quantum information - Is long range entanglement detectable with k-local operators?

Given a state $\psi \in H_1\otimes H_2\otimes ... H_n$, and there is long range entanglement, is it possible to certify this by only using k-local operators where $k < n$?

To make it concrete but less general, an example is asking if the GHZ state, $\vert\psi\rangle = \frac{1}{\sqrt{2}}(\vert 00..0\rangle + \vert 11..1\rangle)$ and some less entangled state are indistinguishable if I am allowed to measure only using operators that are of the form, $O_{12}, O_{23}...$ and so on?

Answer

By any measurement on $n-1$ sites, the GHZ state $$ |0,0,\dots,0\rangle+|1,1,\dots,1\rangle $$ and the state $$ |0,0,\dots,0\rangle-|1,1,\dots,1\rangle $$ are indistinguishable, as they have the same $n-1$-site reduced density matrix.

So the answer to your question is No.