title: Permutationally invariant part of a density matrix and nonseparability of N-qubit states.
abstract: We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of N-qubit systems. We show here that the concept is, in fact, basis dependent but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part ρ(PI) of a general (mixed) N-qubit state ρ, we obtain (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multipartite separability criteria (one of which entails a sufficient criterion for genuine N-partite entanglement) that can be experimentally determined by just 2N+1 measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of N qubits, and (iv) an explicit example that shows there are, for increasing N, genuinely N-partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is k nonseparable, then so is the actual state. We further argue to add as requirement on any multipartite entanglement measure E that it satisfy E(ρ)≥E(ρ(PI)), even though the operation that maps ρ→ρ(PI) is not local.
authors.affiliation: College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA.
authors.forename: Ting
authors.initials: T
authors.lastname: Gao
authors.affiliation: College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA.
authors.forename: Fengli
authors.initials: F
authors.lastname: Yan
authors.affiliation: Department of Physics, University of Oregon, Eugene, Oregon 97403, USA.
authors.forename: S J
authors.initials: SJ
authors.lastname: van Enk