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.
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.
Modern approaches to causal modeling give a central role to interventions, which require the active input of an observer and introduces an explicit `causal arrow of time'. Causal models typically adopt a mechanistic interpretation, according to which the direction of the causal arrow is intrinsic to the process being studied. Here we investigate whether the direction of the causal arrow might be a contribution from the observer, rather than an intrinsic property of the process. Working within a counterfactual and non-mechanistic interpretation of causal modeling developed in arXiv:1806.00895, we propose a definition of a `quantum observational scheme' that we argue characterizes the observer-invariant properties of a causal model. By restricting to quantum processes that preserve the maximally mixed state (unbiasedness) we find that the statistics is symmetric under reversal of the time-ordering. The resulting model can therefore accommodate the idea that the causal arrow is observer-dependent, indicating a route towards reconciling the causal arrow with time-symmetric laws of physics.
It is known that the classical framework of causal models is not general enough to allow for causal reasoning about quantum systems. While the framework has been generalized in a variety of different ways to the quantum case, much of this work leaves open whether causal concepts are fundamental to quantum theory, or only find application at an emergent level of classical devices and measurement outcomes. Here, we present a framework of quantum causal models, with causal relations defined in terms intrinsic to quantum theory, and the central object of study being the quantum process itself. Following Allen et al., Phys. Rev. X 7, 031021 (2017), the approach defines quantum causal relations in terms of unitary evolution, in a way analogous to an approach to classical causal models that assumes underlying determinism and situates causal relations in functional dependences between variables. We show that any unitary quantum circuit has a causal structure corresponding to a directed acyclic graph, and that when marginalising over local noise sources, the resulting quantum process satisfies a Markov condition with respect to the graph. We also prove a converse to this statement. We introduce an intrinsically quantum notion that plays a role analogous to the conditional independence of classical variables, and (generalizing a central theorem of the classical framework) show that d-separation is sound and complete for it in the quantum case. We present generalizations of the three rules of the classical do-calculus, in each case relating a property of the causal structure to a formal property of the quantum process, and to an operational statement concerning the outcomes of interventions. In addition, we introduce and derive similar results for classical split-node causal models, which are more closely analogous to quantum causal models than the classical causal models that are usually studied.
Though the topic of causal inference is typically considered in the context of classical statistical models, recent years have seen great interest in extending causal inference techniques to quantum and generalized theories. Causal identification is a type of causal inference problem concerned with recovering from observational data and qualitative assumptions the causal mechanisms generating the data, and hence the effects of hypothetical interventions. A major obstacle to a theory of causal identification in the quantum setting is the question of what should play the role of "observational data," as any means of extracting data at a certain locus will almost certainly disturb the system. Hence, one might think a priori that quantum measurements are already too much like interventions, so that the problem of causal identification trivializes. This is not the case. Fixing a limited class of quantum instruments (namely the class of all projective measurements) to play the role of "observations," we note that as in the classical setting, there exist scenarios for which causal identification is not possible. We then present sufficient conditions for quantum causal identification, starting with a quantum analogue of the well-known "front-door criterion" and finishing with a broader class of scenarios for which the effect of a single intervention is identifiable. These results emerge from generalizing the process-theoretic account of classical causal inference due to Jacobs, Kissinger, and Zanasi beyond the setting of Markov categories, and thereby treating the classical and quantum problems uniformly.
Quantum theory can be viewed as a generalization of classical probability
theory, but the analogy as it has been developed so far is not complete.
Classical probability theory is independent of causal structure, whereas the
conventional quantum formalism requires causal structure to be fixed in
advance. In this paper, we develop the formalism of quantum conditional states,
which unifies the description of experiments involving two systems at a single
time with the description of those involving a single system at two times. The
analogies between quantum theory and classical probability theory are expressed
succinctly within the formalism and it unifies the mathematical description of
distinct concepts, such as ensemble preparation procedures, measurements, and
quantum dynamics. We introduce a quantum generalization of Bayes' theorem and
the associated notion of Bayesian conditioning. Conditioning a quantum state on
a classical variable is the correct rule for updating quantum states in light
of classical data, regardless of the causal relationship between the classical
variable and the quantum degrees of freedom, but it does not include the
projection postulate as a special case. We show that previous arguments that
projection is the quantum generalization of conditioning are based on
misleading analogies. Since our formalism is causally neutral, conditioning
provides a unification of the predictive and retrodictive formalisms for
prepare-and-measure experiments and leads to an elegant derivation of the set
of states that a system can be steered to by making measurements on a remote
system.
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: Channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.
Recent years have seen great interest in extending causal inference concepts developed in the context of classical statistical models to quantum theory. So far, this program has only barely addressed causal identification, a type of causal inference problem concerned with recovering from observational data and qualitative assumptions the causal relationships generating the data, and hence the effects of hypothetical interventions. A major obstacle to a theory of causal identification in the quantum setting is the question of what should play the role of “observational data,” as any means of extracting data at a locus will almost certainly disturb the system. One might think that quantum measurements are already too much like interventions, so that the problem of causal identification is trivialized. This is not the case: when we fix a limited class of quantum measurement instruments (namely, the class of all projective measurements) to play the role of “observations,” there exist scenarios for which causal identification is impossible. In this paper, we present a framework, based on process theories (also known as strict symmetric monoidal categories), for studying quantum causal identification scenarios on the same footing as their classical counterparts. Within this framework, we present sufficient conditions for quantum causal identification in networks with unobserved confounding systems, including quantum analogs of the well-known “back-door” and “front-door” criteria. These results arise from a type of causal model designed to facilitate the transfer of inference techniques from the classical to the quantum setting. Published by the American Physical Society 2024