Polynomial inequalities for multipartite quantum systems in mixed states

Abstract

Entanglement in a multipartite quantum system represents non-local correlations between the system parts. This phenomenon is not affected by the action of a symmetry group, which acts separately on each part of the multipartite composite system. The space of the invariants of the action is a natural domain of the definition for all possible measures of the entanglement. Allowed values of the invariants are restricted by a set of inequalities, which is analyzed in this paper. The inequalities ensue from the semipositivity and Hermicity of the density matrix for the system as well as from the orbit space description in terms of the invariant theory.