scholarly journals Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 280 ◽  
Author(s):  
Elie Wolfe ◽  
David Schmid ◽  
Ana Belén Sainz ◽  
Ravi Kunjwal ◽  
Robert W. Spekkens
Keyword(s):  
Lower Bound ◽  
Causal Model ◽  
Causal Structure ◽  
Bell Inequalities ◽  
Resource Theory ◽  
Theoretic Approach ◽  
Binary Outcome ◽  
Common Cause ◽  
Complete Set ◽  
Probabilistic Processes

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.

scholarly journals The Inflation Technique for Causal Inference with Latent Variables

2019 ◽  
Vol 7 (2) ◽  
Author(s):  
Elie Wolfe ◽  
Robert W. Spekkens ◽  
Tobias Fritz
Keyword(s):  
Probability Distribution ◽  
Causal Inference ◽  
Latent Variables ◽  
Causal Model ◽  
Causal Structure ◽  
Bell Inequalities ◽  
Difficult Case ◽  
Common Causes ◽  
Multiple Copies ◽  
New Inequalities

AbstractThe problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the inflation technique for tackling this problem. An inflation of a causal structure is a new causal structure that can contain multiple copies of each of the original variables, but where the ancestry of each copy mirrors that of the original. To every distribution of the observed variables that is compatible with the original causal structure, we assign a family of marginal distributions on certain subsets of the copies that are compatible with the inflated causal structure. It follows that compatibility constraints for the inflation can be translated into compatibility constraints for the original causal structure. Even if the constraints at the level of inflation are weak, such as observable statistical independences implied by disjoint causal ancestry, the translated constraints can be strong. We apply this method to derive new inequalities whose violation by a distribution witnesses that distribution’s incompatibility with the causal structure (of which Bell inequalities and Pearl’s instrumental inequality are prominent examples). We describe an algorithm for deriving all such inequalities for the original causal structure that follow from ancestral independences in the inflation. For three observed binary variables with pairwise common causes, it yields inequalities that are stronger in at least some aspects than those obtainable by existing methods. We also describe an algorithm that derives a weaker set of inequalities but is more efficient. Finally, we discuss which inflations are such that the inequalities one obtains from them remain valid even for quantum (and post-quantum) generalizations of the notion of a causal model.

scholarly journals Experimental Connection between the Instrumental and Bell Inequalities

Proceedings ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 27
Author(s):  
Iris Agresti ◽  
Gonzalo Carvacho ◽  
Davide Poderini ◽  
Leandro Aolita ◽  
Rafael Chaves ◽  
...  
Keyword(s):  
Causal Model ◽  
Probability Distributions ◽  
Causal Structure ◽  
Joint Probability ◽  
Bell Inequalities ◽  
Quantum Correlations ◽  
Causal Processes ◽  
Chsh Inequality ◽  
Joint Probability Distributions ◽  
Selection Of

An investigated process can be studied in terms of the causal relations among the involved variables, representing it as a causal model. Some causal models are particularly relevant, since they can be tested through mathematical constraints between the joint probability distributions of the observables. This is a valuable tool because, if some data violates the constraints of a causal model, the implication is that the observed statistics is not compatible with that causal structure. Strikingly, when non-classical correlations come to play, a discrepancy between classical and quantum causal predictions can arise, producing a quantum violation of the classical causal constraints. The simplestscenario admitting such quantum violation is given by the instrumental causal processes. Here, we experimentally violate an instrumental test on a photonic platform and show how the quantum correlations violating the CHSH inequality can be mapped into correlations violating an instrumental test, despite the different forms of non-locality they display. Indeed, starting from a Bell-like scenario, we recover the violation of the instrumental scenario through a map between the two behaviours, which includes a post-selection of data and then we test an alternative way to violate the CHSH inequality, adopting the instrumental process platform.

scholarly journals Cyclic quantum causal models

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Jonathan Barrett ◽  
Robin Lorenz ◽  
Ognyan Oreshkov
Keyword(s):  
Causal Reasoning ◽  
Causal Structure ◽  
Bell Inequalities ◽  
Quantum Correlations ◽  
Quantum Systems ◽  
Causal Models ◽  
Causal Order ◽  
Quantum Processes ◽  
Causal Explanations ◽  
Causal Structures

AbstractCausal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory encompassing quantum systems and gravity is expected to allow causally nonseparable processes featuring operations in indefinite causal order, defying that events be causally ordered at all. The first challenge has been addressed through the recent development of intrinsically quantum causal models, allowing causal explanations of quantum processes – provided they admit a definite causal order, i.e. have an acyclic causal structure. This work addresses causally nonseparable processes and offers a causal perspective on them through extending quantum causal models to cyclic causal structures. Among other applications of the approach, it is shown that all unitarily extendible bipartite processes are causally separable and that for unitary processes, causal nonseparability and cyclicity of their causal structure are equivalent.

scholarly journals Invariant Causal Prediction for Nonlinear Models

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Christina Heinze-Deml ◽  
Jonas Peters ◽  
Nicolai Meinshausen
Keyword(s):  
Linear Models ◽  
Causal Model ◽  
Nonlinear Models ◽  
Causal Structure ◽  
Nonparametric Tests ◽  
Population Projections ◽  
World Population ◽  
Residual Distribution ◽  
Using Data

AbstractAn important problem in many domains is to predict how a system will respond to interventions. This task is inherently linked to estimating the system’s underlying causal structure. To this end, Invariant Causal Prediction (ICP) [1] has been proposed which learns a causal model exploiting the invariance of causal relations using data from different environments. When considering linear models, the implementation of ICP is relatively straightforward. However, the nonlinear case is more challenging due to the difficulty of performing nonparametric tests for conditional independence.In this work, we present and evaluate an array of methods for nonlinear and nonparametric versions of ICP for learning the causal parents of given target variables. We find that an approach which first fits a nonlinear model with data pooled over all environments and then tests for differences between the residual distributions across environments is quite robust across a large variety of simulation settings. We call this procedure “invariant residual distribution test”. In general, we observe that the performance of all approaches is critically dependent on the true (unknown) causal structure and it becomes challenging to achieve high power if the parental set includes more than two variables.As a real-world example, we consider fertility rate modeling which is central to world population projections. We explore predicting the effect of hypothetical interventions using the accepted models from nonlinear ICP. The results reaffirm the previously observed central causal role of child mortality rates.

A Game Theoretic Approach to Multi-Period Newsvendor Problems with Censored Markovian Demand

2019 ◽  
Vol 01 (01) ◽  
Author(s):  
Farzaneh Mansourifard ◽  
Parisa Mansourifard ◽  
Bhaskar Krishnamachari
Keyword(s):  
Lower Bound ◽  
Convex Combination ◽  
Repeated Game ◽  
Theoretic Approach ◽  
Worst Case ◽  
Point Distribution ◽  
Demand Distribution ◽  
Deterministic Solution ◽  
Uncertainty Set ◽  
Game Theoretic

This paper studies the Newsvendor problem for a setting in which (i) the demand is temporally correlated, (ii) the demand is censored, (iii) the distribution of the demand is unknown. The correlation is modeled as a Markovian process. The censoring means that if the demand is larger than the action (selected inventory), only a lower bound on the demand can be revealed. The uncertainty set on the demand distribution is given by only the upper and lower bound on the amount of the change from a time to the next time. We propose a robust approach to minimize the worst-case total cost and model it as a min-max zero-sum repeated game. We prove that the worst-case distribution of the adversary at each time is a two-point distribution with non-zero probabilities at the extrema of the uncertainty set of the demand. And the optimal action of the decision-maker can have any of the following structures: (i) a randomized solution with a two-point distribution at the extrema, (ii) a deterministic solution at a convex combination of the extrema. Both above solutions balance over-utilization and under-utilization costs. Finally, we extend our results to uni-model cost functions and present numerical results to study the solution.

scholarly journals Defining Metabolic Syndrome: Which Kind of Causality, if any, is Required?

Disputatio ◽  
2017 ◽  
Vol 9 (47) ◽  
pp. 553-580
Author(s):  
Margherita Benzi
Keyword(s):  
Metabolic Syndrome ◽  
Causal Model ◽  
Causal Structure ◽  
Causal Knowledge ◽  
Underlying Mechanisms ◽  
Definition Of ◽  
Type 3 ◽  
Causal Specificity

Abstract The definition of metabolic syndrome (MetS) has been, and still is, extremely controversial. My purpose is not to give a solution to the associated debate but to argue that the controversy is at least partially due to the different ‘causal content’ of the various definitions: their theoretical validity and practical utility can be evaluated by reconstructing or making explicit the underlying causal structure. I will therefore propose to distinguish the alternative definitions according to the kinds of causal content they carry: (1) definitions grounded on associations, (2) definitions presupposing a causal model built upon statistical associations, and (3) definitions grounded on underlying mechanisms. I suggest that analysing definitions according to their causal content can be helpful in evaluating alternative definitions of some diseases. I want to show how the controversy over MetS suggests a distinction among three kinds of definitions based on how explicitly they characterise the syndrome in causal terms, and on the type of causality involved. I will call ‘type 1 definitions’ those definitions that are purely associative; ‘type 2 definitions’ the definitions based on statistical associations, plus generic medical and causal knowledge; and ‘type 3 definitions’ the definitions based on (hypotheses about) mechanisms. These kinds of definitions, although different, can be related to each other. A definition with more specific causal content may be useful in the evaluation of definitions characterised by a lower degree of causal specificity. Moreover, the identification of the type of causality involved is of help to constitute a good criterion for choosing among different definitions of a pathological entity. In section (1) I introduce the controversy about MetS, in section (2) I propose some remarks about medical definitions and their ‘causal import’, and in section (3) I suggest that the different attitudes towards the definition of MetS are relevant to evaluate their explicative power.

scholarly journals An Information-Theoretic Approach on Causal Structure Learning for Heterogeneous Data Characteristics of Real-World Scenarios

2021 ◽  
Author(s):  
Johannes Huegle
Keyword(s):  
Real World ◽  
Structure Learning ◽  
Causal Structure ◽  
Heterogeneous Data ◽  
Theoretic Approach ◽  
Causal Relationships ◽  
Information Theoretic ◽  
Essential Components ◽  
Discrete Manufacturing ◽  
Information Theoretic Approach

While the knowledge about the structures of a system’s underlying causal relationships is crucial within many real-world scenarios, the omnipresence of heterogeneous data characteristics impedes applying methods for causal structure learning (CSL). In this dissertation project, we reduce the barriers for the transfer of CSL into practice with threefold contributions: (1) We derive an information-theoretic conditional independence test that, incorporated into methods for CSL, improves the accuracy for non-linear and mixed discrete-continuous causal relationships; (2) We develop a modular pipeline that covers the essential components required for a comprehensive benchmarking to support the transferability into practice; (3) We evaluate opportunities and challenges of CSL within different real-world scenarios from genetics and discrete manufacturing to demonstrate the accuracy of our approach in practice.

Sign in / Sign up

Export Citation Format

Share Document