Reasoning about Bell nonlocality from the correlations observed in post-selected data is always a matter of concern. This is because conditioning on the outcomes is a source of non-causal correlations, known as a selection bias, rising doubts whether the conclusion concerns the actual causal process or maybe it is just an effect of processing the data. Yet, even in the idealised case without detection inefficiencies, post-selection is an integral part of every experimental design, not least because it is a part of the entanglement generation process itself. In this paper we discuss a broad class of scenarios with post-selection on multiple spatially distributed outcomes. A simple criterion is worked out, called the all-but-one principle, showing when the conclusions about nonlocality from breaking Bell inequalities with post-selected data remain in force. Generality of this result, attained by adopting the high-level diagrammatic tools of causal inference, provides safe grounds for systematic reasoning based on the standard form of multipartite Bell inequalities in a wide array of entanglement generation schemes without worrying about the dangers of selection bias.
Reasoning about Bell nonlocality from the correlations observed in post-selected data is always a matter of concern. This is because conditioning on the outcomes is a source of non-causal correlations, known as a selection bias, rising doubts whether the conclusion concerns the actual causal process or maybe it is just an effect of processing the data. Yet, even in the idealised case without detection inefficiencies, post-selection is an integral part of every experimental design, not least because it is a part of the entanglement generation process itself. In this paper we discuss a broad class of scenarios with post-selection on multiple spatially distributed outcomes. A simple criterion is worked out, called the all-but-one principle, showing when the conclusions about nonlocality from breaking Bell inequalities with post-selected data remain in force. Generality of this result, attained by adopting the high-level diagrammatic tools of causal inference, provides safe grounds for systematic reasoning based on the standard form of multipartite Bell inequalities in a wide array of entanglement generation schemes without worrying about the dangers of selection bias.
Reasoning about Bell nonlocality from the correlations observed in post-selected data is always a matter of concern. This is because conditioning on the outcomes is a source of non-causal correlations, known as a selection bias , rising doubts whether the conclusion concerns the actual causal process or maybe it is just an effect of processing the data. Yet, even in the idealised case without detection inefficiencies, post-selection is an integral part of experimental designs, not least because it is a part of the entanglement generation process itself. In this paper we discuss a broad class of scenarios with post-selection on multiple spatially distributed outcomes. A simple criterion is worked out, called the all-but-one principle, showing when the conclusions about nonlocality from breaking Bell inequalities with post-selected data remain in force. Generality of this result, attained by adopting the high-level diagrammatic tools of causal inference, provides safe grounds for systematic reasoning based on the standard form of multipartite Bell inequalities in a wide array of entanglement generation schemes, without worrying about the dangers of selection bias. In particular, it can be applied to post-selection defined by single-particle events in each detection chanel when the number of particles in the system is conserved.
As with entanglement, different forms of Bell nonlocality arise in the multipartite scenario. These can be defined in terms of relaxations of the causal assumptions in local hidden-variable theories. However, a characterisation of all the forms of multipartite nonlocality has until now been out of reach, mainly due to the complexity of generic multipartite causal models. Here, we employ the formalism of Bayesian networks to reveal connections among different causal structures that make a both practical and physically meaningful classification possible. Our framework holds for arbitrarily many parties. We apply it to study the tripartite scenario in detail, where we fully characterize all the nonlocality classes. Remarkably, we identify new highly nonlocal causal structures that cannot reproduce all quantum correlations. This shows, to our knowledge, the strongest form of quantum multipartite nonlocality known to date. Finally, as a by-product result, we derive a non-trivial Bell-type inequality with no quantum violation. Our findings constitute a significant step forward in the understanding of multipartite Bell nonlocality and open several venues for future research.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated \nmeasurements on entangled quantum systems violate a Bell inequality [Bell1964]. \nWe put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate \nparticular maps, or functions to particular theories. The violation of a Bell \ninequality then implies the ability to perform some functions, or computations \nthat classical, or more generally, local hidden variable (LHV) theories cannot. \nWe derive an in�finite class of Bell inequalities that establish a link to so-called "non-local games" [Cleve2004]. We then make the connection between \nRaussendorf and Briegel's formulation of Measurement-based Quantum Computing (MBQC) [Raussendorf2001], and these non-local games. Not only can we show that a quantum violation implies a computational advantage in this model, \nwe show that adaptive measurements are required to perform all quantum computations. \nFinally, we explore post-selection of data in Bell tests from both a practical and \nconceptual point-of-view, with particular consideration to so-called "loopholes". \nLoopholes allow LHV theories to simulate quantum correlations through post-selection. \nWe give a computational description of how loopholes can emerge in diff�erent post-selection scenarios. This motivates us to �find a form of post-selection \nthat does not lead to loopholes. Central again to this discussion is the \ndescription of LHV theories in terms of computations. \nInterestingly, quantum correlators can be made more "non-classical" with this \nloophole-free post-selection. This method of post-selection also can simulate \ninformation processing tasks, such as MBQC, that have time-like separated components. \nThis opens up new avenues for the study of time-like tasks studied \nwithin the space-like separated scenario of the Bell test.
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
Discarding undesirable measurement results in Bell experiments opens the detection loophole that prevents a conclusive demonstration of nonlocality. As closing the detection loophole represents a major technical challenge for many practical Bell experiments, it is customary to assume the so-called fair sampling assumption (FSA) that, in its original form, states that the collectively postselected statistics are a fair sample of the ideal statistics. Here, we analyze the FSA from the viewpoint of causal inference: We derive a causal structure that must be present in any causal model that faithfully encapsulates the FSA. This provides an easy, intuitive, and unifying approach that includes different accepted forms of the FSA and underlines what is really assumed when using the FSA. We then show that the FSA can not only be applied in scenarios with non-ideal detectors or transmission losses, but also in ideal experiments where only parts of the correlations are postselected, e.g., when the particles' destinations are in a superposition state. Finally, we demonstrate that the FSA is also applicable in multipartite scenarios that test for (genuine) multipartite nonlocality.
The no-signalling principle lies at the heart of Bell-type experiments involving spacelike-separated parties. Although not all no-signalling correlations can be realized within quantum theory, the no-sinalling principle itself remains fundamental even for post-quantum theories, since its violation would imply superluminal signalling. In the spirit of device independent nature of the Bell inequalities, there has recently been growing interest in causal inequalities, which identify correlations admitting no causal explanation. Such correlations have been rigorously studied within the process-matrix formalism, which is built upon a constraint analogous to spacelike separation in Bell scenarios. We refer to this constraint as the single-interaction constraint, whereby each party interacts with the environment only once. Despite considerable effort, no fundamental guiding principle is currently known that characterizes the restrictions this constraint imposes on the correlations generated by Process Matrices. In this work, we address this gap by proposing a generalized causality principle for the bipartite case. We show that for certain cases, the generalised causality principle defines a strict subset of the set of all signalling correlations. This principle thus provides a fundamental limit on quantum and even post-quantum theories subject to the single-interaction constraint. By analogy with no-signalling, whose violation excludes certain causal structures among the parties, a violation of the generalised causality principle rules out a corresponding class of causal structures(causal structures with the single interaction constraint). More broadly, our techniques provide a general framework for deriving generalised causal constraints, applicable to a wide class of indefinite-causal-order scenarios.