Manifolds of interconvertible pure states

TL;DR

This paper analyzes the geometric structure of pure states in bipartite quantum systems, focusing on the dimensions of their local orbits, including separable and maximally entangled states, based on Schmidt decomposition.

Contribution

It provides explicit calculations of the dimensions of local orbits for pure states, highlighting the dependence on Schmidt coefficient degeneracy and characterizing special state manifolds.

Findings

Generic orbit has 2N^2 - N - 1 dimensions

Separable states form a 4(N-1) dimensional manifold

Maximally entangled states form an N^2 - 1 dimensional manifold

Abstract

Local orbits of a pure state of an N x N bi-partite quantum system are analyzed. We compute their dimensions which depends on the degeneracy of the vector of coefficients arising by the Schmidt decomposition. In particular, the generic orbit has 2N^2 -N-1 dimensions, the set of separable states is 4(N-1) dimensional, while the manifold of maximally entangled states has N^2-1 dimensions.