linear_inversion.jl

"""
    LinearInversion(povm, basis=gell_mann_matrices(size(first(povm), 1)))

Construct a linear inversion method for quantum state tomography.
"""
struct LinearInversion{T1,T2}
    pseudo_inv::Matrix{T1}
    basis::Array{T2,3}
    function LinearInversion(povm, basis=gell_mann_matrices(size(first(povm), 1)))
        pseudo_inv = stack(F -> real_orthogonal_projection(F, basis), povm, dims=1) |> pinv
        T = eltype(pseudo_inv)
        new{T,complex(T)}(pseudo_inv, basis)
    end
end

"""
    prediction(outcomes, method::LinearInversion)

Predict the quantum state from the outcomes of a tomography experiment using the [`LinearInversion`](@ref) method.
"""
function prediction(outcomes, method::LinearInversion)
    xs = method.pseudo_inv * vec(normalize(outcomes, 1))
    linear_combination(xs, method.basis) |> project2density
end