Geometric and Operational Characterization of Two-Qutrit Entanglement

TL;DR

This paper provides a comprehensive geometric and operational analysis of entanglement in two-qutrit pure states, introducing new invariants and linking them to optical interferometry to better understand entanglement structure.

Contribution

It introduces a rank-sensitive geometric invariant based on the coefficient matrix determinant and relates it to I-concurrence, advancing the understanding of two-qutrit entanglement.

Findings

The determinant invariant vanishes for rank-2 states and is nonzero for rank-3 entangled states.

Derived an explicit analytic relation between the determinant invariant and I-concurrence.

Numerical results confirm the theoretical framework and complementarity relations.

Abstract

We investigate the entanglement structure of bipartite two-qutrit pure states from both geometric and operational perspectives.Using the eigenvalues of the reduced density matrix, we analyze how symmetric polynomials characterize pairwise and genuinely three-level correlations. We show that the determinant of the coefficient matrix defines a natural, rank-sensitive geometric invariant that vanishes for all rank-2 states and is nonzero only for rank-3 entangled states. An explicit analytic constraint relating this determinant-based invariant to the I-concurrence is derived, thereby defining the physically accessible region of two-qutrit states in invariant space. Furthermore, we establish an operational correspondence with three-path optical interferometry and analyze conditional visibility and predictability in a qutrit quantum erasure protocol, including the effects of unequal path transmittances. Numerical demonstrations confirm the analytic results and the associated complementarity relations. These findings provide a unified geometric and operational framework for understanding two-qutrit entanglement.