  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probability theory are superseded by those of quantum mechanics.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.
