Is there any correlation between the energy density fluctuations of two separate systems in a vacuum state? I think the title says it all.  What I am curious to find out is if there are any observable changes in the fluctuations of zero-point energy in a vacuum state system that are the consequence of operations performed at a separate vacuum state system.  I am not simply asking if there are correlations naturally, I am asking if there is any operation/setup/configuration that could be implemented to derive such a result.         -Thanks-  
 A: I've assumed that by a "separate vacuum state system" you mean two systems of measurement or preparation operators, $\hat\phi(x_1)\cdots\hat\phi(x_m)$ and $\hat\phi(y_1)\cdots\hat\phi(y_n)$, where the positions in the $x$ group and the positions in the $y$ group are separated by a relatively large space-like distance.
In that case, No, at least not if the vacuum state $\omega$ of the quantum field satisfies Cluster Decomposition, which is the requirement that for any space-like 4-vector $V$,
\begin{eqnarray}\lim\limits_{\lambda\rightarrow\infty}\omega(\hat\phi(x_1)\cdots\hat\phi(x_m)\hat\phi(y_1+\lambda V)\cdots\hat\phi(y_n+\lambda V))=\omega(\hat\phi(x_1)\cdots\hat\phi(x_m))\omega(\hat\phi(y_1)\cdots\hat\phi(y_n)).
\end{eqnarray}
This ensures that measurements at large space-like separation are independent. Weinberg, for example, emphasizes that for him this is a Fundamental Principle by giving it its own chapter in Volume $\scriptstyle{\mathrm{I}}$ of his "The Quantum Theory of Fields".
The Cluster Decomposition Condition or Principle, however, is precisely introduced to ensure that there are no correlations at large space-like separation in models of quantum field theory. Within the Wightman axioms, we can prove that if the other axioms that define a quantum field are satisfied, then Cluster Decomposition is equivalent to the vacuum state being unique.
We can introduce models that do not have a unique vacuum state, and people have done so, but we then have to look to experiment to determine whether, in the vacuum state, there are the long-range correlations that a particular model predicts. In non-vacuum states, however, there are states such as Bell states in which there are non-local correlations at arbitrary space-like separation, so the simple discovery of long-range correlations may simply mean that we are not in the vacuum state; the observed long-range correlations must also be Lorentz and translation invariant for them to be considered a property of the vacuum.
A: The paper you mentioned is following, right?
http://pra.aps.org/abstract/PRA/v84/i3/e032336
http://xxx.yukawa.kyoto-u.ac.jp/PS_cache/arxiv/pdf/1109/1109.2203v1.pdf
In the paper, some comments are made on topics discussed in this thread. The Hall edge current system has an effective description by quantum chiral boson fields in 1+1 space-time dimensions. So the ground state of the system is called "the vacuum state", and this correspondence is certainly vaild in the context of effective field theory. The authors claim that thermal noise effect at mK temperature is too small to affect the observation of energy teleportation in the ground state. 
The answer of your question seems trivial. Because quantum noises of two separate vacuums are independent each other, a noise measurement of one vacuum does not provide any information about noise of another vacuum. So we cannot make control of the separate noise fluctuaion to suppress its amplitude and extract a part of zero-point energy. 
As opposed to the original setup in the paper, quantum energy teleportation will not be attained in your scheme.
Though you might have already noticed, more information about quantum energy teleportation is available in a review article by Hotta,
http://www.tuhep.phys.tohoku.ac.jp/~hotta/extended-version-qet-review.pdf
