Geometric and operational structure of two qutrit entanglement
We investigate bipartite two-qutrit pure states and analyze their entanglement properties using eigenvalue-based quantities of the reduced density matrix. In addition to the I-concurrence, we show that the determinant of the two-qutrit coefficient matrix defines a rank-sensitive geometric invariant proportional to √ λ1λ2λ3, where λi are the eigenvalues of the reduced density matrix. Unlike the I-concurrence, this invariant vanishes identically for all rank-2 states and is nonzero only for rank-3 states, thereby providing a direct characterization of genuine three-level correlations. Furthermore, we establish an operational interpretation of the invariant in terms of triple-path interference visibility in three-path optical interferometry. These results provide a compact geometric framework for identifying rank-structured entanglement in bipartite two-qutrit systems and reveal a direct connection between high-dimensional quantum correlations and multipath interference phenomena.