  We introduce a novel geometric approach to characterize entanglement
relations in large quantum systems. Our approach is inspired by Schumacher's
singlet state triangle inequality, which used an entropic-based distance to
capture the strange properties of entanglement using geometric-based
inequalities. Schumacher uses classical entropy and can only describe the
geometry of bipartite states. We extend his approach by using von Neumann
entropy to create an entanglement monotone that can be generalized for higher
dimensional systems. We achieve this by utilizing recent definitions for
entropic areas, volumes, and higher-dimensional volumes for multipartite
quantum systems. This enables us to differentiate systems with high quantum
correlation from systems with low quantum correlation and differentiate between
different types of multi-partite entanglement. It also enables us to describe
some of the strange properties of quantum entanglement using simple geometrical
inequalities. Our geometrization of entanglement provides new insight into
quantum entanglement. Perhaps by constructing well-motivated geometrical
structures (e.g. relations among areas, volumes ...), a set of trivial
geometrical inequalities can reveal some of the complex properties of
higher-dimensional entanglement in multi-partite systems. We provide numerous
illustrative applications of this approach, and in particular to a random
sample of a thousand density matrices.
