Dynamics and Geometry of Entanglement in Many-Body Quantum Systems

TL;DR

This paper introduces the Quantum Correlation Transfer Function (QCTF), a novel framework that geometrically characterizes entanglement dynamics in many-body quantum systems without explicit time evolution calculations.

Contribution

The paper develops the QCTF framework, providing an algebraic and geometric description of many-body entanglement, linking it to observable quantities and simplifying complex calculations.

Findings

Entanglement expressed as areas in the Hilbert space, equivalent to second-order Re9nyi entropy.

QCTF encodes entanglement dynamics via residues, avoiding explicit time dependence.

Geometric measures relate to experimentally observable features.

Abstract

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 \textit{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 R\'enyi 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.