Dynamics and Geometry of Entanglement in Many-Body Quantum Systems

TL;DR

The paper addresses the challenge of tracking entanglement dynamics in many-body quantum systems by introducing the Quantum Correlation Transfer Function ($QCTF$), a two- or three-variable Z-transform that maps the density matrix or wavefunction into analytic space where entanglement is encoded in residues and poles rather than explicit time evolution. By formulating a basis-dependent or basis-free (tensorial) description, the $QCTF$ enables direct, residue-based extraction of entanglement measures and reveals a geometric, exterior-algebraic structure to multipartite entanglement. A key result is that the entanglement measure for a subsystem, expressed in the Laplace domain as $ ilde{Q}_M(s)$, can be written in a basis-invariant form through local overlaps and eigenmode contributions, and is related to the second-order Rényi entropy via $ ilde{Q}_M(s)=s_2= frac{1}{2}(1- ext{Tr}( ilde{ ho}^2))= frac{1}{2}(1-2^{-S^{(2)}})$. The framework generalizes to arbitrary subsystems, providing a geometric interpretation where entanglement equals the sum of squared areas spanned by marginal wavefunctions, with potential extensions to time-dependent Hamiltonians and experimental measurements via accessible overlaps.

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 Rényi 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.

Figures (1)

• Figure 1: The area spanned by pairs of marginal (or projected) wave functions is depicted. These vectors, given by (\ref{['expansion']}), belong to $\mathbb{C}^d$, which is the vector-space underlying subsystem $\mathcal{R}$. Equivalently, up to normalization, marginal wave-function $\ket{\hat{j} \psi}$ is equal to the post-measurement (i.e., the projected wave function of subsystem $\mathcal{M}$, after projective measurement of subsystem $\mathcal{M}$, in the $\{\ket{\hat{j}}\}$ basis) wave-function of subsystem $\mathcal{R}$. The total squared areas (i.e., the three shaded regions in the Figure) gives the entanglement between the subsystems.