A graph-separation theorem for quantum causal models

Random variables, or sets of random variables, are denoted by capital roman letters, e.g. $X,Y,Z$, which take values from a set of possible outcomes. If ${{\mathcal{E}}}_{X}$ is the space of all possible outcomes of X, let P(X) represent a probability distribution on ${{\mathcal{E}}}_{X}$ and $P(X=x)$ the probability that the variable X takes the value $x\in {{\mathcal{E}}}_{X}$. In many cases, we will use the term P(X) also to represent $P(X=x)$, except where it might cause confusion. The statement $X=P(X)$ means that the random variable X is distributed over its outcomes according to the distribution P(X). The joint probability $P(X,Y,Z)$ represents a probability distribution over all the possible values of the random variables $\{X,Y,Z\}$. The conditional probability $P(X| Y)$ is a set of probability distributions defined on ${{\mathcal{E}}}_{X}$, for the possible values of Y. Given a joint distribution $P(X,Y)$, a marginal probability P(Y) is defined by summing over all possible values of the other variables, i.e.

$$\begin{eqnarray}P(Y) & = & \displaystyle \sum _{X}P(X,Y)\\ & := & \displaystyle \sum _{x\in {{\mathcal{E}}}_{X}}P(X=x,Y).\end{eqnarray} \tag{ 1 }$$

These concepts are united by the law of total probability, which states that $P(X,Y)=P(X| Y)P(Y)$. Unless otherwise specified, we consider only variables with discrete outcome spaces.

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 CI relations:

Definition 1. CI relations. Let $X,Y,Z$ be three disjoint sets of variables. The sets X and Y are said to be conditionally independent given Z if knowing Y provides no new information about X given that we already know Z (i.e. Z 'screens-off' Y and X from each other). We write this as $(X\perp Y| Z)$, which stands for the assertion that $P(X| Y,Z)=P(X| Z)$. We will often use the shorthand $(X\cup W\perp Z| Y):= ({XW}\perp Z| Y)$ when dealing with set unions in CI relations.

Any joint probability distribution P can be conveniently characterized 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:

• Symmetry: $(X\perp Y| Z)\iff (Y\perp X| Z)$
• Decomposition: $(X\perp {YW}| Z)\Rightarrow (X\perp Y| Z)$
• Weak union: $(X\perp {YW}| Z)\Rightarrow (X\perp Y| {ZW})$
• Contraction: $(X\perp Y| {ZW})\mathrm{and}(X\perp W| Z)\Rightarrow (X\perp {YW}| Z)$ .

Note that if $(X\perp Y| Z)$ and $(W\perp Y| Z)$ both hold for disjoint sets $X,Y,Z,W$, then $({XW}\perp Y| Z)$ does not necessarily hold. This might seem counter-intuitive, but examples where it fails are easy to construct³ .

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 [13], (chapter 1.1):

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

Definition 2. Semi-graphoid closure. Given any set S of CI relations, the closure of S is the set $\bar{S}$ that includes all CI relations derivable from S using the axioms 1.a–d.

Given a joint probability distribution P, let $\bar{C}(P)$ denote the complete closed set of CI relations obtainable from P. In general, the CI relations do not uniquely fix the probability distribution; there may exist two distinct joint distributions P and P' for which $\bar{C}(P)=\bar{C}(P\prime )$. 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 CCM.

A CCM provides us with an algorithm to generate the precise 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 ${L}_{{\mathcal{O}}}$, and a set of functions F called the model parameters.

Definition 3. Causal input list. Let ${\mathcal{O}}$ be an ordering that assigns a unique integer in $\{1,2,\ldots ,N\}$ to each member of a set of N variables. Consider an ordered set of variables $\{{X}_{i};i=1,2,\ldots ,N\}$, where the subset of variables X_j with $j\lt i$ are called the predecessors of X_i. A causal input list is the ordered set of CI relations of the form:${L}_{{\mathcal{O}}}:= \{({X}_{i}\perp R({X}_{i})| {\bf{pa}}({X}_{i})):i=1,2,\ldots ,N\}$, where each set ${\bf{pa}}({X}_{i})$ is a subset of the predecessors of X_i called the parents of X_i, and $R({X}_{i})$ are the remaining predecessors of X_i excluding the parents.

Definition 4. Ancestors and descendants. Given a causal input list ${L}_{{\mathcal{O}}}$, consider the set of parents of X_i, their parents' parents, and so on. These are called the ancestors of X_i. Similarly, the descendants of X_i are all variables for which X_i is an ancestor. We will use ${\bf{an}}(X)$ to denote the union of the ancestors of a set X.

Definition 5. Model parameters. Given a causal input list${L}_{{\mathcal{O}}}$, the model parameters are a set$F:= \{{F}_{i}:i=1,2,\ldots ,N\}$ consisting of N probabilistic functions F_i. Each F_i(X) is equivalent to applying a deterministic function ${f}_{i}(X,{U}_{i})$ with probability $P({U}_{i})$ for some auxiliary variable U_i. The U_i are sometimes called error variables and by definition have no parents. Each function F_i determines the probability of X_i conditional on the values of its parents:

$$\begin{eqnarray}&&P({X}_{i}| {\bf{pa}}({X}_{i}))={F}_{i}({\bf{pa}}({X}_{i}))={f}_{i}({\bf{pa}}({X}_{i}),{U}_{i})\\ &&\quad \mathrm{with}\ \mathrm{probability}P({U}_{i}).\end{eqnarray} \tag{ 2 }$$

For variables without any parents, called exogenous variables, the function F_i just specifies a probability distribution over the possible values of X_i, i.e. ${F}_{i}(\varnothing ):= P({X}_{i})$. We assume that all exogenous variables, including any error variables U_i, are independently distributed (the Markovian assumption).

Definition 6. Classical causal model. A classical causal model on N variables is a pair$\{{L}_{{\mathcal{O}}},F\}$ containing a causal input list${L}_{{\mathcal{O}}}$ and model parameters F defined on those variables. Alternatively, a CCM can be specified by the pair$\{{G}_{L},F\}$, where G_L is the graph generated by${L}_{{\mathcal{O}}}$ (see section 2.4).

Given a CCM, we can construct a joint probability by generating random variables in the order specified by ${\mathcal{O}}$ and using the functions F 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:

$$\begin{eqnarray}&&P({X}_{1},{X}_{2},\ldots ,{X}_{N})=\displaystyle \prod _{j=1}^{N}P({X}_{j}| {X}_{j+1},\ldots ,{X}_{N}).\end{eqnarray} \tag{ 3 }$$

The joint probability obtained in this way from a CCM ${\mathcal{M}}$ is said to be generated by ${\mathcal{M}}$ and is denoted ${P}^{{\mathcal{M}}}$. It satisfies the following property:

Causal Markov condition: given that ${P}^{{\mathcal{M}}}$ is generated by a CCM ${\mathcal{M}}$, each variable X_i in ${P}^{{\mathcal{M}}}$ is independent of its non-descendants, conditional on its parents.

Note: our definition of the causal Markov condition follows Pearl ([13]), 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.

Example 1. Consider three variables $X,Y,Z$. Suppose we have the ordering ${\mathcal{O}}:\{X,Y,Z\}\to \{3,2,1\}$, and the causal input list indicates that ${\bf{pa}}(X)=\{Y,Z\}$; ${\bf{pa}}(Y)=\{Z\}$; and ${\bf{pa}}(Z)=\varnothing$. It will be shown in section 2.4 that this generates the graph shown in figure 1. Suppose the model parameters are $F=\{{f}_{x},{F}_{y},{f}_{z}\}$, where ${f}_{x},{f}_{z}$ are deterministic and ${F}_{y}(A):= {f}_{y}(A,{U}_{Y})$ with probability $P({U}_{Y})$. Then we obtain the joint probability $P(X,Y,Z)$ as follows: first, generate the lowest variable in the ordering, Z, using the random function P(Z). Next, generate U_Y using $P({U}_{Y})$ and then apply ${f}_{y}(Z,{U}_{Y})$ to obtain the value of Y. Finally, use ${f}_{x}(Y,Z)$ to obtain the value of X, the last variable in the ordering. The statistics generated by this procedure are given by:

$$\begin{eqnarray*}&&P(X,Y,Z)=\displaystyle \sum _{{U}_{Y}}\;P(X| Y,Z)P(Y| Z,{U}_{Y})P(Z)P({U}_{Y}),\end{eqnarray*}$$

where

$$\begin{eqnarray}&&P(X| Y,Z)=P(X=x| Y=y,Z=z)=\delta (x,{f}_{x}(y,z)),\\ &&P(Y| Z,{U}_{Y})=P(Y=y| Z=z,{U}_{Y}={u}_{y})=\delta (y,{f}_{y}(z,{u}_{y})).\end{eqnarray} \tag{ 4 }$$

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In the previous section, we gave a formal definition of a CCM 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 1. 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 1. We therefore define the direct causes of X_i as the parents of X_i and the indirect causes as the remaining (non-parental) ancestors of X_i; 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 [14]. We note that there exist in the literature numerous definitions of the RCCP, so our chosen definition deserves clarification. It was pointed out in [15] 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 'factorization principle' or FP). Our definition of the RCCP refers only to the factorization property FP. By contrast, [9] takes the RCCP as being equivalent to just the PCC, while the FP is regarded as a separate additional property that happens to hold for classical correlations. Note that the FP and the PCC are both consequences of the definition of a CCM—they follow directly from what we have called the assumption of Markovianity and the fact that the variables are only functionally dependant on their parents.

It is a topic of debate in the literature whether (and to what extent) Reichenbach's principle applies to quantum mechanics, particularly in light of Bell's theorem, where the factorization property often appears as an explicit assumption governing the statistics of a local hidden-variable model [2, 16]. As discussed in section 2.6, we will take the position that the violation of Bell's inequalities in experiments shows that the correlations produced by quantum entanglement cannot be given a causal explanation using the 'local' causal graph of figure 4 (a), if one also assumes that (i) there is no fine-tuning of the data (section. 2.5); (ii) standard probability theory holds; (iii) the FP holds for quantum systems. In this work we will pursue the possibility—though it is by no means the only possibility—that it is precisely (iii) that is violated by quantum mechanics. Hence our framework does call for a rejection of the RCCP⁴ . Note that the PCC need not be violated, since (without conditioning on effects) two independent quantum systems can only become correlated through direct interactions, hence they share a common source.

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A natural question to ask is whether some other physics principle might serve to distinguish quantum correlations from generalized probabilistic theories. Interesting work along these lines can be found in [17–20]. Here we are concerned with finding a graphical separation rule for causal models that could accommodate such relaxations in which the FP no longer applies. The reasons why a new graph-separation rule is useful in this context will be discussed further in section 2.6.

Finally, we note that one can interpret the ordering of variables ${\mathcal{O}}$ as representing the time-ordering of the variables, such that each variable represents a physical quantity localized 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 [21].

It is useful to represent ${L}_{{\mathcal{O}}}$ using a DAG, which can be thought of as the causal 'skeleton' of the model. In the DAG representation of ${L}_{{\mathcal{O}}}$, there is a node representing each variable and a directed arrow pointing to the node from each of its parents. The DAG G_L constructed in this way is said to be generated by the causal input list ${L}_{{\mathcal{O}}}$. 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 X are represented by nodes in the DAG that each have a directed path (i.e. a path along which all arrows are connected head-to-tail) leading to X, and the descendants of X are those nodes that can be reached by a directed path from X. In example 1, the system is represented by the DAG shown in figure 1.

To establish a correspondence between a DAG and the causal input list ${L}_{{\mathcal{O}}}$ that generates it, we need an algorithm for reconstructing a list of CI relations ${L}_{{\mathcal{O}}}$ from a DAG, such that the list generates the DAG. For this purpose, one uses d-separation (see figure 2).

Definition 7. d-separation. Given a set of variables connected in a DAG, two disjoint sets of variables X and Y are said to be d-separated by a third disjoint set Z, denoted ${(X\perp Y| Z)}_{d}$, 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 X to a member of Y is rendered inactive by a member of Z. A path connecting two nodes is rendered inactive by a member of Z if and only if:

• (i)  
  the path contains a chain $i\to m\to j$ or a fork $i\leftarrow m\to j$ such that the middle node m is in Z, or
• (ii)  
  the path contains an inverted fork (head-to-head) $i\to m\leftarrow j$ such that the node m is not in Z, and there is no directed path from m to any member of Z.

A path that is not rendered inactive by Z is said to be active.

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 ${L}_{{\mathcal{O}}}$ 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 Z in precisely those situations where we would physically expect the two variables to be independent conditional on Z: 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:

$$\begin{eqnarray}&&{(X\perp Y| Z)}_{d}\Rightarrow (X\perp Y| Z),\end{eqnarray} \tag{ 5 }$$

i.e. we assume that if X and Y are d-separated by Z in a DAG, then they are conditionally independent given Z in the semi-graphoid closure of any list ${L}_{{\mathcal{O}}}$ that generates the DAG.

Formally, let G be a DAG, and let C(G) be the set of CI relations obtainable from G using d-separation, and $\bar{C}(G)$ the closure of this set. We then have the following theorem:

Theorem 1 let G_L be the DAG generated by ${L}_{{\mathcal{O}}}$. Then $C({G}_{L})={\bar{L}}_{{\mathcal{O}}}$. That is, a CI relation is implied by the DAG G_L if and only if it is deducible from ${L}_{{\mathcal{O}}}$ and the semi-graphoid axioms.

Theorem 1 implies that d-separation is sound, since every CI relation obtainable from the DAG is in the closure of the causal input list ($C({G}_{L})\subseteq {\bar{L}}_{{\mathcal{O}}}$), and complete, since there are no CI relations implied by the causal input list that are not obtainable from the DAG (${\bar{L}}_{{\mathcal{O}}}\subseteq C({G}_{L})$). It also follows that the set of CI relations obtainable from a DAG G is closed under the semi-graphoid axioms, $C(G)=\bar{C}(G)$.

A DAG G defines a partial ordering of the variables. A partial ordering is a weaker notion than a total ordering: given any pair of variables $X,Y$, it either assigns a strict ordering, $X\lt Y$ or $Y\lt X$, or else leaves them unordered, $X\sim Y$. In the case of a DAG, if two variables connected by a directed path then they are strictly ordered according to the direction of the arrows, otherwise they are left unordered.

One can always find a causal input list that generates G as follows: choose a total ordering ${\mathcal{O}}$ that is consistent with the partial ordering imposed by G, and then write down the ordered list of CI relations of the form $({X}_{i}\perp R({X}_{i})| {\bf{pa}}({X}_{i}))$, where the parents of each variable are the same as the parents of its representative node in G. Moreover, the list obtained in this way is unique, modulo some freedom in the ordering of causally separated events (e.g. 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 6).

Example 2. Consider the DAG shown in figure 3. The graph assigns parents as follows:

$$\begin{eqnarray}{\bf{pa}}(X) & = & \varnothing ,\\ {\bf{pa}}(V) & = & X,\\ {\bf{pa}}(Y) & = & X,\\ {\bf{pa}}(W) & = & V,\\ {\bf{pa}}(Z) & = & V\cup Y.\end{eqnarray} \tag{ 6 }$$

A total ordering consistent with this graph is ${\mathcal{O}}:\{X,V,W,Y,Z\}\to \{1,2,3,4,5\}$. Therefore, we obtain the causal input list:

$$\begin{eqnarray}{L}_{{\mathcal{O}}} & = & \{(X\perp \varnothing | \varnothing ),(V\perp \varnothing | X),(W\perp X| V),\\ & & (Y\perp \varnothing | X),(Z\perp {XW}| {YV})\}.\end{eqnarray} \tag{ 7 }$$

Finally, we have the following useful definitions:

Definition 8. Independence maps and perfect maps. Given P(X) and a DAG G on the same variables, we will call G an independence map (I-map) of P iff $\bar{C}(G)\subseteq \bar{C}(P)$. If equality holds, $\bar{C}(G)=\bar{C}(P)$, then G is called a perfect map of P.

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 dependence in the statistics. Interestingly, there exist distributions for which no DAG is a perfect map. This fact forms the basis for the criterion of faithfulness of a CCM, discussed in the following section.

Suppose we obtain the values of some physical observables over many runs of an experiment and 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 might be propagating 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 QCM 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 [13] for inference in the classical case, and [23, 24] for attempts at quantum generalizations).

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 P(X) 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 P(X) to some larger distribution $P\prime (X,\lambda )$ that includes latent variables λ, such that the observed statistics are the marginal probabilities obtained by summing over the unobserved variables:

$$\begin{eqnarray}&&P(X)=\displaystyle \sum _{\lambda }P\prime (X,\lambda ).\end{eqnarray} \tag{ 8 }$$

If there exists a CCM ${\mathcal{M}}(X,\lambda )$ such that ${P}^{{\mathcal{M}}}(X,\lambda )=P\prime (X,\lambda )$, then we can say that this CCM explains the probability distribution P(X) with the aid of the latent variables λ. The admittance of hidden variables in causal models seems to lead to a problem: it turns out that every probability distribution P(X) 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 9. Faithfulness. Consider a distribution P(X) and a CCM ${\mathcal{M}}(X)=\{G,F\}$ that generates P(X). The explanation offered by ${\mathcal{M}}(X)$ is called faithful to P(X) iff the DAG derived from ${\mathcal{M}}(X)$ is a perfect map of P(X), i.e.$\bar{C}(G)=\bar{C}(P)$.

Latent variables: Suppose there is no CCM ${\mathcal{M}}(X)$ that is faithful to P(X). Consider instead a CCM ${\mathcal{M}}(X,\lambda )=\{G\prime ,F\prime \}$, which obtains P(X) by summing the generated distribution $P\prime (X,\lambda )$ over the hidden variables λ. This extended CCM is considered faithful to P(X) iff every CI relation in P(X) is implied by the extended DAG G', i.e. $\bar{C}(P)\subseteq \bar{C}(G\prime )$.

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 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; this will be relevant when we consider quantum mechanics in section 2.6.

Geiger and Pearl [25] showed that, for every DAG G, one can explicitly construct a distribution P such that G is faithful for P, i.e. such that $\bar{C}(G)=\bar{C}(P)$ holds. Furthermore, if there exists a DAG G that is faithful for a given P, and if we assume there are no latent variables, then it can be shown that there exists a set of model parameters F such that the CCM $\{G,F\}$ generates P [26].

Finally, faithfulness can be equivalently defined as the rejection of fine-tuning of the model parameters, as noted in [2, 13]. In particular, if a DAG is unfaithful, this implies that there exists at least one CI 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 [13]. Thus, the only way that such additional CI relations could arise is by a kind of 'conspiracy' of the model parameters F to ensure that the extra CI 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 3. Recall the system of example 1 and suppose that we observe $(X\perp Y| Z)$ in P. This CI relation is not found in ${\bar{L}}_{{\mathcal{O}}}$; in fact, there is a directed edge from Y to X in the DAG of ${L}_{{\mathcal{O}}}$ (figure 1). The only way to account for this discrepancy is if the model parameters F are chosen such that the predicted signal from Y to X is obscured. This could happen if $X,Y,Z$ are positive integers and we choose model parameters ${f}_{y}[z,{u}_{y}]={u}_{y}+z$ and ${f}_{x}[y,z]=y+z-k$ for some integer constant k, and F_y is restricted such that $P({u}_{y}=k)=1$. Then we obtain the joint probability:

$$\begin{eqnarray}P(X,Y,Z) & = & \displaystyle \sum _{{U}_{Y}}\;\delta (x,{f}_{x}[\;y,z]\;)\;\delta (y,{f}_{y}[\;z,{u}_{y}]\;)P({U}_{Y})P(Z),\\ & = & \delta (x,y+z-k\;)\;\delta (y,k+z\;)P(Z),\\ & = & \delta (x,2\;z)\;\delta (y,k+z\;)P(Z),\end{eqnarray} \tag{ 9 }$$

which factorizes into $P(X| Z)P(Y| Z)P(Z)$. We see that the value of X tells us nothing about the value of Y, because the k's conveniently cancel out in $P(X| {YZ})$, leading to the observed independence $(X\perp Y| Z)$. This can be understood as fine-tuning of F_y such that its error variable is constant. Indeed, if $P({U}_{Y})$ was distributed over several values, then X would still carry information about Y in accordance with the causal structure. Hence, absence of fine-tuning in P with respect to a causal input list ${L}_{{\mathcal{O}}}$ can also be defined as the requirement that the CI relations observed in P should be robust under changes in the model parameters consistent with ${L}_{{\mathcal{O}}}$ [13].

Consider a probability distribution $P(A,B,S,T)$, satisfying the generating set of CI relations $K:= \{(S\perp T| \varnothing ),(A\perp T| S),(B\perp S| T)\}$. Suppose that the closure of this set contains all the CI relations satisfied by P, i.e. $\bar{K}=\bar{C}(P)$. This represents a generic Bell-type experiment: the setting variables $S,T$ are independent of each other, and there is no signalling from S to B or from T to A, but the outputs $A,B$ are correlated. (Here, the absence of signalling is 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 nonlocal hidden variables, etc.)

Note that we cannot explain these correlations by a CCM without latent variables, because A and B are correlated without a common cause. Hence let us consider the extended distribution $P\prime (A,B,S,T,\lambda )$ satisfying ${\displaystyle \sum }_{\lambda }P\prime (A,B,S,T,\lambda )=P(A,B,S,T)$. Of course, we require that $\bar{K}\subseteq \bar{C}(P\prime )$, but we can impose additional physical constraints on the hidden variable λ. In particular, we expect λ to be independent of the settings $S,T$ 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 $K\prime := \{(S\perp T\lambda | \varnothing )$, $(T\perp S\lambda | \varnothing ),(A\perp T| S\lambda )$, $(B\perp S| T\lambda )\}$ and we assume that $\bar{K\prime }\subseteq \bar{C}(P\prime )$, 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 A and B are independent conditional on the hidden variable λ in the distribution P', i.e. if $(A\perp B| \lambda )$ holds in $\bar{C}(P\prime )$, then it is easy to see that λ qualifies as a common cause of A and B and the correlations can be explained by a CCM with the DAG G' shown in figure 4 (a). It can be shown that this occurs whenever P satisfies Bell's inequality [2]. Conversely, Wood and Spekkens showed that if P violates Bell's inequality, there is no CCM that can faithfully explain P, even allowing hidden variables. Of course, one can find numerous un-faithful explanations, such as the DAG in figure 4 (b) and fine-tuning of the model parameters to conceal the causal influence of S on B. This result implies that, in general, CCMs cannot faithfully explain the correlations seen in entangled quantum systems.

Example 4. 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 figure 4(b). Here, λ is a hidden variable that carries information about the setting S faster than light to the outcome B. The model posits a CCM that generates a distribution $P\prime (A,B,S,T,\lambda )$ and the observed statistics $P(A,B,S,T)$ are interpreted as the marginal obtained from this distribution by summing over λ. The no-signaling CI relations $(A\perp T| S),(B\perp S| T)$ hold in the observed statistics, however they do not follow from the DAG figure 4(b) which includes the hidden variables, hence the CCM that generates P' using this graph is not a faithful explanation for P. In general, the deBroglie–Bohm interpretation and its variants appear to require fine-tuning [2].

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 figure 4(a) as a quantum circuit, where λ 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 $(A\perp B| \lambda )$ to hold in the extended set of constraints K' for the 'common cause' λ, and hence that we abandon RCCP (specifically, the factorization property). Since we aim to use a DAG like figure 4(a) as the explanation, this implies that d-separation is no longer the correct criterion for reading CI relations from the graph, now that 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 QCM.

We begin by considering quantum networks modelled as a DAGs, in which the nodes represent state preparations, 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 e.g. [27, 28].

The general theory of quantum networks as given in [7] 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 three classes: as before, those with only outgoing edges are called exogenous; those with only ingoing edges are called drains; those with ingoing and outgoing edges are called intermediates. We will typically restrict our attention to connected graphs, i.e. graphs in which there are no isolated nodes having no parents or children. Such variables can usually be ignored since they are irrelevant to all other variables of interest. However, in cases where they do arise (e.g. when removing nodes in the proof of theorem 3 later on in section 3.3), they can be interpreted as indicating an operation performed on a system not connected to anything else—like the value of a dial on a broken telephone.

We assign the following interpretations to the elements of the DAG:

Edges: every edge in a DAG is associated with a Hilbert space of finite dimension $d\gt 1$. Thus, we can associate a physical system with finite degrees of freedom to every edge. The Hilbert space dimension is allowed to be different for different edges in the DAG.

Exogenous nodes: every exogenous node is associated with a random variable. Each possible value of the variable corresponds to the preparation of a normalized quantum state, represented by a density matrix. The set of 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 ${{\mathcal{H}}}^{(\mathrm{out})}$, which is the tensor product of the Hilbert spaces of all the outgoing edges.

Intermediate nodes: every node with both incoming and outgoing edges is associated with a random variable. Each value of the variable represents a general transformation (a CPT map) ${\mathcal{C}}:{{\mathcal{H}}}^{(\mathrm{in})}\to {{\mathcal{H}}}^{(\mathrm{out})}$ from the input to output Hilbert spaces.

Drains: every drain is associated with a random variable, which may be classed as either a setting or an outcome (see section 3.2). For outcomes, each value of the variable corresponds to one of the possible outcomes of a general measurement (POVM) on the incoming systems ${{\mathcal{H}}}^{(\mathrm{in})}$.

In the alternative case where the drain node is a setting, it is treated just like an intermediate node (each value corresponds to a CPT map on ${{\mathcal{H}}}^{(\mathrm{in})}$) except that the outgoing systems are discarded. Since these variables will turn out to have no dependence on any other variables in the system, they could be ignored just like isolated nodes, although we will find it useful to include them in the proof of theorem 3 in section 3.3.

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.

Example 5. Consider the circuit in figure 5(a). This describes the preparation of two qubits as mixtures ${\rho }_{1}={\gamma }_{1}| {\psi }_{0}\rangle \langle {\psi }_{0}| +(1-{\gamma }_{1})| {\psi }_{1}\rangle \langle {\psi }_{1}|$ and ${\rho }_{2}={\gamma }_{2}| {\psi }_{0}\rangle \langle {\psi }_{0}| +(1-{\gamma }_{2})| {\psi }_{1}\rangle \langle {\psi }_{1}|$ in an arbitrary orthogonal basis $\{| {\psi }_{0}\rangle ,| {\psi }_{1}\rangle \}$. These are followed by a measurement of the first qubit in the computational ${\sigma }_{Z}$ basis $\{| 0\rangle ,| 1\rangle \}$ and the subsequent application of a ${\sigma }_{X}$ 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 ${U}_{1},{U}_{2}$ or U₃) to a third ancilla qubit $| \phi \rangle$. The ancilla is traced out and the remaining qubit measured in the ${\sigma }_{Z}$ basis. In figure 5(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 ${\sigma }_{Z}$ basis, whose outcome can be ignored. In this form, the circuit can be cast directly into a DAG, as shown in figure 5(c). The variables X and Y take values corresponding to the basis states $\{| {\psi }_{0}\rangle ,| {\psi }_{1}\rangle \}$, distributed with probabilities so as to produce the mixed states ${\rho }_{1},{\rho }_{2}$. Z and S are single-valued, corresponding to the state $| \phi \rangle$ and the unitary CNOT respectively. T has three values corresponding to the three possible unitaries, and is distributed according to the probability of each unitary being implemented. Finally, $U,V,W$ are all binary-valued, corresponding to outcomes $\in \{0,1\}$ and whose probabilities are given by quantum mechanics (see section 3.2).

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The example illustrates that it is also possible (although not necessary) to restrict all operations to pure states, unitaries and projective measurements and treat more general operations as epistemic mixtures of these. In the rest of the paper we allow the model parameters to be arbitrary, without confining ourselves to any particular convention.

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

Recall that the classical causal input list ${L}_{{\mathcal{O}}}$ represents a set of CI 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 section 2.3, 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 S_i and outcomes O_i (we continue to use X to denote a generic variable or set of variables). The settings S_i determine the states produced at the sources and the transformations applied in the network, while the O_i represent the outcomes of general measurements. Since the settings S_i 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 [9] (recall the discussion of section 2.3).

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 transformation. 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 transformation are assumed to be independent, they cannot carry any information about each other, nor can the variable representing the transformation 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 2. The possible causes of an outcome are its ancestors. (Compare to assumption 1.)

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 [9] that there is no reason to maintain the distinction between direct and indirect causes in any generalized model that goes beyond classical correlations.

It is clear from our discussions in section 2 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 factorization property of the RCCP (recall section 2.3). 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 O and settings S. The following definitions will also be useful (see figure 6):

Definition 10. O'-chain. Given a set of outcomes O', two other sets of outcomes O₁ and O₂ are said to be connected by an O'-chain iff O₁ shares an ancestor with a member of O' that shares an ancestor with another member of O', (etc), that shares an ancestor with O₂. A set of settings S₁ is linked to O₂ by an O'-chain iff S₁ has a descendant inO₂, or a descendant O' that is connected by an O'-chain to O₂. Similarly,S₁ andS₂ are connected by an O'-chain iff they both have descendants in O' that are connected in this way.

Definition 11. O'-detached. Given a set of outcomes O' and some variables V, the set of all variables not connected to V by an O'-chain are said to be O'-detached from V, denoted ${{\bf{dt}}}_{{O}^{\prime }}(V)$.

Note that when O' is the empty set, the detached variables ${{\bf{dt}}}_{\varnothing }(V)$ are just those outcomes that do not share an ancestor with outcomes in V, and those settings that are not ancestors of outcomes in V—which are precisely the variables identified by the quantum causality condition when applied to the outcomes in V.

Intuitively, the O'-chain tells us when conditioning on O' will make two independent variables dependant. If two variables in a DAG are connected by an O'-chain, assuming there is no directed path from one to the other, it means either they are connected by a path on which every collider has a directed path to O', or they are both settings whose descendants in O' share a common cause. Hence, the mutually detached variables are those nodes in the graph for which every path between them contains at least one collider that does not lead to O', or else both nodes are settings and at least one of them has no descendants in O'. This will be useful later when we consider graph separation.

Notation: let $\neg X$ denote the complement of a set X, let $X/S$ denote only the outcomes in X, and let ${\neg }_{S}X$ be all settings not in X, i.e. ${\neg }_{S}X:= S\cap \neg X$. Under a choice of ordering ${\mathcal{O}}$, let $S(\lt {X}_{i})$ denote the set of predecessors of X_i in S. Using these definitions, we propose the following characterization of the CI relations in a quantum network:

Definition 12. Quantum input list. A quantum input list ${Q}_{{\mathcal{O}}}$ is a pair$\{{\mathrm{PA}}_{{\mathcal{O}}},Q\}$, containing:

• (i)  
  An ordered list of parents, ${\mathrm{PA}}_{{\mathcal{O}}}:= \{{\bf{pa}}({X}_{i}):i=1,2,\ldots ,N\}$, where each set of parents ${\bf{pa}}({X}_{i})$ is a subset of $S(\lt {X}_{i})$. Members of O cannot have children. Ancestors, descendants, etc, are defined from the list of parents in the usual way.
• (ii)  
  A set of CI relations denoted Q, constructed as follows. For every subset of settings S' and outcomes O', there is a CI relation of the form $(S\prime \;\perp \;{{\bf{dt}}}_{{O}^{\prime }}(S\prime )| O\prime )$ and a CI relation of the form $(O\prime \;{\bf{an}}(O\prime )\perp \;{{\bf{dt}}}_{\varnothing }(O\prime )| \varnothing )$ in Q.

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 conditioned on an O'-chain that connects them. The second CI relation simply expresses the quantum causality condition, since ${{\bf{dt}}}_{\varnothing }(O\prime )$ contains both the 'non-causes' of O', ${\neg }_{S}{\bf{an}}(O\prime )$, and the outcomes that do not share a common cause with O'.

The quantum input list ${Q}_{{\mathcal{O}}}$ is said to be compatible with a given probability distribution P iff $\bar{Q}\subseteq \bar{C}(P)$. 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 G_Q constructed in this way is said to be generated by the list ${Q}_{{\mathcal{O}}}$. As usual, the ancestors of X are those nodes in the graph that have a directed path leading to X. The quantum input list defines the causal constraints on the variables, based on their interpretation as settings and outcomes in a quantum network. All conditional independencies in this list are expected to hold in a quantum circuit, when the circuit is expressed as a DAG as outlined in section 3.1. In addition, we conjecture that the list captures all such relations:

Conjecture: if a CI relation holds in every quantum network represented by a DAG G, then it is implied by Q in any quantum input list ${Q}_{{\mathcal{O}}}$ that generates G.

One way to prove this conjecture would be to show that it is always possible to find parameters (i.e. states, operations and measurements) on a quantum network that violate any relation not implied by its quantum input list. We content ourselves with the following plausibility argument: suppose that there exists some CI relation R that holds for a quantum circuit with DAG G, but which does not appear in a quantum input list that generates G. This suggests that R is not implied by the quantum causality condition, hence that it expresses the independence of two outcomes that share a common ancestor, or the independence of an outcome and a setting that is its ancestor. There seems nothing preventing us from making a new circuit with the same DAG in which all outcomes with a common ancestor are correlated with each other, and in which each outcome is dependent on all of its ancestors (assuming no restrictions on the size of the Hilbert spaces involved). Since this would invalidate R, we conclude that there is no relation that holds in all circuits with the same DAG, but does not appear in the quantum input list, vindicating the conjecture. However, a rigorous proof is still needed.

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, 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 13. Quantum model parameters. Consider a set of variables X_i with outcome spaces${{\mathcal{E}}}_{{X}_{i}}$, with the drain nodes partitioned into settings or outcomes, connected in a DAG G_Q. Then the quantum model parameters F_q consist of:

• (i)  
  For each edge, a Hilbert space of finite dimension $n\gt 1$;
• (ii)  
  For every outcome drain node X_i, a POVM with an outcome for every value in ${{\mathcal{E}}}_{{X}_{i}}$;
• (iii)  
  For every exogenous node X_i, a normalized quantum state for every value in ${{\mathcal{E}}}_{{X}_{i}}$;
• (iv)  
  For every intermediate node (and every setting drain node) X_i, a CPT map ${\mathcal{C}}:{{\mathcal{H}}}^{(\mathrm{in})}\to {{\mathcal{H}}}^{(\mathrm{out})}$ for every value in ${{\mathcal{E}}}_{{X}_{i}}$;
• (vi)  
  A marginal probability distribution on the outcome space ${{\mathcal{E}}}_{{X}_{i}}$ of every variable that is not an outcome drain 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. The distributions given in (vi) represent the experimenters' setting 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 14. Quantum causal model. A QCM on a set of variables X is a pair $\{{Q}_{{\mathcal{O}}},{F}_{q}\}$ consisting of a quantum input list ${Q}_{{\mathcal{O}}}$ for the set X, and a set of quantum model parameters F_q for the DAG G_Q generated by the input list.

Every QCM defines a joint probability distribution over its variables according to the following procedure. Consider a QCM on N ordered variables $\{{X}_{i}:i=1,2,\ldots ,N\}$, and partition of the set $\{1,2,\ldots ,N\}:= S\cup O$ such that $i\in S$ labels the exogenous variables, intermediate variables, and setting drain variables, and $i\in O$ labels the variables corresponding to outcome drains in the DAG G_Q. From F_q we obtain the mutually independent marginal distributions $P({X}_{i})\;\forall i\in S$, which includes all setting variables. The joint probability of the outcomes conditional on the settings, $P({\cup }_{i\in O}{X}_{i}| {\cup }_{i\in S}{X}_{i})$, is computed in the usual way from the quantum circuit obtained from the DAG G_Q using the states and transformations associated with the settings $\{{\cup }_{i\in S}{X}_{i}\}$. One thus obtains the total joint probability:

$$\begin{eqnarray}&&P(X)=\displaystyle \prod _{i\in S}\;P({X}_{i})\;P({\cup }_{j\in O}{X}_{j}| {\cup }_{j\in S}{X}_{j}).\end{eqnarray} \tag{ 10 }$$

Note: Our definition of a QCM on a DAG G can be regarded as a concrete example of the more general notion of a quantum correlation on the graph G, as defined by Fritz [9]. In particular, working in the category ${\mathcal{C}}$ 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 ${\mathcal{C}}$-instruments in the language of [9], allowing us to compute probabilities. Our model distinguishes these functions according to the placement of their nodes in the graph (exogenous, intermediate or drain). This convention does not represent a limitation 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 generalization to continuous variables following [9] 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 ${Q}_{{\mathcal{O}}}$ from the DAG G_Q. This is proposed in the next section.

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 section 2.3). 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 S₁ to the outcome O₂ and no chain of conditioned effects, one would expect the two to be independent. We must be careful, however: it is also possible for O₂ to be correlated with another outcome that is descended from S₁, such that conditioning on the latter outcome correlates S₁ with the causally separated outcome O₂. 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 15. q-separation. Given a DAG representing a quantum network, two disjoint sets of variables X and Y are said to be q-separated by a third disjoint set Z, denoted ${(X\perp Y| Z)}_{q}$, iff every undirected path between X and Y is rendered inactive by a member of Z. A path connecting two variables is rendered inactive by Z iff at least one of the following conditions is met (see figure 7):

• (i)  
  both variables are settings, and at least one of the settings has no directed path to any outcome in Z;
• (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 Z;
• (iii)  
  the path contains a collider $i\to m\leftarrow j$ where m is not an outcome in Z, and there is no directed path from m to any outcome in Z.

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.

In this section we prove the soundness and completeness of q-separation. The proof approximately follows that of Pearl and Verma [22] for d-separation. Consider the set of CI relations obtainable from a DAG G using the q-separation criterion (definition 15) and assuming ${(X\perp Y| Z)}_{q}\Rightarrow (X\perp Y| Z)$. Let this set be denoted C_q(G), with ${\bar{C}}_{q}(G)$ its closure. In fact, we prove in appendix that q-separation satisfies the semi-graphoid axioms, and therefore this set is equal to its closure: ${C}_{q}(G)={\bar{C}}_{q}(G)$. Hence, C_q(G) and ${\bar{C}}_{q}(G)$ are used interchangeably in what follows. If we replace d-separation with q-separation in definition 8, we obtain analogous criteria for G to be an I-map or a perfect map of a given distribution P. We can now prove the following useful theorem:

Theorem 2. : Let the DAG G be a perfect map of a distribution P(X) under q-separation, i.e. ${\bar{C}}_{q}(G)=\bar{C}(P)$. Then there is a quantum input list ${Q}_{{\mathcal{O}}}$ compatible with P that generates the DAG G.

Proof. The DAG G imposes a partial order on the variables X. Choose any total order ${\mathcal{O}}$ that is consistent with this. Label the nodes in G as outcomes ${O}_{i}\in O$ if they are drains, and settings ${S}_{i}\in S$ otherwise. Define the parents ${\bf{pa}}({X}_{i})$ in ${\mathrm{PA}}_{{\mathcal{O}}}$ to be the nodes with directed edges pointing to X_i in the graph. A path between a setting and any other variable is rendered inactive by outcomes O' if there is at least one collider on the path not in O' and with no directed path to O', or if the other variable is also a setting and O' does not contain descendants of both settings (condition (i) of q-separation). This is true for all variables that are O'-detached from the setting, hence the CI relation $(S\prime \;\perp \;{{\bf{dt}}}_{{O}^{\prime }}(S\prime )| O\prime )$ is implied by G. For each set of outcome nodes and their ancestors, $O\prime {\bf{an}}(O\prime )$, a path from this set to the non-ancestors ${\neg }_{S}{\bf{an}}(O\prime )$ can only be activated by conditioning on an outcome. Furthermore, a path from $O\prime {\bf{an}}(O\prime )$ to ${{\bf{dt}}}_{\varnothing }(O\prime )/S$ must contain a collider, or else O' and ${{\bf{dt}}}_{\varnothing }(O\prime )/S$ 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 since ${\neg }_{S}{\bf{an}}(O\prime )\cup {{\bf{dt}}}_{\varnothing }(O\prime )/S={{\bf{dt}}}_{\varnothing }(O\prime )$,the CI relation $(O\prime {\bf{an}}(O\prime )\perp \;{{\bf{dt}}}_{\varnothing }(O\prime )| \varnothing )$ is implied by G. According to definition 12, these ingredients are sufficient to specify a quantum input list. By construction, this list also generates the DAG G. □

Corollary 1. Let ${Q}_{{\mathcal{O}}}$ be any quantum input list that generates a DAG G. Then every relation in the closure $\bar{Q}$ is implied by G via q-separation, i.e. $\bar{Q}\subseteq {\bar{C}}_{q}(G)$. To see why, let ${Q}_{{\mathcal{O}}}^{\prime }$ be the input list generated from G by the method described in theorem 2. According to that theorem, $\bar{Q\prime }\subseteq {\bar{C}}_{q}(G)$ for this input list. The corollary follows from the fact that every quantum input list that generates G must be equivalent to ${Q}_{{\mathcal{O}}}^{\prime }$, up to a choice of total ordering consistent with G.

The next theorem provides the key result.

Theorem 3. Given a quantum input list ${Q}_{{\mathcal{O}}}$ that is compatible with some distribution P(X), the DAG G generated by ${Q}_{{\mathcal{O}}}$ is an I-map of P, that is, ${\bar{C}}_{q}(G)\subseteq \bar{C}(P)$.

Proof. We prove the result by induction on the number of variables. First we show that the result holds for k variables, given that it holds for $k-1$ variables. Then we note that the result holds trivially for one variable; hence, by induction, it holds for any number of variables.

Let P be a distribution on k variables and ${Q}_{{\mathcal{O}}}$ a compatible quantum input list, which generates the DAG G. Let n be the last variable in the ordering ${\mathcal{O}}$; let $\bar{C}(P-n)$ be the closed set of CI relations formed after removing from $\bar{C}(P)$ all CI relations involving n; let $P-n$ be any probability distribution having exactly the closed set of CI relations $\bar{C}(P-n)$ (such a distribution can always be constructed [13]); and let $G-n$ be the DAG formed by removing the node n and all its connected edges from the graph G.

Consider the list obtained from ${Q}_{{\mathcal{O}}}$ by removing every CI relation involving n from Q and removing ${\bf{pa}}(n)$ from the list of parents, and let the reduced list of parents be denoted ${\mathrm{PA}}_{{\mathcal{O}}}-n$ (Note that this procedure might result in one or more settings that are isolated or drain nodes in the graph). Let the resulting list be denoted $Q-n$. Then the pair ${Q}_{{\mathcal{O}}}-n:= \{{\mathrm{PA}}_{{\mathcal{O}}}-n,Q-n\}$ is a valid quantum input list on $k-1$ variables. Furthermore, by construction, ${Q}_{{\mathcal{O}}}-n$ generates the DAG $G-n$.

Let us now assume that $G-n$ is an I-map of $P-n$: ${\bar{C}}_{q}(G-n)\subseteq \bar{C}(P-n)$. We aim to prove that, under this assumption, G is also an I-map of P, ${\bar{C}}_{q}(G)\subseteq \bar{C}(P)$. To do so, we will consider each CI relation of ${\bar{C}}_{q}(G)$ and show that it exists also in $\bar{C}(P)$.

The CI relations of ${\bar{C}}_{q}(G)$ can be divided into three cases:

• (1)  
  n does not appear in the CI relation;
• (2)  
  n appears in the first position in the CI relation, e.g. $({nX}\perp Y| Z)$;
• (3)  
  n appears in the last position in the CI relation, e.g. $(X\perp Y| {nZ})$.

Note that, if n 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 $X,Y,Z$ be disjoint sets of variables and let $R\in {\bar{C}}_{q}(G)$ be a relation of the form $(X\perp Y| Z)$ that does not contain the variable n. Then $R\in \bar{C}(P)$.

Proof. Since R is in ${\bar{C}}_{q}(G)$, it must also be in ${\bar{C}}_{q}(G-n)$. If it were not, then there would be a path between X and Y that is active in ${\bar{C}}_{q}(G-n)$ but rendered inactive by Z in ${\bar{C}}_{q}(G)$. 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 $G-n$ is an I-map of $P-n$, the relation R must also be contained in $\bar{C}(P-n)$, and since $\bar{C}(P-n)$ is a subset of $\bar{C}(P)$, R is also contained in $\bar{C}(P)$.

For the remaining two cases, we first consider the special instance where n is a setting variable:

Lemma 2. Let $R\in {\bar{C}}_{q}(G)$ be either of the form $({nX}\perp Y| Z)$ or $(X\perp Y| {nZ})$ and n is a setting variable. Then $R\in \bar{C}(P)$.

Proof. If either choice of R holds in G, then $(X\perp Y| Z)$ must hold in G. If this were not true for $R=({nX}\perp Y| Z)$, there would be an active path between X and Y conditional on Z, but this would imply an active path between nX and Y conditional on Z, contradicting R. It must also be true for $R=(X\perp Y| {nZ})$ because conditioning on a setting n cannot deactivate a previously active path. Since $(X\perp Y| Z)$ holds in G, by lemma 1 it holds in $\bar{C}(P)$. Let ${Z}_{O}({Z}_{S})$ denote the outcomes (settings) in Z. Since n is a setting with no descendants, it is O'-detached from any subset of variables, for any set of outcomes O'; in particular, $(n\;\perp \;{{\bf{dt}}}_{{Z}_{O}}(n)| {Z}_{O})$ holds in the quantum input list ${Q}_{{\mathcal{O}}}$, hence in $\bar{C}(P)$. Noting that XYZ_S is a subset of ${{\bf{dt}}}_{{Z}_{O}}(n)$ and using the sem-graphoid axioms, we obtain:

$$\begin{eqnarray*}&&(n\perp {{\bf{dt}}}_{{Z}_{O}}(n)| {Z}_{O})\in \bar{C}(P),\\ &&\quad \Rightarrow (n\perp {{XYZ}}_{S}| {Z}_{O})\in \bar{C}(P),\\ &&\quad \Rightarrow (n\perp Y| {XZ})\in \bar{C}(P).\end{eqnarray*}$$

Combining this with $(X\perp Y| Z)\in \bar{C}(P)$ by the contraction axiom we obtain $({nX}\;\perp \;Y| Z)\in \bar{C}(P)$, which also implies $(X\;\perp \;Y| {nZ})\in \bar{C}(P)$, so either way R is in $\bar{C}(P)$.

We now examine the last two cases under the assumption that n is an outcome.

Lemma 3. Let $R\in {\bar{C}}_{q}(G)$ be a relation of the form $({nX}\perp Y| Z)$. Then $R\in \bar{C}(P)$.

Proof. First, we partition the sets $X,Y,Z$ into disjoint sets of outcomes and settings, e.g. $Z={Z}_{O}\cup {Z}_{S}$ where Z_O contains only outcomes and Z_S only settings.

Define the set ${\bf{zo}}(O\prime )$ as the members of Z_O that are connected to outcomes O' by a Z_O-chain. Let ${\bf{-zo}}(O\prime )$ denote its complement in Z_O. Next, consider ${O}_{x}:= {{nX}}_{O}{\bf{zo}}({{nX}}_{O})$.

We can write the ancestors ${\bf{an}}({O}_{x})$ as the union of four disjoint sets, ${\bf{an}}({O}_{x})={A}_{X}\cup {A}_{Y}\cup {A}_{Z}\cup A$, where ${A}_{X}:= {\bf{an}}({O}_{x})\cap X$, and similarly for A_Y and A_Z. Any remaining members of ${\bf{an}}({O}_{x})$ not contained in any of $X,Y,Z$ are contained in A. Likewise, let us decompose ${\neg }_{S}{\bf{an}}({O}_{x})$ into disjoint sets: ${\neg }_{S}{\bf{an}}({O}_{x})={B}_{X}\cup {B}_{Y}\cup {B}_{Z}\cup B$ where e.g. ${B}_{X}:= {\neg }_{S}{\bf{an}}({O}_{x})\cap X$ and similarly for ${B}_{Y},{B}_{Z}$ (see figure 8). Note that ${X}_{S}={A}_{X}\cup {B}_{X}$ and analogously for Y and Z. The CI relation $({O}_{x}\;{\bf{an}}({O}_{x})\perp {{\bf{dt}}}_{\varnothing }({O}_{x})| \varnothing )$ must hold in Q and hence in $\bar{C}(P)$. Using the above definitions:

$$\begin{eqnarray}&&({O}_{x}\;{\bf{an}}({O}_{x})\perp \;{{\bf{dt}}}_{\varnothing }({O}_{x})| \varnothing )\in \bar{C}(P)\\ &&\quad \Rightarrow ({O}_{x}\perp {\neg }_{S}{\bf{an}}({O}_{x})\;[{{\bf{dt}}}_{\varnothing }({O}_{x})/S]| {\bf{an}}({O}_{x}))\in \bar{C}(P)\\ &&\quad \Longleftrightarrow ({O}_{x}\perp {B}_{X}{B}_{Y}{B}_{Z}B\;[{{\bf{dt}}}_{\varnothing }({O}_{x})/S]| {A}_{X}{A}_{Y}{A}_{Z}A)\in \bar{C}(P).\end{eqnarray} \tag{ 11 }$$

The set ${\bf{-zo}}({{nX}}_{O})$ must be a subset of ${{\bf{dt}}}_{\varnothing }({O}_{x})/S$ (since the members of ${\bf{-zo}}({{nX}}_{O})$ by definition cannot share an ancestor with O_x). The same goes for Y_O, otherwise there would be a path connecting Y_O to ${{nX}}_{O}$ on which every collider is in Z_O or has a directed path to Z_O, and they could not be q-separated in G as is required for R to be true. Hence, using the semi-graphoid axioms:

$$\begin{eqnarray}&&({O}_{x}\perp {B}_{X}{B}_{Y}{B}_{Z}B\;{{\bf{dt}}}_{\varnothing }({O}_{x})/S| {A}_{X}{A}_{Y}{A}_{Z}A)\in \bar{C}(P)\\ &&\quad \Rightarrow ({O}_{x}\perp {B}_{Y}{B}_{Z}\;{Y}_{O}\;{\bf{-zo}}({{nX}}_{O})| {X}_{S}{A}_{Y}{A}_{Z})\in \bar{C}(P)\\ &&\quad \Rightarrow (n\perp {B}_{Y}{Y}_{O}| {{XZA}}_{Y})\in \bar{C}(P).\end{eqnarray} \tag{ 12 }$$

No member of Y can be an ancestor of ${{nX}}_{O}$ in Q, or else there would be a directed path from a setting in Y_S to an outcome in ${nX}$ and they could not be q-separated by Z in G, contradicting our initial premise R. Therefore ${A}_{Y}=\varnothing$ and ${B}_{Y}={Y}_{S}$, and (12) implies $(n\perp Y| {XZ})\in \bar{C}(P)$. The relation R implies $(X\perp Y| Z)\in {\bar{C}}_{q}(G)$ and hence (by lemma 1): $(X\perp Y| Z)\in \bar{C}(P)$. Combining this with (12) and the semi-graphoid axioms, we obtain the desired result $({nX}\perp Y| Z)\in \bar{C}(P)$.

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Lemma 4. Let $R\in {\bar{C}}_{q}(G)$ be a relation of the form $(X\perp Y| {nZ})$. Then $R\in \bar{C}(P)$.

Proof. Note that n cannot share an ancestor with both ${X}_{O}{\bf{zo}}({X}_{O})$ and ${Y}_{O}{\bf{zo}}({Y}_{O})$, or else there would be a path connecting X_O to Y_O on which every collider has a descendant in ${{nZ}}_{O}$, preventing them from being q-separated in G. We therefore assume without loss of generality that n does not share an ancestor with ${Y}_{O}{\bf{zo}}({Y}_{O})$. By a similar argument, no member of X_O can share an ancestor with ${Y}_{O}{\bf{zo}}({Y}_{O})$; hence ${{nX}}_{O}{\bf{-zo}}({Y}_{O})\in {{\bf{dt}}}_{\varnothing }({Y}_{O}{\bf{zo}}({Y}_{O}))$. Now, either n shares an ancestor with ${X}_{O}{\bf{zo}}({X}_{O})$, or it does not. If it does, then Y cannot contain any ancestors of O_x (defined as in the previous Lemma). Furthermore Y_O cannot share any ancestors with O_x, since by assumption it shares no ancestors with ${{nX}}_{O}$, and if it shared an ancestor with ${\bf{zo}}({{nX}}_{O})$ it could not be q-separated from X given nZ, contradicting the premise R. Hence we can use the same procedure as in lemma 3 to obtain the result $({nX}\perp Y| Z)\in \bar{C}(P)$, and use semi-graphoid axiom 1.c. to obtain the desired result: $(X\perp Y| {nZ})\in \bar{C}(P)$.

In the remaining case, n does not share an ancestor with ${X}_{O}{\bf{zo}}({X}_{O})$ so we have the relation ${{nY}}_{O}{\bf{-zo}}({X}_{O})\in {{\bf{dt}}}_{\varnothing }({X}_{O}{\bf{zo}}({X}_{O}))$. Let ${O}_{{xz}}:= {X}_{O}{\bf{zo}}({X}_{O})$ and consider the relation $({O}_{{xz}}\;{\bf{an}}({O}_{{xz}})\perp \;{{\bf{dt}}}_{\varnothing }({O}_{{xz}})| \varnothing )$ that holds in Q and hence in $\bar{C}(P)$. Using the above properties, and the fact that Y cannot contain any ancestors of O_xz (for the usual reason that this would imply an active path between X and Y in G), we obtain:

$$\begin{eqnarray}&&({O}_{{xz}}\;{\bf{an}}({O}_{{xz}})\perp {\neg }_{S}{\bf{an}}({O}_{{xz}})\;{{\bf{dt}}}_{\varnothing }({O}_{{xz}})/S| \varnothing )\in \bar{C}(P)\\ &&\quad \Rightarrow ({X}_{O}\;{\bf{zo}}({X}_{O})\perp {Y}_{S}\;{{nY}}_{O}\;{\bf{-zo}}({X}_{O})| {X}_{S}{Z}_{S})\in \bar{C}(P)\\ &&\quad \Rightarrow ({X}_{O}\perp Y| {{nX}}_{S}\;Z)\in \bar{C}(P).\end{eqnarray} \tag{ 13 }$$

Let us partition ${Z}_{S}={D}_{Z}\cap {E}_{Z}$, where D_Z contains the members of Z_S that are detached from X_S by ${{nZ}}_{O}$, and E_Z contains the rest. Consider the CI relation $({X}_{S}{E}_{Z}\perp {{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S}{E}_{Z})| {{nZ}}_{O})$ that holds in Q and hence in $\bar{C}(P)$. The set ${{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S}{E}_{Z})$ must contain ${{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S})$. If not, there would be a member of ${{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S})$ that is not detached from ${X}_{S}{E}_{Z}$, hence it would be connected by an ${{nZ}}_{O}$-chain to E_Z. But since E_Z has a chain to X_S, it could not be a member of ${{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S})$, implying a contradiction. Hence $({X}_{S}{E}_{Z}\perp {{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S})| {{nZ}}_{O})\in \bar{C}(P)$. But note that Y must be detached from X_S by ${{nZ}}_{O}$, otherwise there would be a path connecting X to Y in G on which every collider has a directed path to ${{nZ}}_{O}$, contradicting R. Thus ${D}_{Z}Y\in {{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S})$, and we obtain:

$$\begin{eqnarray}&&({X}_{S}{E}_{Z}\perp {{\bf{dt}}}_{{{nZ}}_{O}}({X}_{S})| {{nZ}}_{O})\in \bar{C}(P)\\ &&\quad \Rightarrow ({X}_{S}\perp Y| {nZ})\in \bar{C}(P).\end{eqnarray} \tag{ 14 }$$

Combining this result with (13) and the contraction axiom 1.d, we obtain $(X\perp Y| {nZ})\in \bar{C}(P)$ as desired.

Lemmas 1–4 together imply that G is an I-map of P, i.e. ${\bar{C}}_{q}(G)\subseteq \bar{C}(P)$, provided that $G-n$ is an I-map of $P-n$. The latter condition can be guaranteed using the same logic: $G-n$ is an I-map of $P-n$ provided $G-n-m$ is an I-map of $P-n-m$, where m is now the second-last variable in the chosen ordering. Continuing this process, every graph in the hierarchy is an I-map of its corresponding distribution provided that we can prove that G for a single variable is an I-map of a probability distribution P on a single variable. But ${\bar{C}}_{q}(G)\subseteq \bar{C}(P)$ is trivially satisfied for a single variable, because both these sets are empty. This completes the proof of theorem 3.

The result of theorems 2 and 3 is that q-separation is sound and complete for quantum input lists. That is, if ${Q}_{{\mathcal{O}}}$ is a quantum input list, the DAG G generated by ${Q}_{{\mathcal{O}}}$ is a perfect map of the semi-graphoid closure $\bar{Q}$. Hence a CI relation follows from the DAG using q-separation if and only if it can be obtained from Q using the semi-graphoid axioms.

In a quantum circuit, one can obtain a 'classical limit' by restricting all states and operators to a subspace of Hilbert space. In order to define the classical limit of a QCM, we must ensure that, in this limit, we recover the assumptions listed in section 2. We expect that Reichenbach's common cause principle will be recovered by restricting our operations to a classical subspace of Hilbert space, since this will rule out the possibility of entanglement. However, the causal Markov condition requires that direct causes 'screen-off' indirect causes, which will not in general be true after restricting the circuit to a classical subspace. To recover this principle, therefore, we need to transform the DAG of the QCM into a form that respects this property. This can be done by assigning the ancestors of every outcome (the 'possible causes' by assumption 2) to either direct or indirect causes. There may be many ways to do this, so we will adopt the simplest solution and make them all direct causes. This procedure is described in the following definition:

Definition 16. Classical limit for DAGs. Given a DAG G interpreted as a quantum network, the classical limit of G is a new DAG G_C obtained by the following procedure:

• (1)  
  Draw a directed edge from every setting S_i (non-drain node) to every outcome (drain node) O_i that is descended from S_i in G_C, unless such an edge already exists.
• (2)  
  Eliminate all edges that connect pairs of settings to each other.

The first step makes every setting a direct cause of every outcome that is descended from it, while the second step uses the fact that the settings are mutually independent to eliminate redundant edges. The resulting DAG is consistent with the causal structure of the original DAG in the following sense: there is a causal chain from one variable to another in G_C only if there existed such a chain in the DAG of the quantum network G. The the screening-off property is enforced since there are no intermediate nodes left in the DAG. This allows us to recover d-separation in the classical limit:

Theorem 4. Given any DAG G with classical limit G_C, every CI relation obtainable from G_C by q-separation is also implied by d-separation, i.e. ${\bar{C}}_{q}({G}_{C})\subseteq {\bar{C}}_{d}({G}_{C})$.

Proof. Suppose ${({S}_{A}\perp {S}_{B}| V)}_{q}$ holds under q-separation for disjoint sets ${S}_{A},{S}_{B},V$ where ${S}_{A},{S}_{B}$ are settings. Since all paths connecting two settings in G_C must contain at least one collider that is an outcome, q-separation implies that at least one of these colliders is not in V, but this also implies that the settings are d-separated. Similarly, if ${({S}_{A}\perp {O}_{B}| V)}_{q}$ holds under q-separation for a set of outcomes O_B, this implies that O_B is not descended from S_A. It follows that every path between them must contain at least one collider, and at least one of these colliders is not in V, which also implies d-separation. Finally, if ${({O}_{A}\perp {O}_{B}| V)}_{q}$ holds for two sets of outcomes, then every path connecting them must contain a collider that is not in V, again implying d-separation.

Given this result, it is straightforward to convert a QCM into a CCM: we restrict the quantum model parameters to a classical subspace and obtain a classical circuit. This circuit defines a set of functions F that determine the values of the outcomes given the values of their ancestors (these become their parents in G_C obtained from G_Q). The pair $\{{G}_{C},F\}$ then satisfies the requirements of a CCM.

We note that the 'Bell-type' experiment described in section 2.6 also applies more generally to any joint probability distribution with settings and outcomes that obey the no-signalling criterion. In particular, one can find a joint probability distribution P on the variables $A,B,S,T$ that satisfies the CI relations K (implied by setting independence and no-signalling) but which cannot be generated by any QCM defined on the same variables. For example, let all variables be binary variables taking values in $\{0,1\}$, let ⨁ represent addition modulo 2, and consider the joint distribution:

$$\begin{eqnarray*}&&P(A,B,S,T):= P(S)P(T)P(A,B| S,T),\ \ \mathrm{where}\end{eqnarray*}$$

$$\begin{eqnarray}&&P(A,B| S,T)=\displaystyle \frac{1}{2}(1\oplus A\oplus B)({ST}\oplus 1)+\displaystyle \frac{1}{2}(A\oplus B){ST}.\end{eqnarray} \tag{ 15 }$$

One can check that this distribution satisfies the CI relations K and that the probabilities sum to one. This distribution characterizes a Popescu–Rorlich box (PR-box), also called a nonlocal or non-signalling box [29]. A PR-box defines correlations that are stronger than quantum correlations, in the sense that they violate Bell's inequality to its algebraic maximum. How do super-quantum correlations fit into the present framework of QCMs?

There is no QCM on just the variables $A,B,S,T$ that provides a faithful explanation of the probability distribution P. This is trivially true because the constraints in K require that there is no directed edge from S to B or from T to A in the corresponding DAG (or else they would not be q-separated by any subset of variables) and no outgoing edges from A and B, which are outcomes—but this splits the DAG into two disconnected parts, which are necessarily independent, implying no correlations between A and B. As before, we can try to explain the correlations by introducing a hidden variable λ and extending the constraints to the set K' (see section 2.6). For a QCM, this is equivalent to supposing that the outcomes A and B can depend on a shared entangled resource, in a pure state specified by λ; the corresponding DAG is shown in figure 9(a). It should therefore come as no surprise that a QCM based on this DAG fails to reproduce the distribution P. Whichever pure states one chooses for the values of λ, the resulting statistics obtained from the QCM must obey quantum mechanics and hence the joint probability generated by the QCM must be able to violate Bell's inequality only up to Tsirelson's bound.

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Of course, there is no reason to restrict ourselves to this DAG—just as in the classical case, we can consider DAGs such as the one shown in figure 9(b), in which there is a signal from S to B, representing a 'hidden' link in the underlying quantum network (Note: if we interpret this DAG as a CCM, it also serves as an alternative to figure 9(b) for explaining quantum or PR-Box correlations using fine-tuning). This additional link can be used to send a single bit of information from S to B, which can be exploited to simulate the desired probability distribution. But, just as in the classical case, this would imply that S and B should not be q-separated, so the CI relation $(S\perp B| T\lambda )\in \bar{K}\prime$ can only be due to fine-tuning of the model parameters and the explanation is not faithful. Just as a CCM cannot faithfully explain quantum correlations, a QCM also cannot faithfully explain super-quantum correlations.

Note that this observation follows from the definition of post-quantum theories which violate Bell's inequalities up to the algebraic maximum while obeying the same causal constraints. It is a topic for future work to find out whether information theoretic or entropic constraints characteristic of quantum correlations, such as those discussed in [17–20], can be incorporated into the graphical representation in a useful way.