  Algebro-geometric methods have proven to be very successful in the study of
graphical models in statistics. In this paper we introduce the foundations to
carry out a similar study of their quantum counterparts. These quantum
graphical models are families of quantum states satisfying certain locality or
correlation conditions encoded by a graph. We lay out several ways to associate
an algebraic variety to a quantum graphical model. The classical graphical
models can be recovered from most of these varieties by restricting to quantum
states represented by diagonal matrices. We study fundamental properties of
these varieties and provide algorithms to compute their defining equations.
Moreover, we study quantum information projections to quantum exponential
families defined by graphs and prove a quantum analogue of Birch's Theorem.
