Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)

TL;DR

This paper explores the geometric structure of four-qubit entangled states using algebraic geometry and invariant theory, identifying key varieties and entanglement classes with algorithms for classification.

Contribution

It provides a detailed geometric classification of four-qubit entanglement states, introducing 47 varieties and algorithms for their identification.

Findings

Describes the nullcone and third secant variety for four-qubit systems.

Identifies 47 non-equivalent entanglement patterns.

Provides algorithms for classifying states into these patterns.

Abstract

We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.