  We recently introduced a method to approximate functions of Hermitian Matrix
Product Operators or Tensor Trains that are of the form $\mathsf{Tr} f(A)$.
Functions of this type occur in several applications, most notably in quantum
physics. In this work we aim at extending the theoretical understanding of our
method by showing several properties of our algorithm that can be used to
detect and correct errors in its results. Most importantly, we show that there
exists a more computationally efficient version of our algorithm for certain
inputs. To illustrate the usefulness of our finding, we prove that several
classes of spin Hamiltonians in quantum physics fall into this input category.
We finally support our findings with numerical results obtained for an example
from quantum physics.
