Classifying Quantum Entanglement through Topological Links

TL;DR

This paper introduces a novel topological link-based classification scheme for quantum entanglement, using polynomial invariants to categorize multipartite entangled states, demonstrated on three- and four-qubit systems.

Contribution

It presents a new topological approach to classify quantum entanglement, linking physical states to mathematical link polynomials, and applies this to multipartite qubit systems.

Findings

Classifies multipartite entanglement using topological links.

Develops a polynomial formalism for link equivalence classes.

Demonstrates the scheme on three- and four-qubit states.

Abstract

We propose a new classification scheme for quantum entanglement based on topological links. This is done by identifying a non-rigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for a given number of rings. To determine all different possibilities, we develop a formalism which associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. In order to demonstrate the use of this classification scheme, we choose qubit quantum states as our example of physical system. A possible procedure to obtain qubit states from the polynomials is also introduced, providing an example state for each link class. We apply the formalism for the quantum systems of three and four qubits, and demonstrate the potential of these new tools in a context of qubit networks.