Hi good question. Getting the von Neumann entropy of a mixed state \rho which is in the form of an MPO is not an easy task, and how best to do it is somewhat of an open research question. But obtaining the Renyi entropy, especially the second Renyi entropy S_2 is something straightforwardly computable with MPOs.
For others reading this discussion, recall that S_2 = -\log(\text{Tr}[ \rho_A^2]) where \rho_A = \text{Tr}_B[\rho] is the partial trace of \rho taken over region B which is the complement of region A.
Here is then a sketch of \rho, \rho_A and \text{Tr}[ \rho_A^2] for the case where A is the first half of the sites of the MPO:
So the main steps to computing the second Renyi entropy of an MPO mixed state or density matrix are:
- partial-trace the MPO (for more, see this question trace and partial trace of MPO)
- square the resulting MPO of \rho_A with itself
- trace the result of that
Then you just take the negative log of the result.
One thing I like about this tensor network viewpoint of the second Renyi entropy is that it makes it clear what people are talking about when they say it corresponds to a “double sheeted path integral” or some similar language to that. Really it just means the diagram above where the two density matrices are traced individually on region B, but then together on region A.