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.