Reichenbach’s principle assumes that it cannot happen both that $Y$ is a cause of $Z$ and that $Z$ is a cause of $Y$. This is natural if $Y$ and $Z$ are physical variables pertaining to systems that are localized in space and time. But it is an assumption that may not hold for the generic case in which causal explanations are sought for statistical data, since there can be causal feedback loops: partaking of an addiction, say, may cause a low mood, which in turn may worsen the addiction. Ultimately, it is the specific application that will determine whether adoption of the qualitative part of Reichenbach’s principle (more generally, the formalism of directed acyclic graphs introduced below) is appropriate.

In the community studying classical causal inference, a deterministic causal dependence such as $Y=f(X,\lambda )$ is termed a structural equation, and a causal model wherein all causal dependences are deterministic is termed a functional causal model (see, e.g., Ref. [8]).

This is because any other $\lambda$ would necessarily introduce new common causes for $Y$ and $Z$ that are not screened through $X$, which would violate the assumption that $X$ is a complete common cause.

One could equally well evaluate the conditional mutual information on any distribution $\widehat{P}(XYZ)$ obtained from $P(YZ|X)$ and an input distribution $P(X)$ that has full support; again, the statement $I(Y:Z|X)=0$ is equivalent to the conditional independence of $Y$ and $Z$ given $X$. We consider the particular case of a uniform distribution over $X$ in order to maintain the strongest possible analogy with the quantum case.

The analogous mixed notation also appears in Fig. 5 for the classical case.

In introducing the circle notation, we define the action of gates on the left or right factors in such a way that coherence between different subspaces in the direct sum can be maintained. This corresponds to the fact that the condition of decomposing into the appropriate form is applied to the unitary or Kraus operators, rather than to the channel operator itself. We have done this on the grounds that with this definition, the circle notation is most likely to be useful in future applications. In the lower two circuits of Fig. 6, however, note that coherence between the different subspaces is lost. In the lower right circuit, coherence is lost when the partial traces are performed on the extra outgoing wires. In the lower left circuit, the final output admits a global factorization of the form $B\otimes C$, and output wires carrying an $i$ index do not even appear, indicating that this degree of freedom has been traced out. Each Kraus operator, in this case, must act nontrivially only on the $i$th subspace, for some $i$, and one may deduce that ${\rho }_{B|A}$ is of the form ${\sum }_i{\rho }_{B|A_i^L}\otimes I_{A_i^R}$, and similarly ${\rho }_{C|A}$, consistently with condition (4) of Theorem 3.

Clearly, the notation can be extended in various ways to include circles with multiple output wires, circles indicating a further decomposition following another circle, and so on. A fully general interpretation and calculus for these extended circuit diagrams is left for future work.

The quantum version in Fig. 9 was studied for similar reasons in Ref. [33], though from a different perspective.

It is interesting to consider an exactly analogous scenario, as it arises in the toy theory of Ref. [36]. Here, a system analogous to a qubit can exist in one of four distinct classical states (the ontic states of the system). But an agent who prepares systems and measures them can only ever have partial information about which of the four ontic states a system is in. The toy equivalent of a cnot gate corresponds to a reversible deterministic map, i.e., a permutation of the ontic states. By considering the probability distribution over ontic states of the various systems, one may verify directly that the ontic states of toy systems $B$ and $C$ are not determined by the ontic state of toy system $A$. Rather, the ontic states of $B$ and $C$ depend also on the ontic state of $\lambda$. Furthermore, the analogue of a pure quantum state for $\lambda$ is a probability distribution on $\lambda$ that is not a point distribution. In this way, statistical correlations between $B$ and $C$ can be underwritten by statistical variation in the ontic state of $\lambda$.

This might be seen as a generalization of standard usage, since in most treatments of quantum theory, the state describes a collection of systems at a single time, i.e., with none being causal descendants of others.

Classical interventional models, as we describe them, seem rarely to be studied in full generality in the classical literature. However, among the possible intervention schemes are included the following special cases: ignoring $X_i^I$ and repreparing $X_i^O$ with a fixed value $x$ of which one keeps a record, corresponding to $P(k_i,X_i^O|X_i^I)={\delta }_{k_i,x}{\delta }_{X_i^O,x}$ (the standard notion of intervention, as set out, e.g., in Ref. [8]); ignoring $X_i^I$ and repreparing $X_i^O$ with a value that is sampled randomly and independently of $X_i^I$ and of which one keeps a record, corresponding to $P(k_i,X_i^O|X_i^I)={\delta }_{k_i,X_i^O}P(X_i^O)$ (a randomized trial); observing $X_i^I$, keeping a record of this value, and preparing $X_i^O$ to have this value, corresponding to $P(k_i,X_i^O|X_i^I)={\delta }_{k_i,X_i^I}{\delta }_{X_i^O,X_i^I}$ (passive observation of $X_i$); observing $X_i^I$, keeping a record of its value, and repreparing $X_i^O$ to have a fixed value $x$, corresponding to $P(k_i,X_i^O|X_i^I)={\delta }_{k_i,X_i^I}{\delta }_{X_i^O,x}$ (the sort of intervention considered in single-world intervention graphs [72]); simply letting the value of $X_i^O$ track the value of $X_i^I$, corresponding to $k_i$ being trivial, and $P(X_i^O|X_i^I)={\delta }_{X_i^O,X_i^I}$ (no observation being made); and many others besides.