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.
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.
Can an agent's predictions reveal whether its reasoning is based on a classical or a quantum model? Using known results on matrix factorizations, here we show that quantum Bayesian retrodiction can produce one-step-ahead forecasts incompatible with every classical Bayesian explanation based on learning about a fixed but unknown sampling law from conditionally independent observations. The separation has a sharp threshold in the number of outcomes: for any measurement with at most four outcomes, the quantum predictions admit a classical realization, irrespective of the prior state and the finite Hilbert-space dimension, whereas a five-outcome measurement on a single qubit already violates an explicit classical bound. This bound holds for arbitrary classical latent states, prior probabilities, and sampling laws, and involves only the agent's initial and updated predictions. Thus, in this setting, quantum Bayesian retrodiction is strictly more expressive than classical Bayesian learning, and this difference can be witnessed without observing or specifying the agent's internal description.
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.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.
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.
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.
We consider the problem of gambling on a quantum experiment and enforce rational behavior by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalized to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalized Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalization (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.