I wonder how I can compute the sum of the norm of a matrix represented by an MPO, namely, \sum_{ij} |a_{ij}|, where a_{ij} is the matrix element of the matrix A, represented by a MPO (in reality, it will be a pure state density matrix).
Below is my thought:
Mathematically, I could represent \sum_{ij} |a_{ij}| = \vec{1}^\intercal |A |~ \vec{1}, where \vec{1} is a ‘all-one vector’ \vec{1} =[1,1,1,...]^\intercal . Then I can simply compute the expected value of A to get the sum of norm.
So now the problem is broken into 2 steps, which both I don’t know to implement…
I need to find a way to apply the absolute value to MPO A, such that I get a new MPO A', where the matrix element a'_{ij} = |a_{ij}|. (I checked the literature a bit about Quantic Tensor Train but did not find it.)
How to efficiently represent the all one vector using MPS. I used the following built-in method to construct the MPS, and find that the bond dimension scales exponentially. Is there a clever way to represent this all one vector?
L=10
sites=siteinds(2,L)
T=ITensor(sites)
for index in eachindex(T)
T[index]=1.
end
M =MPS(T,sites)
For the all ones vector, it should be the case that the MPS is just a product state, so like [1,1] [1,1] [1,1] … where I’m writing it out there for the case of a physical dimension of 2.
Unfortunately, the part about taking the element-wise absolute value is not easy. In general there isn’t a direct or guaranteed efficient way to apply element-wise non-linear functions to tensor networks. The only way I can think of would be to apply the “tensor cross interpolation” (TCI) algorithm, and when it computes a value of the “target tensor” (output tensor) you would compute that value by taking a certain element of your MPO then returning its absolute value. That approach might work well if the absolute value of your MPO is another MPO, which is just something that would have to be determined on a case-by-case basis.
In general, many of the quantities proposed by the theory community (e.g. entanglement of formation, or I suppose also coherence which I had not heard of before) are difficult to compute in practice.
Thanks for your reply! Is it possible to ask for some literature on tensor cross interpolation that may implement an “abs” function? I understand that this may not be the case in the end, as the |a_ij| may not be an efficient representation of MPO. However, the idea is still very intriguing, and I would like to give it a try.