Is Schmidt Decomposition well defined for periodic matrix product representation(MPS)? We are able to perform Schmidt decomposition for open-boundary MPSs with dimension of boundary bond $m_0 = 1$, $|\psi\rangle=\sum w_{a_l}|a_l\rangle_L|a_l\rangle_R$. Because we can make $|a_l\rangle_{L,R}$ orthorgonal.
However, I think it is hard to make a Schmidt decomposition for periodic chain with boundary dimension $m_0>1$, for we need to cut the chain twice to make if bipartite, and how can we keep the orthogonality of states during such kind of operations? 
In other words, can we make a Schmidt decomposition with orthogonal bulk states as environment?
Example, we want to make a mixed ensemble consisted of $|a_l\rangle_L$ and $|a_r\rangle_R$ out of MPS $\Psi_{a_l,a_r}|a_l\rangle_L|a_la_r\rangle_C|a_r\rangle_R$ by tracing out the center part, we must make the center part orthogonal, which is not as easy as making the right or left ensembles orthogonal, the latter can be achieved by making MPS left or right canonical. 
 A: One possible solution: 
I figured out a possible solution to this problem.
1. Get the inner product of center block $M_{a_la_r,a_l'a_r'}=\Psi_{a_l,a_r}^{\sigma_c}\Psi^{\sigma_c}_{a_l',a_r'}$, here $\sigma_c$ is the physical indices for center block.
2. perform cholesky(if not able to perform it in rank revealing way, use eigen-value decomposition instead) decomposition, get something like $C_{a_la_r,k} C^\dagger_{k,a_l'a_r'}=M_{a_la_r,a_l'a_r'}$. Then $C^{-1}$ matrix is what diagonalize the central block, it's rank $r(C)\le min(d^{number\;of\;site},a_la_r)$. $C^{-1}$(not well defined in general?) is what normalize the center block noticing the normalization condition $C^{-1}MC^{-1\dagger}=1$. 
3. Insert $CC^{-1}$ in to original MPS, $C^{-1}$ normalizes the enviroment, so we drop $C^{-1}M$(trace procedure) and replace the center block $\Psi\rightarrow C^\dagger$ and get the MPS representation of quantum mixture. 
The drawback is that we can not always get the inverse of $C$ due to rank deficiency, but the result seems do not rely on this property, I believe there is a way to circumvent this issue in deduction.
Still, I call for a smart way to do this, the above way is computational costly: $dm^5$ for m the states kept.
