Are uncorrelated states separable?

Question

Consider spin population observables, a separable state can be correlated or uncorrelated. For example, if I consider the two particle state $$|+\rangle|-\rangle$$ it is separable and spin measurements will be correlated, because measuring the spin of the first particle will always give +1, while for spin two we will always get -1: here spin populations are perfectly anticorrelated. However $$|+\rangle(|+\rangle+|-\rangle)$$ shows no correlations between spin measurement.

The question is : Are uncorrelated states (for two particles or more, spin half or more) always separable ?

Answer 1

In the bipartite case (only two particle), correlation in spin measurements implies a state of the form $$ |\psi\rangle = \frac{1}{\sqrt{2}} (|+\rangle|-\rangle+|-\rangle|+\rangle) , $$ for example, because the measurement outcome after the first measurement already tells you what the measurement outcome of the second measurement will be. As such, it represents an entangled state. When you have only one of these terms, you cannot really talk about a correlation, because each measurement only ever produces one specific value.

Bipartite entanglement always implies some form of correlation. By simple logic implication, the lack of correlation in a bipartite system must then imply separability.

As argued in the comments, the situation for more than two particles is more complicated. If you only measure the spin of two particles and don't see any correlations, you may still have a system with some form of multipartite entanglement.