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arXiv:2607.27876

Bayesian Networks of Density Operators

Steffen Lauritzen, Piotr Zwiernik

quant-phmath-phSubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations
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Abstract

We study quantum analogues of Bayesian networks on a directed acyclic graph (DAG), distinguishing two constructions for positive definite density operators on finite-dimensional tensor-product Hilbert spaces. The intrinsic construction starts from a joint state and its conditional-independence properties. The extrinsic construction assembles a state sequentially from prescribed local quantum kernels, following an ordering compatible with the arrows of the DAG. For the intrinsic construction, we prove the equivalence of the ordered, local, and global directed Markov properties, together with entropy, recursive-factorization, and logarithmic characterizations. The extrinsic construction always gives a normalized state and recovers each kernel as a conditional on all preceding systems. The same kernel, however, need not be recovered from the marginal on the vertex and its parents; a three-qubit example exhibits this obstruction. We prove that independence of the chosen topological ordering is sufficient exactly for transitive DAGs: every order-invariant kernel family then yields an intrinsically directed Markov state. Finally, we associate a logarithmic candidate with every positive definite state and DAG, prove that it is subnormalized, and show that the trace-one candidate is a directed Markov state. Both the candidate and the excess global information are invariant under DAG Markov equivalence.

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We study quantum analogues of Bayesian networks on a directed acyclic graph (DAG), distinguishing two constructions for positive definite density operators on finite-dimensional tensor-product Hilbert spaces. The intrinsic construction starts from a joint state and its conditional-independence properties. The extrinsic construction assembles a state sequentially from prescribed local quantum kernels, following an ordering compatible with the arrows of the DAG. For the intrinsic construction, we prove the equivalence of the ordered, local, and global directed Markov properties, together with entropy, recursive-factorization, and logarithmic characterizations. The extrinsic construction always gives a normalized state and recovers each kernel as a conditional on all preceding systems. The same kernel, however, need not be recovered from the marginal on the vertex and its parents; a three-qubit example exhibits this obstruction. We prove that independence of the chosen topological ordering is sufficient exactly for transitive DAGs: every order-invariant kernel family then yields an intrinsically directed Markov state. Finally, we associate a logarithmic candidate with every positive definite state and DAG, prove that it is subnormalized, and show that the trace-one candidate is a directed Markov state. Both the candidate and the excess global information are invariant under DAG Markov equivalence.
We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a canonical logarithmic construction T(R)T(\mathcal R)T(R), the noncommutative analogue of the junction-tree formula for decomposable graphical models. Unlike in the classical case, this formal construction may fail: noncommutativity can prevent it from being a normalized state with the prescribed marginals. We prove that this obstruction is captured exactly by a trace condition. For two overlapping marginals, and for clique marginals on a chordal graph, the condition Tr(T(R))=1{\rm Tr}(T(\mathcal R))=1Tr(T(R))=1 is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to T(R)T(\mathcal R)T(R), and selected by the maximum entropy principle. In the two-clique case, we also give an equivalent conditional reconstruction characterization: the two natural one-sided sandwich reconstructions agree if and only if the trace condition holds. We introduce the global quantum information gI(G)ρg{\rm I}(\mathcal{G})_ρgI(G)ρ​ associated with a chordal graph G\mathcal{G}G and show that it is a relative-entropy discrepancy from ρρρ to the logarithmic candidate, with a trace correction when the candidate is not normalized. We also prove an intersection property for strictly positive quantum conditional independence. Three-qubit Pauli examples illustrate how the quantum obstructions are real: local consistency, feasibility, Markov feasibility, and maximum entropy can all separate.
We show that any quantum density matrix can be represented by a Bayesian network (a directed acyclic graph), and also by a Markov network (an undirected graph). We show that any Bayesian or Markov net that represents a density matrix, is logically equivalent to a set of conditional independencies (symmetries) satisfied by the density matrix. We show that the d-separation theorems of classical Bayesian and Markov networks generalize in a simple and natural way to quantum physics. The quantum d-separation theorems are shown to be closely connected to quantum entanglement. We show that the graphical rules for d-separation can be used to detect pairs of nodes (or of node sets) in a graph that are unentangled. CMI entanglement (a.k.a. squashed entanglement), a measure of entanglement originally discovered by analyzing Bayesian networks, is an important part of the theory of this paper.
Roughly a quarter of a century ago, the author proposed "Observable System Theory" (OST) as a foundational theory of systems theory. OST describes both statistics and quantum mechanics within a single framework of state–observable–measured value, governed by two explicitly formulated principles: the Born principle and the causal principle. This paper examines Bayesian networks (BNs) from the standpoint of OST, with particular attention to the following fundamental question: what, ultimately, do DAG calculations compute? The answer proposed here is that DAG calculations ultimately compute the probabilistic causality of measured values. Accordingly, OST admits two distinct notions of causality. The first is the causal evolution of states—equivalently, the causal evolution of observables in the Heisenberg picture—governed by OST's causal principle. The second is the probabilistic dependence structure among measured values, represented by a directed acyclic graph (DAG) together with local conditional probability distributions. The main result of this paper is the following fundamental theorem: for every finite Bayesian network specified by a DAG and local conditional probability distributions, there exists a corresponding measurement in OST whose outcome distribution coincides exactly with the joint probability distribution generated by the Bayesian network. We also aim to dispel the notion that theories involving multiple, distinct kinds of causal relation—such as Aristotle's doctrine of the four causes—are inherently untenable. Keywords: Observable System Theory; Bayesian network; directed acyclic graph; measurement; measured value; measurement probability.
We report a scalable hybrid quantum-classical machine learning framework to build Bayesian networks (BN) that captures the conditional dependence and causal relationships of random variables. The generation of a BN consists of finding a directed acyclic graph (DAG) and the associated joint probability distribution of the nodes consistent with a given dataset. This is a combinatorial problem of structural learning of the underlying graph, starting from a single node and building one arc at a time, that fits a given ensemble using maximum likelihood estimators (MLE). It is cast as an optimization problem that consists of a scoring step performed on a classical computer, penalties for acyclicity and number of parents allowed constraints, and a search step implemented using a quantum annealer. We have assumed uniform priors in deriving the Bayesian network that can be relaxed by formulating the problem as an estimation Dirichlet parameters. We demonstrate the utility of the framework by applying to the problem of elucidating the gene regulatory network for the MAPK/Raf pathway in human T-cells using proteomics data where the concentration of proteins, nodes of the BN, are interpreted as probabilities.
This work develops a quantum–statistical framework for analyzing non-local linear kernels by embedding them as Gibbs states of self-adjoint Hamiltonians on a tensor-product Hilbert space. In the symmetric setting, the construction yields a well-defined equilibrium ensemble and separates exact operator-level identities from conditional diagnostics that depend on explicit spectral assumptions. In the classical limit, the conditional Gibbs marginals recover row-wise softmax weights by construction, while a finite kinetic scale introduces a controlled within-block deformation that preserves block structure. Independent of equilibrium considerations, any kernel defines an operator-induced bipartite state whose reduced spectra are determined by the singular values of the kernel. The paper formulates a quantitative purity flatness condition under which the associated operator entanglement approaches the maximal logarithmic scaling allowed by the natural bipartition. Thermodynamic observables are linked to information geometry by identifying the Fisher–Rao metric along the inverse-temperature direction with the Hamiltonian variance, motivating energetic susceptibility and finite-size scaling diagnostics for large operator families. The framework distinguishes operator-state diagnostics, which are well defined for arbitrary kernels, from equilibrium and spectral diagnostics, which require a Hermitian representative, and provides a set of mathematically explicit tools for studying non-local kernels within quantum statistical mechanics. This is a preprint version.
Abstract We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons, we restrict our analysis to tree-structured sets of bipartite marginals. We focus on the tripartite scenario, where we solve the problem for the couples of measured marginals which are compatible with a quantum Markov chain, providing then an algebraic necessary and sufficient condition for the compatibility to be verified. We introduce the generalization of the procedure to the n-partite scenario, giving some preliminary results. In particular, we prove that if the pairwise Markov condition holds between the subparts then the choice of the best set of tree-structured bipartite marginals can be performed efficiently. Moreover, we provide a new characterization of quantum Markov chains in terms of quantum Bayesian updating processes .