Unable to construct a FiniteMPS that has alternating virtual spaces without specifying all virtual spaces.

[DaanMaertens]

A possible constructor for a FiniteMPS is to specify a (vector of) physical spaces and a single maxvirtspace representing the maximum space one wants on the virtual level, according to

FiniteMPS(rand, ComplexF64, rand(3:2:20) ,pspace, maxvirtspace)

If one however wants the physical space pspace=Rep[SU₂](1//2 => 1) then this constructor will fail because it will try to find the infimum between a space containing integer reps of SU2 and another containing half-integers only. The virtual spaces of the mps should be alternating between integer and half-integer and this can currently only be done by using the constructor that requires all the virtual spaces.

[DaanMaertens]

see commit cad1d97 for my proposed solution.

[Jutho]

What happens if you make maxvirtspace have both integer and halfinteger representations?

[DaanMaertens]

This gives the same result as my implementation. So technically you could already do it, though this way might be a little counterintuitive?

[Jutho]

I agree that being able to specify a vector of maxvirtspaces should be an option.

The simplicity of specifying a single space is of course attractive, and I think it works generally, but indeed, you need to think through the consequences and how to choose/construct it.

[Gertian]

You could also assume that the user defines the maximum dimension of the virtual Hilbert space.
So if you give it maxvirtdim = 5 with e.g. SU(2) sectors it might take two spin 1/2 sectors and a spin 0 sector. Or 5 spin 0 sectors etc...

Something like this could be easily achieved by using the truncdim option in TensorKit.svd() .

[lkdvos]

The problem being that you don't have anything to SVD yet if you create a random state. The singular values will be distributed randomly, and you might end up picking 1 spin 5 sector by accident, instead of an even distribution which is more likely what you want/expect.

I would suggest to modify the current constructor which specifies all virtual spaces, to automatically use these as site-dependent maximal spaces. This way, we can also intercept accidental creation of states that are not full-rank, which are likely to lead to issues further down the line.

[lkdvos]

# constructor with a single physical space, and a single virtual space.
FiniteMPS([f, elt], pspace::VectorSpace, maxvspace::ElementarySpace; length=N, kwargs...)
# constructor with variable physical spaces and variable virtual spaces
FiniteMPS([f, elt], pspaces::Vector{<:VectorSpace}, maxvspaces::Vector{<:ElementarySpace}; kwargs...)

While it might be convenient to automatically infer all other combinations, it might also become quite a mess to keep this implemented nicely, and I would argue that the it is not that much more verbose to type repeat(alternatingspaces, N/2).
In the case where you explicitly wish to construct an MPS which is not full-rank, probably it is better to really require the user to construct the tensors themselves, as it is also less obvious how to gauge them consistently I think.

[Gertian]

Since it's a finiteMPS I'd just build the n-th virtual space as (physical space)^n or a trunctated version thereof.
But indeed it's slightly ill defined since the truncated sectors would be randomly picked.

Perhaps something like : ```FiniteMPS([f, elt], pspaces::Vector{<:VectorSpace}, truncator; kwargs...)''' with truncator an algorithm that is passed to the QR/SVD used in the constructor ? Like this users can freely choose between the available truncation methods from TensorKit.

[lkdvos]

But again, it really does not make sense to truncate a just-initialized state, no matter what truncated algorithm you are using? For example, I could ask for a state filled with zeros, in which case any truncation method fails anyways. If you really want to truncate an MPS, I think it makes more sense to construct it, and then just truncate the result.
The FiniteMPS constructor does not really need to have the responsability of also truncating.