[ITensors] Fix issue with directsum involving EmptyStorage

[kmp5VT]

Description

In this I attempt to fix issue 1430 where there is an issue in directsum when an EmptyTensor is fed in as the first argument instead of the second. In tensor_algebra.jl the directsum code attempts to call one(EmptyNumber).

To fix this code I am modifying this function

function directsum_projectors(
  elt1::Type{<:Number}, elt2::Type{<:Number}, i::Index, j::Index, ij::Index
)
  D1 = zeros_itensor(elt1, dag(i), ij)
  D2 = zeros_itensor(elt1, dag(j), ij)
  directsum_projectors!(tensor(D1), tensor(D2))
  return D1, D2
end

I have 2 fix ideas, the first is slightly intrusive elt = promote_type(elt1, elt2) and use elt instead of elt1 in both D1 and D2. The second is less intrusive, it is

directsum_projectors(
  elt1::Type{<:EmptyNumber}, elt2::Type{<:Number}, i::Index, j::Index, ij::Index) = directsum_projector(elt2, elt1, i, j, ij) 

which is what I have implemented.

How Has This Been Tested?

I added some tests in the test_itensors.jl I test an empty tensor in front and in back.

[mtfishman]

I would prefer a solution based on elt = promote_type(elt1, elt2), it is more general (for example it doesn't require code logic that is hard-coded to EmptyNumber like the current solution) and promote_type(elt1, elt2) should be the resulting element type of the final tensor of the operation anyway.

As another corner case, what happens if both inputs are EmptyStorage with EmptyNumber, i.e.:

using ITensors
i, j, k = Index.(2, ("i", "j", "k"))
A = ITensor(i, j)
B = ITensor(i, k)
directsum(A => j, B => k)

? Probably the result should be EmptyStorage but it may be tricky to make that work with the current code logic. Something to keep in mind, but best to leave that for a future PR to keep this PR simpler and more focused.

[kmp5VT]

@mtfishman Would it be reasonable to have this function for now

directsum_projectors(
  elt1::Type{<:EmptyNumber}, elt2::Type{<:EmptyNumber}, i::Index, j::Index, ij::Index
) = error("It is not possible to call directsum on two tensors with element type EmptyNumber")

[mtfishman]

@mtfishman Would it be reasonable to have this function for now

directsum_projectors(
  elt1::Type{<:EmptyNumber}, elt2::Type{<:EmptyNumber}, i::Index, j::Index, ij::Index
) = error("It is not possible to call directsum on two tensors with element type EmptyNumber")

Sure, that sounds like a good idea. Maybe add a sentence like:

If you are inputting ITensors constructor like ITensor(i, j), try specifying the element type, e.g. ITensor(Float64, i, j), or filling them with zero value, e.g. ITensor(0.0, i, j).

[mtfishman]

Thanks!