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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?

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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.

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  • $\begingroup$ 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 $k<s$? $\endgroup$ – user1936752 Jan 11 at 17:33
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    $\begingroup$ @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. $\endgroup$ – Norbert Schuch Jan 11 at 17:41
  • $\begingroup$ 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 $O_{1..k}, O_{2..k+1}...O_{n-k+1..n}$? $\endgroup$ – user1936752 Jan 11 at 18:17
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    $\begingroup$ Yes, the 50-50 mixture of |0,0...> and |1,1,...> is indistinguishable from the GHZ for all n-1 site measurements. $\endgroup$ – Norbert Schuch Jan 11 at 21:13

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