Effect of indirect dependencies on "Maximum likelihood blind separation of two quantum states (qubits) with cylindrical-symmetry Heisenberg spin coupling"

In a previous paper [1], we investigated the Blind Source Separation (BSS) problem, for the nonlinear mixing model that we introduced in that paper. We proposed to solve this problem by using a maximum likelihood (ML) approach. When applying the ML approach to BSS problems, one usually determines the analytical expressions of the derivatives of the log-likelihood with respect to the parameters of the considered mixing model. In the literature, these calculations were mainly considered for linear mixtures up to now. They are more complex for nonlinear mixtures, due to dependencies between the considered quantities. Moreover, the notations commonly employed by the BSS community in such calculations may become misleading when using them for nonlinear mixtures, due to the above-mentioned dependencies. In this document, we therefore explain this phenomenon, by showing the effect of indirect dependencies on the application of the ML approach to the mixing model considered in [1]. This yields the explicit expression of the complete derivative of the log-likelihood associated to that mixing model.

💡 Research Summary

The paper revisits the blind source separation (BSS) problem for a nonlinear mixing model introduced in a previous work, where two qubits are coupled through a cylindrically‑symmetric Heisenberg interaction. While maximum‑likelihood estimation (MLE) is a standard tool for linear BSS, extending it to nonlinear mixtures is non‑trivial because the observed signals depend on the model parameters both directly and indirectly through intermediate variables. The authors first formalize the mixing model: source vectors s are transformed by a nonlinear function f(s, θ) that incorporates the coupling parameter θ, producing the observations x. The log‑likelihood L(θ) is expressed as a sum over the log‑densities of the observations given θ.

A central contribution is the rigorous derivation of the full derivative dL/dθ. In linear settings the gradient is simply ∂L/∂x·∂x/∂θ, but for the present nonlinear case an additional term arises from the dependence of x on an intermediate function g(s, θ) inside f. By applying the chain rule meticulously, the authors separate the direct component (∂L/∂x·∂x/∂θ) from the indirect component (∂L/∂g·∂g/∂θ). They also clarify a common notation pitfall: the symbol ∂/∂θ in many BSS papers implicitly denotes only the direct partial derivative, whereas the total derivative d/dθ must be used when indirect dependencies exist.

The final gradient expression is:

dL/dθ = Σ_i