An algebraic classification of entangled states

TL;DR

This paper introduces a novel algebraic approach to classify entangled quantum states using discrete invariants derived from algebraic properties of associated linear maps, applicable to multiple subsystems.

Contribution

It develops a new algebraic classification method for entangled states based on invariants, extending to multiple subsystems and providing detailed classifications for three subspaces and four qubits.

Findings

Classified entangled states using algebraic invariants.

Established a correspondence between invariants and equivalence classes.

Identified 27 fundamental classes of four-qubit entanglement.

Abstract

We provide a classification of entangled states that uses new discrete entanglement invariants. The invariants are defined by algebraic properties of linear maps associated with the states. We prove a theorem on a correspondence between the invariants and sets of equivalent classes of entangled states. The new method works for an arbitrary finite number of finite-dimensional state subspaces. As an application of the method, we considered a large selection of cases of three subspaces of various dimensions. We also obtain an entanglement classification of four qubits, where we find 27 fundamental sets of classes.