Bipartite entanglement, spherical actions and geometry of local unitary orbits

TL;DR

This paper explores the geometric structure of entangled quantum states using the moment map, providing a classification framework for entanglement types based on local unitary orbits and symplectic geometry.

Contribution

It introduces a geometric approach using the moment map and Brion's theorem to classify bipartite entanglement and describes the symplectic structure of local orbits for various particle types.

Findings

Partial classification of entanglement classes via coadjoint orbits

Explicit description of symplectic structures for bipartite systems

Application of Brion's theorem to quantum entanglement geometry

Abstract

We use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map on coadjoint orbits of local transformations (unitary transformations which do not change entanglement), thus the geometry of coadjoint orbits provides a partial classification of different entanglement classes. To achieve the full classification a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions and distinguishable particles.