Direct sum feature

[ytdHuang]

Problem Description

QuTiP is also preparing the direct_sum function here: qutip/qutip#2662

I think we should also start to think about implementing it.

Just put some thoughts I have in my mind below.

Proposed Solution

For the internal Space and Dimensions structure

we should add a new

struct SumSpace <: AbstractSpace

same as QuTiP.

For the type field

Seems that QuTiP only allows the output of direct_sum to be SuperOperator().

For the data field

Thanks to the function hvcat in Julia's standard library, we can just add a method to it, and everything seems to be easy:

const _QobjConcatGroup = Union{QuantumObject,Number}
function Base.hvcat(rows::Tuple{Vararg{Int}}, X1::_QobjConcatGroup, X::_QobjConcatGroup...)
    new_data = hvcat(rows, get_data(X1), (get_data.(X))...)
    new_dimensions = # implement with SumSpace
    return Qobj(new_data, SuperOperator(), new_dimensions)
end

QuantumToolbox.get_data(n::Number) = n

some examples:

m11 = Qobj([
    3 3 3;
    3 3 3;
    3 3 3
])
m22 = Qobj([
    2 2;
    2 2
])
m12 = 1im * Qobj([
    1 1;
    1 1;
    1 1
])

[
    m11  m12;
    m12' m22
].data

5×5 Matrix{Complex{Int64}}:
 3+0im  3+0im  3+0im  0+1im  0+1im
 3+0im  3+0im  3+0im  0+1im  0+1im
 3+0im  3+0im  3+0im  0+1im  0+1im
 0-1im  0-1im  0-1im  2+0im  2+0im
 0-1im  0-1im  0-1im  2+0im  2+0im

x = sigmax()
ψ = 2im * basis(2, 1)

[
    x  ψ;
    ψ' 3
].data

3×3 SparseMatrixCSC{ComplexF64, Int64} with 5 stored entries:
     ⋅      1.0+0.0im      ⋅    
 1.0+0.0im      ⋅      0.0+2.0im
     ⋅      0.0-2.0im  3.0-0.0im

Alternate Solutions

One thing I'm not quite sure is the type _QobjConcatGroup = Union{QuantumObject,Number} for hvcat. I think this might cause piracy issues (especially for Number-type).

But we need the Number type for the second example above.
One way to avoid this issue is that we wrap the number as Qobj(3). In QuTiP, this has the type Qobj(3).type = 'scalar'. Not sure what should we call it in Julia, because Scalar() is used by StaticArrays, and ScalarOperator() is used by SciMLOperators.

Additional Context

No response