Expanding quasi-MV algebras by a quantum operator

We investigate an expansion of quasi-MV algebras by a genuine quantum unary operator. The variety $\sqrt{^{\prime }}\mathbb{ QMV}$ of such $\sqrt{^{\prime }}$ \emph{quasi-MV algebras} has a subquasivariety whose members - called \emph{cartesian} - can be obtained in an appropriate way out of MV algebras. After showing that cartesian $\sqrt{ ^{\prime }}$ quasi-MV algebras generate $\sqrt{^{\prime }}\mathbb{QMV}$, we prove a standard completeness theorem for $\sqrt{^{\prime }}\mathbb{QMV}$ w.r.t. an algebra over the complex numbers.