On the permutationally invariant part of a density matrix and nonseparability of N-qubit states

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 $\rho^{{\rm PI}}$ of a general (mixed) $N$-qubit state $\rho$, we obtain: (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multi-partite 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 multi-partite entanglement measure $E$ that it satisfy $E(\rho)\geq E(\rho^{{\rm PI}})$, even though the operation that maps $\rho\rightarrow\rho^{{\rm PI}}$ is not local.