Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory

TL;DR

The paper introduces the Gaussian Entanglement Measure (GEM), a computationally feasible multipartite entanglement measure for Gaussian states, applicable to complex quantum systems including graph states and bosonic field theories, revealing insights into topology and connectivity.

Contribution

It generalizes a geometric entanglement measure to multimode Gaussian states and demonstrates its application to graph states and quantum field theories, providing a new tool for analyzing multipartite entanglement.

Findings

GEM effectively quantifies entanglement in multimode Gaussian states.

Graph state topology ratios reflect graph connectivity properties.

GEM offers insights into quantum field theory beyond bipartite entanglement entropy.

Abstract

Computationally feasible multipartite entanglement measures are needed to advance our understanding of complex quantum systems. An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers, showing several advantages over existing methods, including ease of computation, a deep geometrical interpretation, and applicability to multipartite entanglement. Here, we present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states, based on the purity of fragments of the whole systems. Our analysis includes the application of GEM to a two-mode Gaussian state coupled through a combined beamsplitter and a squeezing transformation. Additionally, we explore 3-mode and 4-mode graph states, where each vertex represents a bosonic mode, and each edge represents a quadratic transformation for various graph topologies. Interestingly, the ratio of the geometric entanglement measures for graph states with different topologies naturally captures properties related to the connectivity of the underlying graphs. Finally, by providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory on $\mathbb R_t\times S^1$, going beyond the standard bipartite entanglement entropy approach between different regions of spacetime. The results presented herein suggest how the GEM paves the way for using quantum information-theoretical tools to study the topological properties of the space on which a quantum field theory is defined.