Bundled matrix product states represent low-energy excitations faithfully

We consider a set of density matrices. All of which are written in the same orbital basis, but the orbital basis size is less than the total Hilbert space size. We ask how each density matrix is related to each of the others by establishing a norm between density matrices based on the truncation error in a partial trace for a small set of orbitals. We find that states with large energy differences must have large differences in their density matrices. Small energy differences are divided into two groups, one where two density matrices have small differences and another where they are very different, as is the case of symmetry. We extend these ideas to a bundle of matrix product states and show that bond dimension of the wavefunction ansatz for two states with large energy differences are larger. Meanwhile, low energy differences can have nearly the same bond dimensions for similar states.