# 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?

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.
• Just to add on, your alternate state is also maximally entangled. Would you be able to comment on the case of whether k-local operators can also distinguish between the GHZ state and one with entanglement, say over s sites and $k<s$? – user1936752 Jan 11 at 17:33
• @user1936752 If you don't restrict to pure state (which seems contrived), then there is a separable state with the same $n-1$ site reduced density matrices. – Norbert Schuch Jan 11 at 17:41
• Yes, but I am allowed to use multiple k-local operators. For it to be indistinguishable, every k-local operator must produce the same measurements. The claim you make, is it still true if I have many operators like $O_{1..k}, O_{2..k+1}...O_{n-k+1..n}$? – user1936752 Jan 11 at 18:17