Hopf Fibration and Quantum Entanglement in Qubit Systems

TL;DR

This paper explores the geometric representation of multi-qubit entanglement using Hopf fibrations and sedenions, proposing new measures and analyzing their properties for three and four qubit systems.

Contribution

It introduces a novel geometric approach to quantify entanglement in multi-qubit systems using Hopf fibrations and sedenions, extending existing measures.

Findings

Connection between Hopf fibration and three-qubit entanglement measure.

Proposal of a new entanglement measure for four-qubit states.

Calculation of entanglement degrees for specific quantum states.

Abstract

Based on the geometry of entangled three and two qubit states, we present the connection between the entanglement measure of the three-qubit state defined using the last Hopf fibration and the entanglement measures known as two- and three-tangle. Moreover, the generalization of the geometric representation of four qubit state and a potential entanglement measure is studied using sedenions for the simplification of the Hilbert space S^31 of the four qubit system. An entanglement measure is proposed and the degree of entanglement is calculated for specific states. The difficulties of a possible generalization are discussed.