Symplectic geometry of entanglement

TL;DR

This paper introduces a symplectic geometric framework to characterize and measure entanglement in quantum systems, providing a new geometric perspective and criteria for separability and entanglement.

Contribution

It develops a symplectic geometric approach to describe entanglement, identifying orbits of states and a new geometric measure based on degeneracy of the symplectic form.

Findings

Separable states form a unique Kaehler orbit.

Entangled states are characterized by degeneracy of the symplectic form.

The method applies to systems with additional symmetries, including indistinguishable particles.

Abstract

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.