Polynomial invariants for discrimination and classification of four-qubit entanglement

TL;DR

This paper develops polynomial invariants to discriminate and classify four-qubit entanglement, providing a unified scheme that combines qualitative and quantitative descriptions, which is crucial given the complexity of entanglement classes.

Contribution

It introduces a general polynomial invariant-based criterion for classifying four-qubit entanglement, reproduces existing classifications, and proposes a new scheme using 'tangle patterns'.

Findings

Reproduces existing SLOCC classifications for four-qubit entanglement.

Introduces a polynomial classification scheme based on 'tangle patterns'.

Provides a criterion applicable to pure N-partite entangled states.

Abstract

It is well known that the number of entanglement classes in SLOCC (stochastic local operations and classical communication) classifications increases with the number of qubits and is already infinite for four qubits. Bearing in mind the rapid evolution of experimental technology, criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore in the focus of intense theoretical research. In this article we develop a general criterion for the discrimination of pure N-partite entangled states in terms of polynomial SL(d,C) invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which families are identified through 'tangle patterns', thus bringing together qualitative and quantitative description of entanglement.