Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)

TL;DR

This paper introduces a geometric and algebraic approach to classify four-qubit entanglement, using invariant theory and algebraic geometry to describe stratifications and normal forms within the Hilbert space.

Contribution

It develops a new geometric framework for four-qubit entanglement classification, interpreting normal forms as algebraic varieties and proposing an algorithm for state identification.

Findings

Normal forms are interpreted as dense subsets of dual varieties.

An algorithm for identifying SLOCC equivalent states is proposed.

The approach provides a stratification of the four-qubit Hilbert space.

Abstract

We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by SLOCC invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete {\em et al.} are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).