Do any entanglement measures for mixed states exist that use only single site correlation functions? For a pure state $\rho_{AB}$, the entropy of entanglement of subsystem $A$ is
\begin{equation}
S( \rho_A) = -tr (\rho_A \log \rho_A)
\end{equation}
where $\rho_A$ is the reduced density matrix of A. 
For a single site of a spin chain, $\rho_A$ can be written in terms of single site correlation functions $\langle \sigma_l^\alpha \rangle$ where $\alpha = x,y,z$.
Are there any entanglement measures for mixed states that use only the same correlation functions, $\langle \sigma_l^\alpha \rangle$ where $\alpha = x,y,z$?
 A: It seems that such a measure for mixed states is fundamentally impossible, since you can have both entangled and separable states which have exactly the local expectation values. For pure states, monogamy of entanglement ensures that the impurity of a reduced density matrix (which can be infered from the expectation values of local Pauli operators) is directly related to entanglement. However for mixed states, this is not the case, as the following example will homefully make clear:
Consider a two qubit system, in which the two reduced density matrices are maximally mixed. In this case, it is possible the system is separable, composed of two copies of the maximally mixed state, or it is maximally entangled, composed of a single EPR pair, or anything in between. 
Thus no function of local expectation values can distinguish separable from entangled states in general.
However, purity (which is a function of single site correlation functions), can indeed be used as a bound on the entanglement of a system, again due to monogomy of entanglement. If the local system is not maximally mixed, then it is not maximally entangled, and hence maximum amount of entanglement possible for a system is a monotonic function of its (im)purity.
A: For any multipartite state $\rho_{ABC...}$ the product state $\rho_A\otimes\rho_B\otimes\rho_C\otimes\ldots$, with $\rho_A$, $\rho_B$, $\rho_C$, ... the local reduced density matrices of $\rho_{ABC...}$, has exactly the same single-site expectation values. So, with only single-site expectation values one cannot say anything about any kind of correlation, not only about entanglement. On the other hand, thanks to the Schmidt decomposition and the invariance under local unitaries of entanglement measures, it is obvious that all bipartite entanglement measures on pure states can only depend on the spectrum of the reduced density matrix.
