A graph-separation theorem for quantum causal models

Pienaar, Jacques; Brukner, Časlav

doi:10.1088/1367-2630/17/7/073020

Cited by 84 publications

(111 citation statements)

References 23 publications

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“…Examples of causal structures that posit superluminal causal influences but no hidden variables to explain Bell correlations. 15 [53] describes a proposal to abandon the standard framework and revise the notion of a common cause in order to secure an explanation of quantum correlations wherein the quantum state acts as a common cause. Our own proposal for how one might modify the standard framework of causal models to secure a causal explanation of quantum correlations will be discussed in the conclusions.…”

Section: Causal Explanations Without Hidden Variablesmentioning

confidence: 99%

The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning

2015

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An active area of research in the fields of machine learning and statistics is the development of causal discovery algorithms, the purpose of which is to infer the causal relations that hold among a set of variables from the correlations that these exhibit . We apply some of these algorithms to the correlations that arise for entangled quantum systems. We show that they cannot distinguish correlations that satisfy Bell inequalities from correlations that violate Bell inequalities, and consequently that they cannot do justice to the challenges of explaining certain quantum correlations causally. Nonetheless, by adapting the conceptual tools of causal inference, we can show that any attempt to provide a causal explanation of nonsignalling correlations that violate a Bell inequality must contradict a core principle of these algorithms, namely, that an observed statistical independence between variables should not be explained by fine-tuning of the causal parameters. In particular, we demonstrate the need for such fine-tuning for most of the causal mechanisms that have been proposed to underlie Bell correlations, including superluminal causal influences, superdeterminism (that is, a denial of freedom of choice of settings), and retrocausal influences which do not introduce causal cycles. 4 Other work in the field of machine learning has appealed to statistical features besides CI relations, but not the features of correlations that are relevant for Bellʼs theorem. Peters et al [8] demonstrate that if one is promised an additive noise model, then features of the joint distribution can often distinguish cause from effect in the case of a distribution on a pair of variables, where there are no CI relations to guide the analysis. Other approaches have appealed to the complexity of conditional distributions [3][4][5]. 5 A defining feature of a common cause is that if the statistical dependence between two variables is to be explained entirely by a common cause, then it must be the case that conditioning on the common cause makes the variables statistically independent. As we will see, this feature is built into the framework of causal models. Statements of Reichenbachʼs principle often assert it explicitly. New J. Phys. 17 (2015) 033002 C J Wood and R W Spekkens New J. Phys. 17 (2015) 033002 C J Wood and R W Spekkens

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Section: Causal Explanations Without Hidden Variablesmentioning

confidence: 99%

The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning

2015

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“…The causal inequality(4) can be interpreted as a bound on the maximal probability of success for a bipartite 'guess your neighbor's input' (GYNI) game [29] with uniform input bits x y , (such that p x y , 1 4 ( ) = ), where Alice and Bob's task is to guess each other's input, i.e., to output a = y and b = x. Implicitly assuming uniform input bits 9 , inequality(4) can indeed be written in a more compact form as…”

Section: The Simplest Causal Polytopementioning

confidence: 99%

“…Using the Choi-Jamiołkowski (CJ) isomorphism [31,32], one can represent Alice's maps as some operators 10 As shown in [12], the assumption of local consistency with quantum theory implies that the probability p a b x y , , ( | )of observing the classical outputs a b , for a choice of instruments labelled by x y , is a bilinear function of Alice and Bob's maps, which can be written as 9 Note that the assumption of uniform inputs is only necessary to justify the shorthand notation p a y b x , ( ) = = for the left-hand side of equation (4), and to interpret it as the success probability for the GYNI game. Whether an inequality written as a combination of conditional probabilities (like (4) or (5) for instance) defines a causal inequality or not depends of course in no way on the distribution of inputs.…”

Section: The Process Matrix Frameworkmentioning

confidence: 99%

The simplest causal inequalities and their violation

et al. 2015

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In a scenario where two parties share, act on and exchange some physical resource, the assumption that the parties' actions are ordered according to a definite causal structure yields constraints on the possible correlations that can be established. We show that the set of correlations that are compatible with a definite causal order forms a polytope, whose facets define causal inequalities. We fully characterize this causal polytope in the simplest case of bipartite correlations with binary inputs and outputs. We find two families of nonequivalent causal inequalities; both can be violated in the recently introduced framework of process matrices, which extends the standard quantum formalism by relaxing the implicit assumption of a fixed causal structure. Our work paves the way to a more systematic investigation of causal inequalities in a theory-independent way, and of their violation within the framework of process matrices.This approach is more powerful as it can detect all causally nonseparable process matrices, while not all causally nonseparable process matrices can violate a causal inequality [13,14]. Furthermore, physical implementations of certain (multipartite) causally nonseparable process matrices, and of corresponding causal witnesses that detect their causal nonseparability, have been proposed [13][14][15] and even realized experimentally [16], while it is still not known whether there actually exist any physically realizable process that violates a causal inequality. Nevertheless, the device-independent approach is still of interest as it relaxes the requirement to trust the functioning and the operations implemented by one's devices in an experiment. It is furthermore also theoryindependent: causal inequalities can in principle be tested, and correlations with no definite causal order can be identified whatever the description of the physical resource is-whether we use the process matrix framework or any other theory to be discovered in the future. A related open question is whether the ability to violate causal inequalities can-in analogy with Bell nonlocality [17]-be exploited as a resource, just like causally nonseparable process matrices provide advantages for information-theoretical [18], computational [19], and communication complexity [20] tasks.Our paper aims at providing a better understanding of the device-independent characterization of correlations that are compatible with a definite causal order or not. We show that bipartite correlations with a definite causal order form a convex polytope, whose facets correspond to causal inequalities (section 2). We characterize this causal polytope in the simplest scenario where the two parties observe correlations with binary inputs and outputs, which gives us two families of new causal inequalities. We then investigate their possible violation in the framework of process matrices, and find that these can indeed be violated (section 3). This provides an example of 'noncausal' process matrix correlations in a simpler scenario than that considered i...

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“…An intrinsically quantum version of the d-separation theorem, by contrast, would be one which concerns the causal relations among quantum systems (see, for instance, Ref. [73]). If a set of nodes representing quantum systems can be described by a joint or conditional state, then one can seek to determine whether factorization conditions on this state are implied by d-separation relations among the quantum systems on the graph.…”

Section: Relation To Prior Workmentioning

confidence: 99%

Causality in the Quantum World

Pienaar¹

2017

Physics

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Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations. Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize. The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution. In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand. Motivated by this, we introduce a quantum counterpart to the principle. We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way. Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models and provide examples of how the formalism works.

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A graph-separation theorem for quantum causal models

Cited by 84 publications

References 23 publications

The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning

The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning

The simplest causal inequalities and their violation

Causality in the Quantum World

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