Quantum Bayesian networks provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks -- prominent graphical models for classical probabilistic reasoning and inference. The goal of this paper is to bring compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems, and which we prove sound and complete with respect to quantum Bayesian networks.
Quantum Bayesian networks provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks -- prominent graphical models for classical probabilistic reasoning and inference. The goal of this paper is to bring compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems, and which we prove sound and complete with respect to quantum Bayesian networks.
Quantum Bayesian networks [9] provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks [17]-prominent graphical models for classical probabilistic reasoning and inference. Our paper brings compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems.
Bayesian networks provide a powerful tool for reasoning about probabilistic causation, used in many areas of science. They are, however, intrinsically classical. In particular, Bayesian networks naturally yield the Bell inequalities. Inspired by this connection, we generalise the formalism of classical Bayesian networks in order to investigate non-classical correlations in arbitrary causal structures. Our framework of `generalised Bayesian networks' replaces latent variables with the resources of any generalised probabilistic theory, most importantly quantum theory, but also, for example, Popescu-Rohrlich boxes. We obtain three main sets of results. Firstly, we prove that all of the observable conditional independences required by the classical theory also hold in our generalisation; to obtain this, we extend the classical d-separation theorem to our setting. Secondly, we find that the theory-independent constraints on probabilities can go beyond these conditional independences. For example we find that no probabilistic theory predicts perfect correlation between three parties using only bipartite common causes. Finally, we begin a classification of those causal structures, such as the Bell scenario, that may yield a separation between classical, quantum and general-probabilistic correlations.
Bayesian networks and their accompanying graphical models are widely used for prediction and analysis across many disciplines. We will reformulate these in terms of linear maps. This reformulation will suggest a natural extension, which we will show is equivalent to standard textbook quantum mechanics. Therefore, this extension will be termed "quantum". However, the term "quantum" should not be taken to imply this extension is necessarily only of utility in situations traditionally thought of as in the domain of quantum mechanics. In principle, it may be employed in any modeling situation, say forecasting the weather or the stock market--it is up to experiment to determine if this extension is useful in practice. Even restricting to the domain of quantum mechanics, with this new formulation the advantages of Bayesian networks can be maintained for models incorporating quantum and mixed classical-quantum behavior. The use of these will be illustrated by various basic examples. Parrondo's paradox refers to the situation where two, multi-round games with a fixed winning criteria, both with probability greater than one-half for one player to win, are combined. Using a possibly biased coin to determine the rule to employ for each round, paradoxically, the previously losing player now wins the combined game with probability greater than one-half. Using the extended Bayesian networks, we will formulate and analyze classical observed, classical hidden, and quantum versions of a game that displays this paradox, finding bounds for the discrepancy from naive expectations for the occurrence of the paradox. A quantum paradox inspired by Parrondo's paradox will also be analyzed. We will prove a bound for the discrepancy from naive expectations for this paradox as well. Games involving quantum walks that achieve this bound will be presented.
Markov categories allow formalization of probabilistic and causal reasoning in a general setting that applies uniformly to many different kinds of classical probabilistic processes. It has so far been challenging, however, to generalize these techniques to reasoning about quantum processes, as the quantum no-cloning theorem forbids "copy" maps of the sort that have been used to axiomatize conditional independence, and the related notions of complete common causes and Markovianity, in classical Bayesian networks. Here, we introduce a new categorical notion of Markovian causal model, according to which a distinguished subcategory of "common cause" maps plays a similar role to that of "copy" maps in the categorical formulation of Bayesian networks. Moreover, defining causal models as second-order processes yields a clean and flexible formulation of interventions. Our formalism is both rich enough to handle "complete common cause" assumptions and general enough to encompass not only standard classical causal identification scenarios, but also quantum causal scenarios and new kinds of classical causal identification based on imperfect observations. Furthermore, we show that one can reason uniformly across all of these cases using string-diagrammatic techniques.
We begin with a review of a well-known class of networks, Classical Bayesian (CB) nets (also called causal probabilistic nets by some). Given a situation which includes randomness, CB nets are used to calculate the probabilities of various hypotheses about the situation, conditioned on the available evidence. We introduce a new class of networks, which we call Quantum Bayesian (QB) nets, that generalize CB nets to the quantum mechanical regime. We explain how to use QB nets to calculate quantum mechanical conditional probabilities (in case of either sharp or fuzzy observations), and discuss the connection of QB nets to Feynman Path integrals. We give examples of QB nets that involve a single spin-[Formula: see text] particle passing through a configuration of two or three Stern—Gerlach magnets. For the examples given, we present the numerical values of various conditional probabilities, as calculated by a general computer program specially written for this purpose.
We begin with a review of a well known class of networks, Classical Bayesian (CB) nets (also called causal probabilistic nets by some). Given a situation which includes randomness, CB nets are used to calculate the probabilities of various hypotheses about the situation, conditioned on the available evidence. We introduce a new class of networks, which we call Quantum Bayesian (QB) nets, that generalize CB nets to the quantum mechanical regime. We explain how to use QB nets to calculate quantum mechanical conditional probabilities (in case of either sharp or fuzzy observations), and discuss the connection of QB nets to Feynman Path integrals. We give examples of QB nets that involve a single spin-half particle passing through a configuration of two or three Stern-Gerlach magnets. For the examples given, we present the numerical values of various conditional probabilities, as calculated by a general computer program especially written for this purpose.