Support type = Composed quantum objects

[ytdHuang]

Problem Description

In QuTiP, the dims describes the Hilbert space information separately for left and right basis of the Operator, where QuantumToolbox.jl does not.

Here, I suggest a new type ComposedQuantumObject (not necessary to be this name), where the definition is as follows

struct ComposedQuantumObject{NL<:Int,NR<:Int} <: QuantumObjectType 
    Ldims::SVector{NL,Int}
    Rdims::SVector{NR,Int}
end
Base.show(io::IO, ::ComposedQuantumObject) = print(io, "Composed")
Composed(Ldims::SVector{N,Int}, Rdims::SVector{N,Int}) where {N<:Int} = ComposedQuantumObject(Ldims, Rdims)

This allows us to support some usage for special purpose.

Proposed Solution

Let me explain it by the following example, partial trace (although we already have ptrace, but I just want to show how I expect type = Composed would work) :

# suppose we have two subsystems A and B
NA = 4
NB = 2
IB = qeye(NB)
rhoAB = rand_dm((NA, NB))
basis_list = [ tensor(basis(NA, i), IB) for i in 0:(NA-1)]

The partial trace can actually be obtained as

$$\rho_B = \textrm{Tr}_A [\rho_{AB}] = \sum_{i = 0}^{N_A - 1} \langle i |_A \otimes \mathbb{1}_B \cdot \rho_{AB} \cdot | i \rangle_A \otimes \mathbb{1}_B.$$

That is,

rhoB = Qobj(zeros(ComplexF64, NB, NB))
for bl in basis_list
    for br in basis_list
        rhoB += bl' * rhoAB * br
    end
end

rhoB ≈ ptrace(rhoAB, 2)

The main thing is to support some basic functions like

• kron for two different Qobj type
• adjoint of Composed
• * between Operator and Composed -> Composed
• * between Composed and Composed -> Operator

Alternate Solutions

No response

Additional Context

No response