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In a previous paper [1], we investigated the Blind Source Separation (BSS)problem, for the nonlinear mixing model that we introduced in that paper. Weproposed to solve this problem by using a maximum likelihood (ML) approach.When applying the ML approach to BSS problems, one usually determines theanalytical expressions of the derivatives of the log-likelihood with respect tothe parameters of the considered mixing model. In the literature, thesecalculations were mainly considered for linear mixtures up to now. They aremore complex for nonlinear mixtures, due to dependencies between the consideredquantities. Moreover, the notations commonly employed by the BSS community insuch calculations may become misleading when using them for nonlinear mixtures,due to the above-mentioned dependencies. In this document, we therefore explainthis phenomenon, by showing the effect of indirect dependencies on theapplication of the ML approach to the mixing model considered in [1]. Thisyields the explicit expression of the complete derivative of the log-likelihoodassociated to that mixing model.

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