A graph-separation theorem for quantum causal models

A graph-separation theorem for quantum causal models

Jacques Pienaar jacques.pienaar@univie.ac.at Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria. Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria.    Časlav Brukner Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria. Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria.
Abstract

A causal model is an abstract representation of a physical system as a directed acyclic graph (DAG), where the statistical dependencies are encoded using a graphical criterion called ‘d-separation’. Recent work by Wood & Spekkens shows that causal models cannot, in general, provide a faithful representation of quantum systems. Since d-separation encodes a form of Reichenbach’s Common Cause Principle (RCCP), whose validity is questionable in quantum mechanics, we propose a generalised graph separation rule that does not assume the RCCP. We prove that the new rule faithfully captures the statistical dependencies between observables in a quantum network, encoded as a DAG, and reduces to d-separation in a classical limit. We note that the resulting model is still unable to give a faithful representation of correlations stronger than quantum mechanics, such as the Popescu-Rorlich box.

I Introduction

An essential problem faced by any scientist trying to make sense of the world is this: how do we infer causal relationships between the observed quantities, based only on information about their statistical dependencies? This problem is well known to statisticians and researchers working on Artificial Intelligence (AI), who have developed causal models as a tool for making causal inferences from a set of observed correlations. In most practical situations, the task is made easier by the availability of additional information and physical intuition. For example, in considering possible explanations for the observed correlation between smoking and cancer, we might consider it plausible that the two are independently caused by a common genetic factor, but few people would advocate the idea that having cancer causes people to smoke – not least because smoking tends to precede the onset of cancer, and we know that an effect cannot precede its cause. If we are simply told that two abstract variables X and Y have correlated values, the task is much more difficult. Such situations arise in theoretical work where one aims to relax the existing framework and construct more general models, or in practical applications like programming an AI to make causal inferences about data that it acquires.

In a causal model, defined in Sec II, the random variables of interest are represented by nodes and causal influences between them are represented by lines with arrows, called directed edges. The laws of physics require that no effect can be its own cause, leading to the requirement that the graph be acyclic (i.e. free of directed loops). The resulting directed acyclic graph (DAG) provides a computationally useful tool for extracting information about the statistical relationships of variables. In particular, it allows us to determine whether one set of variables is independent of any other set, conditional on the values of a third set. This information can be obtained directly from the graph using a simple algorithm, based on a concept called d-separation. Two sets of variables will be independent conditional on a third set of variables if and only if they are d-separated by the third set in the graph.

The proof that d-separation allows one to extract all (and only) correct information about the dependencies of the variables makes causal models particularly powerful tools for representing physical systems. Indeed, we are tempted to interpret the causal structure represented by the graph as “out there in the world” in the same sense as we can take the classical space-time manifold (which encodes causal relations between events) to be an independent and objectively defined structure. However, the program faces significant conceptual difficulties when one attempts to apply it to quantum physics. In principle, any observed probability distribution can be explained by some causal model, if we allow the possibility of hidden variables. However, as first clearly articulated by Bell Bell (1976), hidden-variable accounts of quantum mechanics can be challenged because they imply highly nonlocal behaviour of the model. This feature manifests itself in causal models in the form of fine-tuning, where one is forced to posit the existence of causal effects between variables whose statistics are independent. The fact that causal models of quantum systems require fine-tuning was recently shown by Wood & Spekkens Wood and Spekkens (2012).

These considerations revive an old question: what does causality really mean in the context of quantum mechanics? Do we accept that there exist nonlocal hidden variables whose direct influence is in principle unobservable at the statistical level? Or could it be that the classical concept of causality does not extend to quantum systems, and that we need a completely new way of determining whether two quantum events are causally related? Following the latter point of view, we define a causal model based on quantum networks and use it to derive a graph separation rule analagous to d-separation, for obtaining the conditional independence relations between variables. Our approach differs from previous work that assigns quantum amplitudes to the nodes in the DAG Tucci (2007), or that aims to replace the conditional probabilities at the nodes with some appropriate quantum analog Leifer and Spekkens (2013). Instead, we retain classical probability theory, but seek a physically motivated graphical representation of the causal structure that gives rise to the probability distributions predicted by quantum mechanics. Our approach is more closely aligned with previous work in which quantum network diagrams are used to obtain joint probabilities obeying standard probability theory Coecke and Paquette (2009); Oreshkov et al. (2012); Chiribella et al. (2009); Hardy (2007); Fritz (2014); Laskey (2007); Oreshkov and Cerf (2014). Particularly relevant is the recent work by Fritz Fritz (2014), in which a DAG representation of quantum correlations is proposed that encompasses our concept of a quantum causal model as will be discussed in Sec. III. Our work takes the additional step of defining a specific representation and a graph separation rule within this framework.

Recently, another DAG representation for general networks was proposed by Henson, Lal and Pusey Henson et al. (2014) in which d-separation continues to hold between the observed variables representing classical data. This is achieved by adding extra nodes to the graph representing ‘unobserved’ variables, which ensure that the restriction of the CI relations to just the observed nodes produces the conditional independencies expected of a quantum network (or generalised probabilistic theory). Our approach differs from these authors, in that we consider all nodes to be in principle observable; this leads us instead to modify the criterion for obtaining the CI relations from the graph (see Sec. III.3). The comparison to Ref. Henson et al. (2014) will be discussed further in Sec. IV.

The paper is organised as follows: in Sec. II we give a review of the relevant concepts concerning classical causal models and their graphical representation by DAGs. We include a discussion of the physical motivation for these models, and the meaning of the result in Ref. Wood and Spekkens (2012) that such models cannot faithfully represent quantum correlations. In Sec. III we aim to find such a faithful representation by re-interpreting the DAG as a quantum network. We thereby derive a new graph separation rule that does not obey the version of “Reichenbach’s Common Cause Principle”, which holds in the classical case, but instead obeys a weaker property we call the “Quantum Causality Condition”. We show that the d-separation can be recovered in a suitably defined classical limit, and we observe that super-quantum correlations (i.e. that exceed Tsirelson’s bound) still cannot be explained by our model without fine-tuning. We conclude in Sec. IV with a discussion about the physical interpretation of the result and possible directions for future work.

Ii Review of classical causal models

In this section, we review the basic definition of a causal model, here referred to as a classical causal model (CCM) to emphasise that it is tied to physical assumptions motivated by classical systems. For more details on causal models and inference, see the book by Pearl and references found therein Pearl (2000).

Before discussing the formal elements of these models, let us briefly recap their historical motivation. In science and statistics, one is often faced with the task of determining the causal relationships between random variables, given some sample data. We might observe that two variables are correlated, but this fact alone does not indicate the direction of the causal influence. If we are limited in our resources, we would like to know which set of follow-up experiments will most efficiently identify the direction of the causal influences, and which causal information can already be deduced from the existing data. Correlations between variables can be represented graphically, for example, we can require that be independent of conditional on a set whenever the removal of the nodes and their connections from the graph renders the sets of nodes and disconnected in the resulting graph. Such a rule for obtaining independence relations from a graph is referred to as a ‘graph separation rule’. Such graphs are called semi-graphoids and the independence relations they represent satisfy certain axioms, described in Sec II.2.

Correlations can be regarded as restrictions on the possible causal relationships between the variables. Two variables not connected by an edge cannot be directly causally connected, that is, if there is a causal influence of one on the other, it must be mediated by a third set of variables. One can think of causal relations as defining how the observed statistics change after an intervention on a system. When an external agent intervenes to change the probability distribution of some of the variables at will, the distributions of the remaining variables will be updated depending on the direction of the causal influences between them and the manipulated variables; flicking a switch can cause a light to turn off, but extinguishing the light by other means will not affect the position of the switch. Causal information tells us more about the statistical relationships between the variables than can be obtained from correlations alone. It is therefore useful to design a graphical representation and a graph separation rule that captures causal information, not just correlations.

Figure 1: (a) A semi-graphoid representing correlations between variables. (b) One possible causal graph consistent with these correlations.

The directions of causal influences are represented by adding arrows to the edges in the graph. This supplements the information about correlations with further constraints on the conditional independencies. Every causal graph, up to an absolute ordering of the variables, is in one-to-one correspondence with a list of conditional independence relations called a causal input list. The list can be thought of as a set of instructions for generating a probability distribution: one begins by generating the values of the independent variables, then computes the values of any variables that depend directly on them, then the variables that depend on those, and so forth. Hence every causal graph can be taken to represent a stochastic physical process proceeding over many time steps. In practice, working with the causal input list can be cumbersome, so it is more efficient to obtain the conditional independencies directly from the graph using a graph separation rule called d-separation. In this work we will propose to upgrade the definitions of causal input list and d-separation to quantum systems.

ii.1 Notation

Random variables, or sets of random variables, are denoted by capital roman letters, eg. , which take values from a set of possible outcomes. If is the space of all possible outcomes of , let represent a probability distribution on and the probability that the variable takes the value . In many cases, we will use the term also to represent , except where it might cause confusion. The statement means that the random variable is distributed over its outcomes according to the distribution . The joint probability represents a probability distribution over all the possible values of the random variables . The conditional probability is a set of probability distributions defined on , for the possible values of . Given a joint distribution , a marginal probability is defined by summing over all possible values of the other variables, i.e.

(1)

These concepts are united by the law of total probability, which states that . Unless otherwise specified, we consider only variables with discrete outcome spaces.

ii.2 Formal definitions for causal models

Let us consider a set of random variables whose values are governed by some joint probability function and which in general may be correlated. Formally, the statistical dependencies between variables are given by their conditional independence relations:

Definition II.2: Conditional Independence (CI) relations. Let be three disjoint sets of variables. The sets and are said to be conditionally independent given if knowing provides no new information about given that we already know (i.e. ‘screens-off’ and from each other). We write this as , which stands for the assertion that . We will often use the shorthand when dealing with set unions in CI relations.

Any joint probability distribution can be conveniently characterised by the complete set of CI relations that it implies for the variables. In fact, one only needs to specify a subset of CI relations, from which the rest can be obtained using the semi-graphoid axioms:

Semi-graphoid axioms:
1.a. Symmetry:
1.b. Decomposition:
1.c. Weak union:
1.d. Contraction: .

Note that if and both hold for disjoint sets , then does not necessarily hold. This might seem counter-intuitive, but examples where it fails are easy to construct111Suppose that all sets represent binary variables , and where is addition modulo 2. Clearly, knowledge of does not tell us anything about or individually. But knowing does reduce the set of possibilities for the joint set , for example, if then cannot have and the same..

The semi-graphoid axioms can be derived directly from the axioms of probability theory. The interpretation of the axioms is given by the following excerpt from Pearl Pearl (2000), (Chapter 1.1):

“The symmetry axiom states that, in any state of knowledge , if tells us nothing new about then tells us nothing new about . The decomposition axiom asserts that if two combined items of information are judged irrelevant to , then each separate item is irrelevant as well. The weak union axiom states that learning irrelevant information cannot help the irrelevant information become relevant to . The contraction axiom states that if we judge irrelevant to after learning some irrelevant information , then must have been irrelevant before we learned .”

Definition II.2: Semi-graphoid closure. Given any set of CI relations, the closure of is the set that includes all CI relations derivable from using the axioms 1.a-d.

Given a joint probability distribution , let denote the complete closed set of CI relations obtainable from . In general, the CI relations do not uniquely fix the probability distribution; there may exist two distinct joint distributions and for which . Hence, the CI relations alone do not capture the full information about the statistics. In the following, we will supplement the CI relations with a causal structure and functional relations in the form of a classical causal model (CCM).

A CCM provides us with an algorithm to generate the statistics of the observables. It can therefore be regarded as an abstract description of a physical system: if the predictions match the actual observations, then the CCM provides a possible explanation of the data. Formally, a CCM consists of two ingredients: an ordered set of CI relations , and a set of functions called the model parameters.

Definition II.2: Causal input list. An ordering assigns a unique integer in to each member of a set of variables. Consider an ordered set of variables , where the subset of variables with are called the predecessors of . A causal input list is the ordered set of CI relations of the form: , where each set is a subset of the predecessors of called the parents of , and are the remaining predecessors of excluding the parents.

Definition II.2: Ancestors and descendants. Given a causal input list , consider the set of parents of , their parents’ parents, and so on. These are called the ancestors of . Similarly, the descendants of are all variables for which is an ancestor. We will use to denote the union of the ancestors of a set .

Definition II.2: Model parameters. Given a causal input list , the model parameters are a set consisting of probabilistic functions . Each is equivalent to applying a deterministic function with probability for some auxiliary variable . The are sometimes called error variables and by definition have no parents. Each function determines the probability of conditional on the values of its parents:

(2)

For variables without any parents, called exogenous variables, the function just specifies a probability distribution over the possible values of , i.e. . We assume that all exogenous variables, including any error variables , are independently distributed (the Markovian assumption).

Definition II.2: Classical causal model. A classical causal model on variables is a pair containing a causal input list and model parameters defined on those variables. Alternatively, a CCM can be specified by the pair , where is the graph generated by (see Sec. II.4).

Given a CCM, we can construct a joint probability by generating random variables in the order specified by and using the functions to define the probabilities of each variable given its parents. These can then be used to construct a joint distribution from the CCM according to the law of total probability:

(3)

The joint probability obtained in this way from a CCM is said to be generated by and is denoted . It satisfies the following property:

Causal Markov Condition: Given that is generated by a CCM , each variable in is independent of its non-descendants, conditional on its parents.

Note: Our definition of the Causal Markov Condition follows Pearl (Ref. Pearl (2000)), in which it is proven to hold for any Markovian causal model (i.e. a model that is acyclic and whose exogenous variables are all independent). In the present work, a CCM is Markovian by construction, so the Causal Markov Condition holds. In the next section, we will use the Causal Markov Condition to motivate interpreting the parents of a variable as its direct causes.

Figure 2: A DAG representing a simple classical causal model. is a direct cause of both and ; is a direct cause of only.

Example II.2: Consider three variables . Suppose we have the ordering , and the causal input list indicates that ; ; and . It will be shown in Sec. II.4 that this generates the graph shown in Fig. 2. Suppose the model parameters are , where are deterministic and with probability . Then we obtain the joint probability as follows: first, generate the lowest variable in the ordering, , using the random function . Next, generate using and then apply to obtain the value of . Finally, use to obtain the value of , the last variable in the ordering. The statistics generated by this procedure are given by:

where

ii.3 Physical interpretation

In the previous section, we gave a formal definition of a classical causal model and described how it generates a probability distribution over the outcomes of its random variables. Since these variables represent physical quantities, we would like to supplement this mathematical structure with a physical interpretation of a CCM, as describing the causal relationships between these physical quantities. To do so, we make the following assumption that connects the intuitive concept of a ‘direct cause’ with its mathematical representation.

Assumption II.3. A variable’s parents represent its direct causes.
Physically, we expect that knowledge of the direct causes renders information about indirect causes redundant. Hence the direct causes should screen off the indirect causes in the sense of Definition II.2. We therefore define the direct causes of as the parents of and the indirect causes as the remaining (non-parental) ancestors of ; the screening-off property then follows from the Causal Markov Condition.

The above assumption leads to the following physically intuitive properties of a CCM:

Conditioning on common effects: In a CCM, two variables that are initially independent (i.e. conditional on the empty set) may become dependant conditional on the value of a common descendant. This reflects our intuition that two independent quantities may nevertheless be correlated if one ‘post-selects’ on a future outcome that depends on these quantities. For example, conditional on the fact that two independent coin tosses happened to give the same result, knowing the outcome of one coin toss allows us to deduce the outcome of the other.

Reichenbach’s common cause principle (RCCP): If two variables are initially correlated (i.e. conditional on the empty set) and causally separated (neither variable is an ancestor of the other), then they are independent conditional on the set of their common causes (parents shared by both variables).

It is not immediately obvious that the RCCP follows from the Causal Markov Condition. For a proof using the DAG representation (discussed in the next section) see Ref. Arntzenius (2010). We note that there exist in the literature numerous definitions of the RCCP, so our chosen definition deserves clarification. It was pointed out in Ref. Cavalcanti and Lal (2013) that a general formulation of the principle encompasses two main assumptions. The first states that causally separated correlated variables must share a common cause (called the ‘principle of common cause’, or PCC), and the second states that the variables must be screened-off from each other by their common causes (the ‘factorisation principle’ or FP). Our definition of the RCCP refers only to the factorisation property FP, while the PCC is a consequence of the definition of a CCM – it follows directly from what we have called the assumption of Markovianity and the fact that the variables are only functionally dependant on their parents. By contrast, Ref. Fritz (2014) takes the RCCP as being equivalent to the PCC, while the FP is regarded as a separate property that happens to hold for classical correlations. Note that it is precisely the factorisation property that is violated by quantum mechanics, not the common cause principle (without conditioning on effects, two independent quantum systems can only become correlated through interaction); so it is not surprising that our framework calls for a rejection of the RCCP, while the definition of Ref. Fritz (2014) does not. If one accounts for the difference in definitions, the present work is entirely consistent with Ref. Fritz (2014). We return to this point in Sec. III.

Finally, we note that one can interpret the ordering of variables as representing the time-ordering of the variables, such that each variable represents a physical quantity localised to an event in space-time. However, this interpretation is not strictly necessary for what follows. Indeed, it may be interesting to consider alternative interpretations in which some causal influences run counter to the direction of physical time, such as in the retro-causal interpretation of quantum mechanics Price (2008).

ii.4 The DAG representation of a CCM

It is useful to represent using a directed acyclic graph (DAG), which can be thought of as the causal ‘skeleton’ of the model. In the DAG representation of , there is a node representing each variable and a directed arrow pointing to the node from each of its parents. The DAG constructed in this way is said to be generated by the causal input list . The parents of a node in a DAG are precisely those nodes that are directly connected to it by arrows pointing towards it. It is straightforward to see that the ancestors of are represented by nodes in the DAG that each have a directed path leading to , and the descendants of are those nodes that can be reached by a directed path from . In Example II.2, the system is represented by the DAG shown in Fig. 2.

To establish a correspondence between a DAG and its generating causal input list , we need an algorithm for reconstructing a list of CI relations from a DAG, such that the list generates the DAG. For this purpose, one uses d-separation (see Fig. 3).

Definition II.4: d-separation. Given a set of variables connected in a DAG, two disjoint sets of variables and are said to be d-separated by a third disjoint set , denoted , if and only if every undirected path (i.e. a path connecting two nodes through the DAG, ignoring the direction of arrows) connecting a member of to a member of is rendered inactive by a member of . A path connecting two nodes is rendered inactive by a member of if and only if:
(i) the path contains a chain or a fork such that the middle node is in , or
(ii) the path contains an inverted fork (head-to-head) such that the node is not in , and there is no directed path from to any member of .
A path that is not rendered inactive by is said to be active.

Figure 3: An illustration of d-separation. The table indicates which CI relations are implied by d-separation, and which are not.

By assuming that all d-separated nodes are independent conditional on the separating set (see below), we can then obtain CI relations from the DAG. In principle, the rules for d-separation can be derived from the requirement that it produces the CI relations contained in the list that generates the DAG. However, d-separation also provides an intuitive graphical representation of the physical principles discussed in the previous section. In particular, a path between two nodes in the graph is rendered inactive by a set in precisely those situations where we would physically expect the two variables to be independent conditional on : when we are not conditioning on any common effects (head-to-head nodes); when we are conditioning on a common cause (as in the RCCP); or when we are conditioning on a node that is a link in a causal chain (screening off indirect causes). With the physical interpretation in mind, we are motivated to use d-separation to obtain CI relations using the correspondence:

(5)

i.e. we assume that if and are d-separated by in a DAG, then they are conditionally independent given in the semi-graphoid closure of any list that generates the DAG.

Formally, let be a DAG, and let be the set of CI relations obtainable from using d-separation, and the closure of this set. We then have the following theorem:

Theorem II.4 (Verma & Pearl Verma and Pearl (1988)):
Let be the DAG generated by . Then . That is, the closure of the DAG is equal to the closure of the causal input list, so that every CI relation implied by the DAG also follows logically from and vice-versa.

Theorem II.4 implies that d-separation is sound, since every CI relation obtainable from the DAG is in the closure of the causal input list (), and complete, since there are no CI relations implied by the causal input list that are not obtainable from the DAG ().

Given a DAG , one can always find a causal input list that generates as follows: choose a total ordering that is consistent with the partial ordering imposed by , and then write down the ordered list of CI relations of the form , where the parents of each variable are the same as the parents of its representative node in . Moreover, the list obtained in this way is unique, modulo some freedom in the ordering of causally separated events (eg. if the variables represent events in relativistic spacetime, this freedom corresponds to a choice of reference frame). It is this feature that allows us to replace the causal input list with its corresponding DAG in the definition of a CCM (Definition II.2).

Figure 4: A DAG can also be used to construct a causal input list that generates it (see text).

Example II.4: Consider the DAG shown in Fig. 4. The graph assigns parents as follows:

(6)

A total ordering consistent with this graph is . Therefore, we obtain the causal input list:

(7)

Finally, we have the following useful definitions:

Definition II.4: Independence maps and perfect maps. Given and a DAG on the same variables, we will call an independence map (I-map) of iff . If equality holds, , then is called a perfect map of .

The intuition behind this definition can be understood as follows. A DAG is an independence map of a probability distribution iff every CI relation implied by the DAG is also satisfied by the distribution. That means that if two variables are not causally linked in the DAG, they must be conditionally independent in the distribution. However, the converse need not hold: the arrows in a DAG represent only the possibility of a causal influence. In general, depending on the choice of model parameters, it is possible for two variables to be connected by an arrow and yet still be conditionally independent in the probability distribution. Equivalently, one can find a probability distribution that satisfies more conditional independencies than those implied by its DAG. A DAG is a perfect map iff it captures all of the CI relations in the given distribution, i.e. every causal dependence implied by the arrows in the DAG is manifest as an observed signal in the statistics.

Interestingly, there exist distributions for which no DAG is a perfect map, the key example being any bipartite distribution that violates a Bell inequality. This fact forms the basis for the criterion of faithfulness of a CCM, discussed in the following section.

ii.5 Faithful explanations and fine-tuning

Suppose we obtain the values of some physical observables over many runs of an experiment and that the statistics can be modelled (to within experimental errors) by a CCM. Then we can say that the CCM provides a causal explanation of the data, in the sense that it tells us how physical signals and information propagate between the physical observables. In particular, it allows us to answer counterfactual questions (what would have happened if this observable had taken a different value?) and predict how the system will respond to interventions (how will other quantities be affected if a given variable is forcibly altered?). These notions can be given a rigorous meaning using causal models, and they constitute a formal framework for making causal inferences from observed data. In the present work, we will be primarily concerned with defining a quantum causal model that can be given a useful graphical representation in DAGs, so we will not discuss interventions and inference in causal models (the interested reader is referred to Ref. Pearl (2000) for inference in the classical case).

Before we consider quantum systems, it will be useful to review some caveats to the question of whether a CCM provides an adequate description of some given data, and in particular, whether a given CCM gives a faithful account of the observed statistics. The first caveat has to do with the possibility of hidden, or latent variables. Suppose that we have a probability distribution for which there is no CCM that generates it (this can occur, for example, if some exogenous variables in the model are found to be correlated with each other, thereby violating the basic property of Markovianity required for a CCM). Rather than giving up the possibility of a causal explanation, we might consider that there exist additional variables that have not been observed, but whose inclusion in the data would render the statistics explainable by some CCM. Formally, suppose there is an extension of to some larger distribution that includes latent variables , such that the observed statistics are the marginal probabilities obtained by summing over the unobserved variables:

(8)

If there exists a CCM such that , then we can say that this CCM explains the probability distribution with the aid of the latent variables . The admittance of hidden variables in causal models seems to lead to a problem: it turns out that every probability distribution can be explained by a CCM, with the aid of a sufficient number of hidden variables! For this reason, we further constrain the possible explanations of the observed data by requiring that the models be faithful to the data:

Definition II.5: Faithfulness. Consider a distribution and a CCM that generates . The explanation offered by is called faithful to iff the DAG derived from is a perfect map of , i.e. .
Latent variables: Suppose there is no CCM that is faithful to . Consider instead a CCM , which obtains by summing the generated distribution over the hidden variables . This extended CCM is considered faithful to iff every CI relation in is implied by the extended DAG , i.e. .

The motivation for this definition is that a faithful explanation of the observed statistics is a better candidate for describing the ‘real causal structure’ of the system than an unfaithful explanation, because it accurately captures all causal dependencies in the observed statistics. If there exists no faithful explanation of the observed statistics, but one can obtain a faithful explanation using hidden variables, then we can interpret the statistics of the observed variables as the marginal statistics arising from ignoring the unobserved variables. Note that not all probability distributions can be faithfully reproduced by some CCM, even with the aid of hidden variables.

Geiger & Pearl Geiger and Pearl (1988) showed that, for every DAG , one can explicitly construct a distribution such that is faithful for , i.e. such that holds. Furthermore, it can be shown that if there exists a DAG that is faithful for a given , then there must exist a set of model parameters such that the CCM generates Druzdzel and Simon (1993). However, we have already mentioned that there exist probability distributions for which no DAG is a perfect map; this will be relevant when we consider quantum mechanics in Sec. II.6.

Finally, faithfulness can be equivalently defined as the rejection of fine-tuning of the model parameters, as noted in Refs. Pearl (2000); Wood and Spekkens (2012). In particular, if a DAG is unfaithful, this implies that there exists at least one conditional independence in the statistics that is not implied by the DAG. It can be proven that within the set of probability distributions compatible with the DAG, those that satisfy such additional CI relations are a set of measure zero Pearl (2000). Thus, the only way that such additional CI relations could arise is by a kind of ‘conspiracy’ of the model parameters to ensure that the extra conditional independence holds, even though it is not indicated by the causal structure. In this sense, fine-tuning represents causal influences that exist at the ontological level but cannot be used for signalling at the level of observed statistics, due to the careful selection of the functional parameters; any small perturbation to these parameters would result in a signal.

Example II.5: Recall the system of Example II.2 and suppose that we observe in . This CI relation is not found in ; in fact, there is a directed edge from to in the DAG of (Fig. 2). The only way to account for this discrepancy is if the model parameters are chosen such that the predicted signal from to is obscured. This could happen if are positive integers and we choose model parameters and for some integer constant . For simplicity, suppose the model is deterministic, . If the numbers and just so happen to be equal, then we obtain the joint probability:

(9)

and the value of tells us nothing about the value of , because the ’s conveniently cancel out, leading to the observed independence . This can be understood as a coincidence in the model parameters and , whereby they each depend on the same parameter . Indeed, if the constants and were allowed to differ in each function, then the cancellation would not occur, and would still carry information about in accordance with the causal structure. Hence, absence of fine-tuning in with respect to a causal input list can also be defined as the requirement that the CI relations observed in should be robust under changes in the model parameters consistent with Pearl (2000).

ii.6 Does quantum mechanics require fine-tuning?

Consider a probability distribution , satisfying the generating set of CI relations . Suppose that the closure of this set contains all the CI relations satisfied by , i.e. . This represents a generic Bell-type experiment: the setting variables are independent of each other, and there is no signalling from to or from to , but the outputs are correlated. (Here, the absence of signalling is only a constraint on the allowed probability distributions. It refers to the fact that the marginal distribution of the outcomes on each side must be conditionally independent of the values of the setting variables on the opposite side, a requirement often called ‘signal locality’ in the literature. It does not forbid the possibility of signalling at the ontological level, as in non-local hidden variables, etc.)

Note that we cannot explain these correlations by a CCM without latent variables, because and are correlated without a common cause. Hence let us consider the extended distribution satisfying . Of course, we require that , but we can impose additional physical constraints on the hidden variable . In particular, we expect to be independent of the settings and, in keeping with the no-signalling constraint, to represent a common cause subject to Reichenbach’s principle. This leads to the extended set of constraints and we assume that , i.e. that the extended distribution satisfies at least these constraints. We then ask whether there exists a CCM that can faithfully explain the observed correlations.

If and are independent conditional on the hidden variable in the distribution , i.e. if holds in , then it is easy to see that qualifies as a common cause of and and the correlations can be explained by a CCM with the DAG shown in Fig. 5 (a). It can be shown that this occurs whenever satisfies Bell’s inequality Wood and Spekkens (2012). Conversely, Wood & Spekkens showed that if violates Bell’s inequality, there is no CCM that can faithfully explain , even allowing hidden variables. Of course, one can find numerous un-faithful explanations, such as the DAG in Fig. 5 (b) and fine-tuning of the model parameters to conceal the causal influence of on . This result implies that, in general, CCMs cannot faithfully explain the correlations seen in entangled quantum systems.

Figure 5: (a) A possible DAG for a Bell-type experiment using hidden variables. (b) A DAG that can serve as an unfaithful explanation of Bell-inequality violation.

Example II.6: Consider the deBroglie-Bohm interpretation of quantum mechanics. This interpretation gives a causal account of Bell inequality violation using super-luminal influences, one possible variant of which is depicted in Fig. 5 (b). Here, is a hidden variable that carries information about the setting faster than light to the outcome . The model posits a CCM that generates a distribution and the observed statistics are interpreted as the marginal obtained from this distribution by summing over . The no-signaling CI relations hold in the observed statistics, however they do not follow from the DAG Fig. 5 (b) which includes the hidden variables, hence the CCM that generates using this graph is not a faithful explanation for . In general, the deBroglie-Bohm interpretation and its variants appear to require fine-tuning Wood and Spekkens (2012).

How should we interpret this result? On one hand, we might take it as an indication that faithfulness is too strong a constraint on the laws of physics, and that nature allows hidden variables whose causal influences are concealed at the statistical level by fine-tuning. Alternatively, we could take it to indicate that the class of physical models describable by CCMs is not universal, and that a new type of causal model is needed to give a faithful account of quantum systems. Along these lines, we could choose to interpret Fig. 5 (a) as a quantum circuit, where now stands for the preparation of an entangled pair of quantum systems and the arrows stand for their distribution across space. In doing so, we implicitly shift our perception of quantum mechanics from something that needs to be explained, to something that forms part of the explanatory structure. We no longer seek to explain quantum correlations by an underlying causal mechanism, but instead we incorporate them as a fundamental new addition to our causal structure, which can then be used to model general physical systems. This approach entails that we no longer require to hold for the “common cause” and hence that we abandon Reichenbach’s Common Cause Principle (specifically, the factorisation property). This in turn implies that d-separation is no longer the correct criterion for reading CI relations from the DAG, when the DAG is interpreted as a quantum circuit. In what follows, we will propose a new criterion that serves this purpose, leading to the concept of a quantum causal model.

Iii Quantum Causal Models

iii.1 Preliminaries

We begin by considering quantum networks modelled as a DAGs, in which the nodes represent state preparations, unitary transformations and measurements. Based on this interpretation, we obtain a corresponding notion of a quantum input list and a graph separation criterion that connects the DAG to the list that generates it. We mention that there exist other approaches to quantum computation in which it would be interesting to explore causal relations, such as measurement-based quantum computation. For efforts along these lines, see eg. Morimae (2014); J. Miyazaki and Murao (2013).

The general theory of quantum networks as given in Ref. Chiribella et al. (2009) provides a DAG representation in which nodes represent completely general quantum operations. Below, we define a canonical form of a general quantum network in order to cleanly separate the classical apparatus settings from the measurement outcomes, to facilitate the definition of a graph separation criterion. Given a DAG, we divide the nodes into four classes: as before, those with no incoming edges are called exogenous; those with no outgoing edges are called drains; those with ingoing and outgoing edges are called intermediates. We assign the following interpretations to the elements of the DAG:

Edges: every edge in a DAG is associated with a Hilbert space of dimension , with . Thus, we can associate an integer number of qubits with each edge in the graph. The Hilbert space dimension is allowed to be different for different edges; however, for intermediate nodes we require that the total Hilbert space dimension of the ingoing edges (obtained by multiplying the dimensions of the individual edges) be equal to the total dimension of the outgoing edges. The reason for this constraint is that it allows us to associate a unitary map to each intermediate node.

Exogenous nodes: Every exogenous node is associated with a random variable. Each possible value of the variable corresponds to the preparation of a normalised pure state (a source). The set of pure states need not be orthogonal - in fact they may even be degenerate, with more than one value of the variable corresponding to preparation of the same state. The only requirement is that the states exist in a Hilbert space with dimension equal to , which is the tensor product of the Hilbert spaces of all the outgoing edges.

Drains: Every drain is associated with a random variable. Each value of the variable corresponds to the outcome of a projective measurement on , the tensor product of the Hilbert spaces of all the ingoing edges. The measurement basis is assumed to be fixed by convention. For example, since the dimension of is for some integer , we can always take the measurement basis to be the computational basis of qubits.

Intermediate nodes: Every node with both incoming and outgoing edges is associated with a random variable. Each value of the variable represents a unitary operator on .

The above definitions allow us to associate a quantum network to any DAG. Conversely, every quantum network has a representation as a DAG of this form.

Figure 6: The conversion of a quantum circuit (a) into a DAG, using the recipe outlined in the text. The conversion is performed by replacing feed-forwards with unitary interactions, followed by measurements in a fixed basis; and replacing general measurements with unitary coupling to ancilla states, followed by measurement in a fixed basis, as shown in (b). The resulting DAG is shown in (c).

Example III.1: Consider the circuit in Fig. 6 (a). This describes the preparation of two qubits as mixtures and in an arbitrary orthogonal basis . These are followed by a measurement of the first qubit in the computational basis and the subsequent application of a gate to the second qubit, conditional on the outcome of the first measurement. Finally a POVM is applied to the second qubit by coupling it via unitary interaction (either or ) to a third ancilla qubit . The ancilla is traced out and the remaining qubit measured in the basis. In Fig. 6 (b) the feed-forward has been replaced with a unitary interaction (a CNOT) followed by tracing out the first qubit (all feed-forwards can be described in this way to ensure that the setting variables, representing the choice of input state and unitary, remain independent of each other). The tracing-out of the ancilla qubit is replaced with a measurement in the basis, whose outcome can be ignored. In this form, the circuit can be cast directly into a DAG, as shown in Fig. 6 (c). The variables and take values corresponding to the basis states , distributed with probabilities so as to produce the mixed states . and are single-valued, corresponding to the state and the unitary CNOT respectively. has three values corresponding to the three possible unitaries, and is distributed according to the probability of each unitary being implemented. Finally, are all binary-valued, corresponding to outcomes and whose probabilities are given by quantum mechanics (see Sec III.2).

Note that a single DAG can represent any member of the class of quantum networks with the same basic topology (i.e. the same connections between preparations, unitaries and measurements). Thus, in a quantum causal model, the preparations and unitaries are taken as the model parameters and the DAG provides the causal structure, as explained in the next section.

iii.2 Quantum input lists and model parameters

Recall that the classical causal input list represents a set of conditional independence relations between variables in a CCM, from which a DAG can be easily constructed. The motivation for the causal input list comes from its physical interpretation, discussed in Sec. II.3, which embodies principles like the RCCP that we expect to hold for classical physical systems. Hence, to define the quantum analog of a causal input list, we should begin by asking: for variables in a quantum network, what physical principles constrain their statistical dependencies?

First, we note that the observables in a quantum network fall naturally into two distinct categories: settings and outcomes (we continue to use to denote a generic variable or set of variables). The settings determine the states produced at the sources and the unitaries applied in the network, while the represent the outcomes of measurements in a fixed basis. Since the settings play the same role as the exogenous variables in a CCM, we assume that they are all distributed independently of each other; however, unlike in a CCM, this property now also applies to variables represented by intermediate nodes. This assumption is the analog of the Markovianity assumption for a CCM, and it embodies one aspect of the common cause principle that is retained in quantum mechanics, namely, that correlated variables (conditional on the empty set) must share a common source, or must have interacted previously. It is in this sense that the RCCP can be said to hold for quantum correlations in Ref. Fritz (2014) (recall the discussion of Sec. II.3).

Also as before, we assume an absolute ordering of the variables and enforce the physical assumption of causality (no causal loops) and we again assign a set of parents to each variable, representing the connections in the network and (implicitly) the possibility of a causal influence. However, unlike the case of a CCM, we are not able to interpret the parents of a variable as its direct causes. This is because the values of the settings by definition do not have any causes in the network (they are chosen by external factors, like experimental intervention). Furthermore, the parents of an outcome no longer screen it off from its other ancestors: the influence of an initial state preparation on the measurement outcome cannot in general be screened off by a choice of intermediate unitary. We leave it as an open question whether one can formulate a quantum network in a manner that respects this property of CCMs; we will find it more convenient simply to abandon it. Indeed, since the variables representing the preparation and choice of unitary are assumed to be independent, they cannot carry any information about each other, nor can the variable representing the unitary reveal any information about the quantum system on which it acts. The assignment of parents to the variables therefore places much weaker constraints on the correlations than in the classical case. However, the following physical assumption is still justified in a quantum network:

Assumption III.2. The possible causes of an outcome are its ancestors.

This assumption reflects our intuition that it is only the operations performed on a quantum system leading up to its measurement that can have a causal effect the measurement outcome. Indeed, it is also argued in Ref. Fritz (2014) that there is no reason to maintain the distinction between direct and indirect causes in any generalised model that goes beyond classical correlations.

It is clear from our discussions in Sec. II that the Causal Markov Condition is not expected to hold in a quantum network, since the RCCP no longer holds. Instead, we expect it to be replaced by a weaker property:

Quantum Causality Condition: An outcome is independent (conditional on the empty set) of all settings that are not its causes and all outcomes that do not share a common cause.

This property expresses the fact that outcomes should be independent of any settings from which they are causally disconnected and should be correlated only with other outcomes that share a common cause. This property holds also in the classical case, but unlike the classical case, we now do not require sets of outcomes to be independent of each other conditional on their common causes – instead we allow them to still be dependent, admitting violations of the factorisation property of the RCCP (recall Sec. II.3). In addition to this weakened version of the RCCP, we still have the classical feature that independent variables can become dependant conditional on common effects. Thus, for two variables to be independent, we will still have to avoid conditioning on certain colliders.

To make these ideas formal, let us consider a set of random variables partitioned into outcomes and settings . The following definitions will also be useful:

Definition III.2: -chain. Given a set of outcomes , two other sets of outcomes and are said to be connected by an -chain iff shares an ancestor with a member of that shares an ancestor with another member of , (etc), that shares an ancestor with . A set of settings is linked to by an -chain iff has a descendant in that is connected by an -chain to . Similarly, and are connected by an -chain iff they both have descendants in that are connected in this way.

Definition III.2: -detached. Given a set of outcomes and some variables , the set of all variables not connected to by an -chain are said to be -detached from , denoted .

Figure 7: “Chained” variables: and are each connected to and by a W-chain. However and are -detached from , because the collider is not in the set .

(Note that when is the empty set, the detached variables are just those outcomes that do not share an ancestor with outcomes in ). Intuitively, if two variables in a DAG are connected by an -chain, they are connected by a path on which every collider has a directed path to . Hence, the detached variables are those nodes in the graph for which every path contains at least one collider that does not lead to . This will be useful later when we consider graph separation.

Let denote the complement of a set , and let be the complement of restricted to , i.e. . Under a choice of ordering , let denote the set of predecessors of in . Using these definitions, we propose the following characterisation of the CI relations in a quantum network:

Definition III.2: quantum input list. A quantum input list is a pair , containing:
i. An ordered list of parents, , where each set of parents is a subset of . Members of cannot be parents; in addition, every setting must be a parent of at least one other variable (these conventions ensure that the resulting DAG can be interpreted as a quantum circuit). Ancestors, descendants, etc, are defined from the list of parents in the usual way.
ii. A set of CI relations denoted , constructed as follows. For every subset of settings and outcomes , there is a CI relation of the form and a CI relation of the form in .

The first CI relation in the above definition expresses the physical requirement of setting independence, modulo the possibility of correlating the settings by conditioning on their effects. In particular, it says that settings are guaranteed independent except when connected by an -chain. The second CI relation simply expresses the Quantum Causality Condition.

The quantum input list is said to be compatible with a given probability distribution iff . Given a quantum input list, we can construct a DAG in the usual way, by drawing a directed edge to each variable from each of its parent nodes. The DAG constructed in this way is said to be generated by the list . As usual, the ancestors of are those nodes in the graph that have a directed path leading to . The quantum input list defines the causal constraints on the variables, based on their interpretation as settings and outcomes in a quantum network. We conjecture that this list captures all of the conditional independencies that hold in a general quantum circuit, when the circuit is expressed as a DAG as outlined in Sec III.1:

Conjecture: If a CI relation holds in every quantum network represented by a DAG , then it is implied by in any quantum input list that generates .

So far, we have only specified the causal structure and independence relations. To obtain a full joint probability distribution from a quantum input list, we need to supplement it with model parameters specifying the pure state preparations, unitary transformations, and measurements that correspond to the variables. These parameters define the space of possible quantum circuits that are described by a given DAG:

Definition III.2: quantum model parameters. Consider a set of variables with outcome spaces connected in a DAG representing a quantum network. Then the quantum model parameters consist of:
i. A Hilbert space of dimension for each edge, where is a (possibly different) positive integer for each edge;
ii. A specification of the orthonormal basis in which all projective measurements are made;
iii. For every exogenous node , a pure state for every value in ;
iv. For every intermediate node , a unitary for every value in ;
v. For every drain node , a pure state from for every value in ;
vi. A marginal probability distribution on the outcome space of every variable that is an exogenous node or intermediate node. These marginal distributions are all mutually independent.

The states and operators mentioned above apply to the Hilbert spaces of their respective nodes, as determined using i and the number of outgoing and ingoing edges (Recall that the dimensions assigned to the edges is constrained such that the total dimension of ingoing and outgoing edges for intermediate nodes is the same). The distributions given in vi represent a set of ‘initial conditions’, fixed by the experimenters’ choices and/or environmental conditions. These are used to determine the resulting probability distributions of the outcome variables according to the usual laws of quantum mechanics. This is made precise using the following definition:

Definition III.2: Quantum Causal Model
A quantum causal model (QCM) on a set of variables is a pair consisting of a quantum input list for the set , and a set of quantum model parameters for the DAG generated by the input list.

Every QCM defines a joint probability distribution over its variables according to the following procedure. Consider a QCM on ordered variables , and partition of the set such that labels the exogenous variables, labels the intermediate variables, and labels the variables corresponding to drains in the DAG . From we obtain the mutually independent marginal distributions , which includes all setting variables. The joint probability of the outcomes conditional on the settings, , is computed in the usual way from the quantum circuit obtained from the DAG and the pure states and unitaries associated with the settings . One thus obtains the total joint probability:

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Note: Our definition of a QCM on a DAG can be regarded as a concrete example of the more general notion of a Quantum Correlation on the graph , as defined by Fritz Fritz (2014). In particular, working in the category of completely positive maps, where Hilbert spaces are the objects and CP maps are the morphisms, we assign Hilbert spaces to the edges of the graph and CP maps to the nodes. The model parameters are just the set of functions from outcomes to morphisms that define the -instruments in the language of Ref. Fritz (2014), allowing us to compute probabilities. Our model distinguishes these functions according to the placement of their nodes in the graph (exogenous, intermediate or drain); we also model hidden variables as additional nodes, not as edges. These conventions do not represent limitations of our model, but are used for convenience. By contrast, our restriction to the case of variables with finite outcome spaces is a limitation of our model, but we expect the generalisation to continuous variables following Ref. Fritz (2014) to be straightforward.

Now that we have defined a QCM, we would like to have a graph separation rule analogous to d-separation that would allow us to recover all the CI relations implied by from the DAG . This is proposed in the next section.

iii.3 Graph separation in quantum networks

In general, because of the failure of the RCCP, we can never guarantee that two outcomes will be independent conditional on their common causes. However, there are still situations in which variables are expected to be conditionally independent of each other; we examine the possibilities below.

Two settings are already assumed to be chosen independently, so they can only become dependent on each other by conditioning on a common effect (which is an outcome), or conditioning on a connected chain of such effects. This applies also to conditioning on common effects in a CCM (recall Sec. II.3). In the case of a setting and an outcome, these might be dependent on each other if the setting is already a possible cause of the outcome, since the causal influence cannot in general be screened-off by other variables. On the other hand, if there is no directed path from the setting to the outcome and no chain of conditioned effects, one would expect the two to be independent. We must be careful, however: in quantum mechanics, it is also possible for the outcome to be entangled to another outcome that is descended from the setting, such that conditioning on the latter outcome correlates the setting with the causally separated outcome. To ensure their independence, therefore, one should not also not condition on any outcomes that are descended from the setting. Finally, two outcomes should be independent unless they share a common cause, or are connected by a chain of conditioned effects.

These considerations lead us to the following graph separation criterion:

Definition III.3: q-separation
Given a DAG representing a quantum network, two disjoint sets of variables and are said to be q-separated by a third disjoint set , denoted , iff every undirected path between and is rendered inactive by a member of . A path connecting two variables is rendered inactive by iff at least one of the following conditions is met:
(i) both variables are settings, and at least one of the settings has no directed path to any outcome in ;
(ii) one variable is a setting and the other is an outcome, and there is no directed path from the setting to the outcome, or to any outcome in ;
(iii) the path contains a collider where is not an outcome in , and there is no directed path from to any outcome in .

Figure 8: An illustration of q-separation. The table contains CI relations that are either implied or not implied by the DAG using the rules of q-separation. Note the differences to d-separation (Fig. 3).

Of course, the heuristic motivation given above does not necessarily guarantee that q-separation captures all of the CI relations that are implied by a quantum input list, nor is it obvious that the input list contains all CI relations implied by q-separation. A proof that q-separation is sound and complete for quantum input lists is given in the next section.

iii.4 The q-separation theorem

In this section we prove the soundness and completeness of q-separation. The proof approximately follows that of Pearl & Verma Verma and Pearl (1988) for the classical case. By analogy with d-separation, we will consider the set of CI relations obtainable from a DAG using the q-separation criterion (Definition III.3) and . Let this set be denoted , with its closure. If we replace d-separation with q-separation in Definition II.4, we obtain analogous criteria for to be an I-map or a perfect map of a given distribution . We can now prove the following useful theorem:

Theorem III.4:
Let the DAG be a perfect map of a distribution under q-separation, i.e. . Then there is a quantum input list compatible with that generates the DAG .

Proof: The DAG imposes a partial order on the variables . Choose any total order that is consistent with this. Label the nodes in as outcomes if they are drains, and settings otherwise. Define the parents in to be the nodes with directed edges pointing to in the graph. A path between a setting and any other variable is rendered inactive by outcomes if there is at least one collider on the path not in and with no directed path to . This is true for all variables that are -detached from the setting, hence the CI relation is implied by . For each set of outcome nodes and their ancestors, , a path from this set to the non-ancestors can only be activated by conditioning on an outcome. Furthermore, a path from to must contain a collider, or else and would share an ancestor (a contradiction), so it too can only be activated by conditioning on an outcome. Hence these paths are rendered inactive by the empty set, and the CI relation is implied by . According to Definition III.2, these ingredients are sufficient to specify a quantum input list. By construction, this list also generates the DAG .

The next theorem provides the key result.

Theorem III.4
Given a distribution and a compatible quantum input list , the DAG generated by is an I-map of , that is, .

Proof: We prove the result by induction on the number of variables. First we show that the result holds for variables, given that it holds for variables. Then we note that the result holds trivially for one variable; hence, by induction, it holds for any number of variables.

Let be a distribution on variables and a compatible quantum input list, which generates the DAG . Let be the last variable in the ordering ; let be the closed set of CI relations formed after removing from all CI relations involving ; let be any probability distribution having exactly the closed set of CI relations (such a distribution can always be constructed Pearl (2000)); and let be the DAG formed by removing the node and all its connected edges from the graph .

Consider the list obtained from by removing every CI relation involving from and removing from the list of parents (let the reduced list of parents be denoted ). This procedure might result in one or more settings that are not parents of any other variables, so we must be convert these into outcomes in order to make a valid input list. To do so, we first remove the setting in question (say ) from any sets of settings in which it appears in . Next, we re-label it as an outcome, . This entails that for every subset of outcomes (not containing ) and settings , we must add new CI relations and to . Let the resulting list be denoted . Then the pair is a valid quantum input list on variables. Furthermore, by construction, generates the DAG .

Let us now assume that is an I-map of : . We aim to prove that, under this assumption, is also an I-map of , . To do so, we will consider each CI relation of and show that it exists also in .

The CI relations of can be divided into three cases:

(1) does not appear in the CI relation;
(2) appears in the first position in the CI relation, eg. ;
(3) appears in the last position in the CI relation, eg. .

Note that, if appears in the second position in the CI relation, we can use symmetry (semi-graphoid axiom 1.a.) to move it into the first position and thereby convert it into case (2) above. We now prove the result for each case separately.

Lemma 1: Let be disjoint sets of variables and let be a relation of the form that does not contain the variable . Then .

Proof: Since is in , it must also be in . If it were not, then there would be a path between and that is active in but rendered inactive by in . But this is impossible, because an active path cannot be rendered inactive just by adding a node and its associated edges to the graph. Since is an I-map of , the relation must also be contained in , and since is a subset of , is also contained in .

Lemma 2: Let be a relation of the form . Then .

Proof: First, we partition the sets into disjoint sets of outcomes and settings, eg. where contains only outcomes and only settings.

Define the set as the members of that are connected to outcomes by a -chain. Let denote its complement in . Next, consider .

We can write the ancestors as the union of four disjoint sets, , where , and similarly for and . Any remaining members of not contained in any of are contained in . Likewise, let us decompose into disjoint sets: where eg. and similarly for (see Fig. 9). Note that and analogously for and . The CI relation must hold in and hence in . Using the above definitions:

Figure 9: A partitioning of the variables into disjoint sets. Above: the sets are partitioned with respect to their settings, outcomes and . Below: the ancestors and their complement in are further decomposed into disjoint subsets.

The set must be a subset of (since the members of by definition cannot share an ancestor with ). The same goes for , otherwise there would be a path connecting to on which every collider is in or has a directed path to , and they could not be q-separated in as is required for to be true. Hence, using the semi-graphoid axioms:

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No member of can be an ancestor of in , or else there would be a directed path from a setting in to an outcome in and they could not be q-separated by in , contradicting our initial premise . Therefore and , and (12) implies . The relation implies and hence (by Lemma 1): . Combining this with (12) and the semi-graphoid axioms, we obtain the desired result .

Lemma 3: Let be a relation of the form . Then .

Proof: Note that cannot share an ancestor with both and , or else there would be a path connecting to on which every collider has a descendant in , preventing them from being q-separated in . We therefore assume without loss of generality that does not share an ancestor with . By a similar argument, no member of can share an ancestor with ; hence . Now, either shares an ancestor with , or it does not. If it does, then cannot contain any ancestors of (defined as in the previous Lemma) and we can use the same procedure as before to obtain the desired result: .

In the remaining case, does not share an ancestor with so we have the relation . Let and consider the relation that holds in and hence in . Using the above properties, and the fact that cannot contain any ancestors of (for the usual reason that this would imply an active path between and in ), we obtain:

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Let us partition , where contains the members of that are detached from by , and contains the rest. Consider the CI relation that holds in