Topological phases and multiqubit entanglement

TL;DR

This paper explores topological phases arising from cyclic local SU(2) evolutions in multi-qubit systems, revealing their connection to entanglement and providing methods to determine these phases for systems up to seven qubits.

Contribution

It introduces a general method to find topological phases in multi-qubit systems and provides a comprehensive list for systems with up to seven qubits.

Findings

Topological phases are linked to the topological structure of local SU(2) orbits.

A method to compute topological phases in n-qubit systems is developed.

Complete list of topological phases for systems with up to seven qubits is provided.

Abstract

Global phase factors of topological origin, resulting from cyclic local $\rm{SU}$ evolution, called topological phases, were first described in [Phys. Rev. Lett. {\bf 90}, 230403 (2003)], in the case of entangled qubit pairs. In this paper we investigate topological phases in multi-qubit systems as the result of cyclic local $\rm{SU(2)}$ evolution. These phases originate from the topological structure of the local $\rm{SU(2)}$-orbits and are an attribute of most entangled multi-qubit systems. We discuss the relation between topological phases and SLOCC-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an $n$-qubit system is described and a complete list of these phases for up to seven qubits is given.