Isomorphisms

using PiccoloQuantumObjects
using SparseArrays # for visualization

Linear algebra operations on quantum objects are often performed on real vectors and matrices. We provide isomorphisms to convert between complex and real representations of quantum objects. These isomorphisms are used internally by the QuantumSystem type to perform quantum dynamics.

Quantum state isomorphisms

• ket_to_iso is the real isomorphism of a quantum state ψ ∈ ℂⁿ
• iso_to_ket is the inverse isomorphism of a real vector ψ̃ ∈ ℝ²ⁿ

ψ = [1; 2] + im * [3; 4]
ψ̃ = ket_to_iso(ψ)

4-element Vector{Int64}:
 1
 2
 3
 4

iso_to_ket(ψ̃)

2-element Vector{Complex{Int64}}:
 1 + 3im
 2 + 4im

Quantum operator isomorphisms

We often need to convert a complex matrix U to a real vector Ũ⃗. We provoide the following isomorphisms to convert between the two representations.

• iso_vec_to_operator(Ũ⃗::AbstractVector{ℝ})
• operator_to_iso_vec(U::AbstractVector{ℂ})
• iso_vec_to_iso_operator(Ũ⃗::AbstractVector{ℝ})
• iso_operator_to_iso_vec(Ũ::AbstractMatrix{ℝ})
• iso_operator_to_operator(Ũ::AbstractMatrix{ℝ})
• operator_to_iso_operator(U::AbstractMatrix{ℂ})

In additon, we provide mat(x::AbstractVector) to convert a vector x into a square matrix, as the inverse to Base's vec.

Julia uses column-major order.

U = [1 5; 2 6] + im * [3 7; 4 8]
Ũ⃗ = operator_to_iso_vec(U)

8-element Vector{Int64}:
 1
 2
 3
 4
 5
 6
 7
 8

iso_vec_to_operator(Ũ⃗)

2×2 Matrix{Complex{Int64}}:
 1+3im  5+7im
 2+4im  6+8im

Density matrix isomorphisms

The isomorphisms for density matrices are:

• density_to_iso_vec(ρ::AbstractMatrix{ℂ})
• iso_vec_to_density(ρ̃::AbstractVector{ℝ})

ρ = [1 2; 3 4] + im * [5 6; 7 8]
ρ̃⃗ = density_to_iso_vec(ρ)

8-element Vector{Int64}:
 1
 3
 2
 4
 5
 7
 6
 8

Quantum dynamics isomorphisms

The quantum dynamics isomorphisms, which correspond to these state transformations, are handled internally by the QuantumSystem type.

The Isomorphisms.iso isomorphism of a Hamiltonian $H$ is:

\[\text{iso}(H) := \widetilde{H} = \mqty(1 & 0 \\ 0 & 1) \otimes \Re(H) + \mqty(0 & -1 \\ 1 & 0) \otimes \Im(H)\]

where $\Im(H)$ and $\Re(H)$ are the imaginary and real parts of $H$ and the tilde indicates the standard isomorphism of a complex valued matrix:

\[\widetilde{H} := \mqty(1 & 0 \\ 0 & 1) \otimes \Re(H) + \mqty(0 & -1 \\ 1 & 0) \otimes \Im(H)\]

Hence, the generator Isomorphisms.G associated to a Hamiltonian $H$ is:

\[G(H) := \text{iso}(- i \widetilde{H}) = \mqty(1 & 0 \\ 0 & 1) \otimes \Im(H) - \mqty(0 & -1 \\ 1 & 0) \otimes \Re(H)\]

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