Quantum causal models via quantum Bayesianism

This paper presents a framework for quantum causal modeling based on the interpretation of causality as a relation between an observer's probability assignments to hypothetical or counterfactual experiments. The framework is based on the principle of “causal sufficiency”: that it should be possible to make inferences about interventions using only the probabilities from a single “reference experiment” plus causal structure in the form of a directed acyclic graph. This leads to several interesting results: we find that quantum measurements deserve a special status distinct from interventions, and that a special rule is needed for making inferences about what would happen if they are not performed (“unmeasurements”). One natural candidate for this rule is found to be an equation of importance to the quantum Bayesianism interpretation of quantum mechanics. We find that the causal structure of quantum systems must have a “layered” structure, and that the model can naturally be made symmetric under reversal of the causal arrows.

Figure 1

Three special cases in which the causal Markov condition reduces to simpler principles: (a) variables $X_1,X_2$ only connected through common causes, where CMC reduces to FCC; (b) $X_1,X_2$ only connected through intermediate causes, where CMC reduces to SSO; (c) $X_1,X_2$ connected only through common effects, where CMC reduces to BK.

Figure 2

A causal graph before and after a manipulation (or an intervention) of $W$.

Figure 3

The Bell scenario, also known as the common-cause scenario with variable measurement settings. Assuming no fine-tuning, this causal hypothesis is ruled out as an explanation for quantum systems whose statistics violate Bell inequalities.

Figure 4

(a) A causal diagram and (b) its most general possible circuit realization. Single arrows represent quantum systems and double arrows represent classical data. The boxes SIC instruments. (c) Circuit under an intervention of $W$. As explained in the text, the probabilities in this case are not uniquely specified by those in the preintervention circuit.

Figure 5

(a) A causal graph and (b) a corresponding quantum circuit, similar to that shown in Fig. 4, except with an additional SIC instrument $Z$. In contrast to that case, the result of an intervention on $W$ (or on $Z$) in this circuit can be deduced directly from the reference probabilities $P(A,D,W,Z)$ for an arbitrary circuit. This enables us to derive rules for counterfactual inference.

Figure 6

A causal graph of three layers within a layered DAG. The main text explains how to infer the probabilities for an unmeasurement of $K$ variables in the middle layer.

Figure 7

The causal graph before and after unperforming the measurement $Z$; the causal pathways from its parents to its children are preserved.

Figure 8

Schematic of the two types of LDAG that can support a conditional independence of the form $R=({\mathbf{L}}_1{\mathbf{M}}_1\perp {\mathbf{L}}_2{\mathbf{M}}_2|{\mathbf{L}}_3{\mathbf{M}}_3)$. Note that the arrows are indirect causes, as they are intercepted by the intervening layer V (not shown).