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arXiv:quant-ph/0702072

Markovian Entanglement Networks

Pierfrancesco La Mura, Lukasz Swiatczak

Published Feb 7, 2007• 8 citations
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Abstract

Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncertainty. However, in some domains (most notably, in computational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of classical probability theory are superseded by those of quantum mechanics. In this paper we introduce Markovian Entanglement Networks (MEN), a novel class of graphical representations of quantum-mechanical dependencies in the context of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropriate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical structure of MEN. Specifically, we show that for any wave function there exists a MEN which is a perfect map of its conditional separabilities. Next, we show how the graphical structure of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramatically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of conditional independencies afforded by MEN may not only facilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analysis of quantum algorithms and circuits.

DOI
10.48550/arxiv.quant-ph/0702072

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Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncertainty. However, in some domains (most notably, in computational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of classical probability theory are superseded by those of quantum mechanics. In this paper we introduce Markovian Entanglement Networks (MEN), a novel class of graphical representations of quantum-mechanical dependencies in the context of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropriate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical structure of MEN. Specifically, we show that for any wave function there exists a MEN which is a perfect map of its conditional separabilities. Next, we show how the graphical structure of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramatically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of conditional independencies afforded by MEN may not only facilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analysis of quantum algorithms and circuits.
Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncer-tainty. However, in some domains (most notably, in compu-tational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of clas-sical probability theory are superseded by those of quan-tum mechanics. In this paper we introduce Markov Entan-glement Networks (MEN), a novel class of graphical repre-sentations of quantum-mechanical dependencies in the con-text of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropri-ate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical struc-ture of MEN. Specifically, we show that for any wave func-tion there exists a MEN which is a perfect map of its condi-tional separabilities. Next, we show how the graphical struc-ture of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramat-ically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of con-ditional independencies afforded by MEN may not only fa-cilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analy-sis of quantum algorithms and circuits.
Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.
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.
We investigate a correspondence between two formalisms for discrete probabilistic modeling: probabilistic graphical models (PGMs) and tensor networks (TNs), a powerful modeling framework for simulating complex quantum systems. The graphical calculus of PGMs and TNs exhibits many similarities, with discrete undirected graphical models (UGMs) being a special case of TNs. However, more general probabilistic TN models such as Born machines (BMs) employ complex-valued hidden states to produce novel forms of correlation among the probabilities. While representing a new modeling resource for capturing structure in discrete probability distributions, this behavior also renders the direct application of standard PGM tools impossible. We aim to bridge this gap by introducing a hybrid PGM-TN formalism that integrates quantum-like correlations into PGM models in a principled manner, using the physically-motivated concept of decoherence. We first prove that applying decoherence to the entirety of a BM model converts it into a discrete UGM, and conversely, that any subgraph of a discrete UGM can be represented as a decohered BM. This method allows a broad family of probabilistic TN models to be encoded as partially decohered BMs, a fact we leverage to combine the representational strengths of both model families. We experimentally verify the performance of such hybrid models in a sequential modeling task, and identify promising uses of our method within the context of existing applications of graphical models.
This paper investigates how probabilistic graphical models and their quantum extensions can support decision making under uncertainty in robot–human interaction. Using a simple person-following task as a representative example, the study compares a classical Bayesian Network (BN) with two quantum extensions: a quantum-enhanced Bayesian Network (QeBN) and a full Quantum Bayesian Network (QBN). The purpose of this paper is to examine the expressive limitations of classical Bayesian decision models in contexts involving ambiguous sensing and partially conflicting evidence, and to evaluate whether quantum-inspired representations can provide richer and more flexible decision mechanisms while remaining interpretable. In the classical BN, uncertainty in human motion and robot actions is represented through conditional probability distributions, and action selection is performed by marginalizing over hidden variables. This framework supports decision making by combining prior beliefs and sensor-based likelihoods in a principled and computationally efficient manner. However, because inference relies on additive probabilities and conditional independence, alternative explanations contribute only as weighted mixtures, preventing interaction between competing hypotheses. To address this limitation, the paper reviews a QeBN in which classical probabilities are lifted to complex probability amplitudes while the original graph structure is preserved. This extension retains classical marginals but allows phase-dependent interference during marginalization, enabling non-additive belief updates that capture contextual effects such as sensor fusion ambiguity or conflicting cues in human–robot interaction. Building on this, a full QBN formulation is reviewed in which beliefs and conditional relationships are represented directly by quantum states and density operators. Inference is performed through joint state construction, quantum marginalization via partial trace, and measurement-based conditioning, providing a fully quantum-native alternative to classical Bayesian reasoning. Through analytical walk-through examples, the paper clarifies how each framework supports decision making, highlights their conceptual and mathematical differences, and demonstrates how quantum extensions can enhance expressiveness and context sensitivity beyond classical Bayesian models. The results position BN, QeBN, and QBN as complementary tools along a spectrum of decision-making models, offering increasing representational power for robotic systems operating in uncertain and human-centered environments.
Quantum Markov networks are a generalization of quantum Markov chains to arbitrary graphs. They provide a powerful classification of correlations in quantum many-body systems---complementing the area law at finite temperature---and are therefore useful to understand the powers and limitations of certain classes of simulation algorithms. Here, we extend the characterization of quantum Markov networks and in particular prove the equivalence of positive quantum Markov networks and Gibbs states of Hamiltonians that are the sum of local commuting terms on graphs containing no triangles. For more general graphs we demonstrate the equivalence between quantum Markov networks and Gibbs states of a class of Hamiltonians of intermediate complexity between commuting and general local Hamiltonians.