  A new framework is formulated to study entanglement dynamics in many-body
quantum systems along with an associated geometric description. In this
formulation, called the Quantum Correlation Transfer Function (QCTF), the
system's wave function or density matrix is transformed into a new space of
complex functions with isolated singularities. Accordingly, entanglement
dynamics is encoded in specific residues of the QCTF, and importantly, the
explicit evaluation of the system's time dependence is avoided. Notably, the
QCTF formulation allows for various algebraic simplifications and
approximations to address the normally encountered complications due to the
exponential growth of the many-body Hilbert space with the number of bodies.
These simplifications are facilitated through considering the patterns, in lieu
of the elements, lying within the system's state. Consequently, a main finding
of this paper is the exterior (Grassmannian) algebraic expression of many-body
entanglement as the collective areas of regions in the Hilbert space spanned by
pairs of projections of the wave function onto an arbitrary basis. This latter
geometric measure is shown to be equivalent to the second-order Renyi entropy.
Additionally, the geometric description of the QCTF shows that characterizing
features of the reduced density matrix can be related to experimentally
observable quantities. The QCTF-based geometric description offers the prospect
of theoretically revealing aspects of many-body entanglement, by drawing on the
vast scope of methods from geometry.
