Revisiting the invariant ring of two-qubit mixed states

Local unitary (LU) equivalence classifies quantum states up to changes of local bases and preserves their entanglement properties. Polynomial LU invariants provide an algebraic approach to determining whether two quantum states are LU equivalent. We revisit the ring of complex polynomial LU invariants on the full two-qubit operator space, equivalently the invariant ring under the local conjugation action of (2,)×(2,). For the two-qubit system, King et al. previously determined a generating set of 21 invariants and the Cohen--Macaulay structure of this ring. Using a unified notation, we provide a self-contained derivation of the known Molien series and a systematic graphical reconstruction of the known generators. The Molien series is derived by explicit contour integration over the maximal torus, supplying computational steps not given in the earlier treatment. We systematically reconstruct the known generators using the graphical tensor method and provide explicit algebraic reductions of the candidate invariants. This work clarifies the calculations and graphical construction underlying the established results.