Factorization of Quantum Density Matrices According to Bayesian and Markov Networks

TL;DR

This paper demonstrates that quantum density matrices can be represented using Bayesian and Markov networks, extending classical probabilistic graphical models to quantum physics and linking d-separation to quantum entanglement.

Contribution

It introduces a framework for representing quantum states with graphical models and generalizes classical d-separation theorems to quantum systems, connecting them to entanglement measures.

Findings

Quantum density matrices can be represented by Bayesian and Markov networks.

Quantum d-separation theorems relate to quantum entanglement.

Graphical rules can detect unentangled node pairs.

Abstract

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.