### Abstract

Given a specific ordered spectra of (shifted) single-qubit reduced density matrices, we obtain the space of associated entanglement types for pure three-qubit states up to a homeomorphism. In fact, we discuss that given a specific set of minimal eigenvalues of (shifted) reduced density matrices for three qubits, the non-local properties of pure tripartite states are determined by two real parameters. These parameters are the local unitary invariant polynomials, which are in one-to-one correspondence with the coefficients of the normal form in the generalized Schmidt decomposition for pure three-qubit states.

Other components in our line of reasoning are the moment (Kirwan) polytope for three qubits, singular symplectic orbit reduction and the induced Hilbert map. In the context of symplectic geometry, given the components of the invariant moment map in the associated moment polytope, by using the singular symplectic orbit reduction method and the generalized Schmidt decomposition for three-qubit pure states we construct the image under the induced Hilbert map of the resulting symplectic reduced space.