$\operatorname{SU}(6)$ Casimir invariants and $\operatorname{SU}(2)\otimes\operatorname{SU}(3)$ scalars for a mixed qubit-qutrit states

In the present paper few steps are undertaken towards the description of the “qubit-qutrit” pair – quantum bipartite system composed of two and three level subsystems. Calculations of the Molien functions and Poincaré series for the qubit-qubit and qubit-qutrit “local unitary invariants” are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the enveloping algebra $\mathfrak{su}(6)$.