set of realizable distributions that is characterized by nontrivial inequality constraints, analogous to Bell inequalities and instrumental inequalities. We find evidence that as one increases the number of visible variables, the equivalence classes satisfying nontrivial inequality constraints become ubiquitous (Because such classes are the ones for which there can be a difference in the distributions that are quantumly and classically realizable, this implies that the potential for quantum-classical gaps is also ubiquitous). Furthermore, we find evidence that constraint-based causal discovery algorithms that rely solely on conditional independence constraints have a significantly weaker distinguishing power among observational equivalence classes than algorithms that go beyond these (i.e., algorithms that also leverage nested Markov constraints and inequality constraints).
Abstract A causal scenario is a graph that describes the cause and effect relationships between all relevant variables in an experiment. A scenario is deemed ‘not interesting’ if there is no device-independent way to distinguish the predictions of classical physics from any generalised probabilistic theory (including quantum mechanics). Conversely, an interesting scenario is one in which there exists a gap between the predictions of different operational probabilistic theories, as occurs for example in Bell-type experiments. Henson, Lal and Pusey (HLP) recently proposed a sufficient condition for a causal scenario to not be interesting. In this paper we supplement their analysis with some new techniques and results. We first show that existing graphical techniques due to Evans can be used to confirm by inspection that many graphs are interesting without having to explicitly search for inequality violations. For three exceptional cases—the graphs numbered # 15 , 16 , 20 in HLP—we show that there exist non-Shannon type entropic inequalities that imply these graphs are interesting. In doing so, we find that existing methods of entropic inequalities can be greatly enhanced by conditioning on the specific values of certain variables.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
Analyzing the geometry of correlation sets constrained by general causal structures is of paramount importance for foundational and quantum technology research. Addressing this task is generally challenging, prompting the development of diverse theoretical techniques for distinct scenarios. Recently, novel hybrid scenarios combining different causal assumptions within different parts of the causal structure have emerged. In this work, we extend a graph theoretical technique to explore classical, quantum, and no-signaling distributions in hybrid scenarios, where classical causal constraints and weaker no-signaling ones are used for different nodes of the causal structure. By mapping such causal relationships into an undirected graph we are able to characterize the associated sets of compatible distributions and analyze their relationships. In particular we show how with our method we can construct minimal Bell-like inequalities capable of simultaneously distinguishing classical, quantum, and no-signaling behaviors, and efficiently estimate the corresponding bounds. The demonstrated method will represent a powerful tool to study quantum networks and for applications in quantum information tasks.
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.