How to add two Matrix Product States of different bond dimensions? If I have the MPS representation of two quantum states, how do I add them? Note that the bond -dimensions need not be the same for the two MPSs.
 A: Say you have two mps to be added together called $\psi$ and $\phi$. At each site, each mps has a 3-tensor with 2 bond indices and 1 physical index. Lets call the 3-tensors of $\psi$ to be M and of $\phi$ to be N. For each physical index value we can view these M and N as matrices because they have 2 indices, namely the bond indices. So if the physical index dimension is two, we can say we have 2 matrices per site for each mps.
When we add the two mps together, we end up with a new mps, lets call it $\chi$. Now $\chi$ has two matrices per site lets call them H. For addition of mps, H is the direct sum of M with N, this means M resides in the top left block of H and N resides in the bottom right block of H, such that H is block diagonal. The resulting bond dimension of $\chi$ is the addition of the bond dimensions of $\psi$ and $\phi$. This is because the bond dimension would be the dimension of the matrices on each site, since these matrices have two indices which are the bond indices, and given we form H matrices by stacking in block diagonal form the M and N matrices, we end up with the addition of the initial bond dimensions.
Here is a reference (section 4.3): https://arxiv.org/abs/1008.3477
Note: Above I am saying for example matrices M and what I mean is matrices $M^{\sigma_i}$, where $\sigma_i$ is the physical index that would label the two matrices on each site when the physical index dimension is 2, eg $\sigma_i = 1, \sigma_i = 2$ and i labels which site of the mps we are on.
