Topological phases and multiqubit entanglement

Abstract

Global phase factors of topological origin, resulting from cyclic local $\mathrm{SU}$ 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 $\mathrm{SU}(2)$ evolution. These phases originate from the topological structure of the local $\mathrm{SU}(2)$-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.