Geometry of the local equivalence of states

TL;DR

This paper uses symplectic geometry and the moment map to analyze local equivalence of quantum states, linking geometric properties of orbits to criteria for local unitary equivalence.

Contribution

It introduces a symplectic geometric framework for understanding local state equivalence, connecting orbit properties with criteria for local unitary transformations.

Findings

Characterization of local equivalence via coadjoint orbits

Analysis of symplectic properties like isotropy and coisotropy

Criteria for local unitary equivalence based on orbit geometry

Abstract

We present a description of locally equivalent states in terms of symplectic geometry. Using the moment map between local orbits in the space of states and coadjoint orbits of the local unitary group we reduce the problem of local unitary equivalence to an easy part consisting of identifying the proper coadjoint orbit and a harder problem of the geometry of fibers of the moment map. We give a detailed analysis of the properties of orbits of "equally entangled states". In particular we show connections between certain symplectic properties of orbits such as their isotropy and coisotropy with effective criteria of local unitary equivalence.