Classification and Quantification of Entanglement Through Wedge Product and Geometry

TL;DR

This paper introduces a geometric entanglement measure based on wedge products and higher-dimensional volumes, providing a detailed classification of bipartite qutrit entanglement through geometric structures.

Contribution

It extends wedge product formalism to multi-qudit systems, offering a novel, geometry-based entanglement measure that distinguishes different entangled classes.

Findings

Three geometric classes of bipartite qutrit entanglement identified

Maximizing area and volume correlates with entanglement maximization

Wedge product approach reveals inherent geometric structure of entanglement

Abstract

Wedge product of post-measurement vectors leading to an `area' measure of the parallelogram has been shown to give the generalized I-concurrence measure of entanglement. Extending the wedge product formalism to multi qudit systems, we have presented a modified faithful entanglement measure, incorporating the higher dimensional volume and the area elements of the parallelepiped formed by the post-measurement vectors. The measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We have presented a complete analysis for the bipartite qutrit case considering all possible geometric structures. Three entanglement classes can be identified with different geometries of post-measurement vectors, namely three planar vectors, three mutually orthogonal vectors, and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for characterization and quantification of entanglement in higher dimensional systems.