Admitting I didn't go into the tensor train definition... but basically, in Matlab when you have a N-dimensional array, you can just tell the sum() function which dimension to sum along. See help for sum(). So if you define your multidimensional arrays of the tensor train, summing them is that easy.
For example: A is 10x20x100, you ask B = sum(A, 3) and you'll get B which is a 10x20 array, each element of it a sum of the 100 elements under it in the 3rd dimension.
