Geometric and Operational Characterization of Two-Qutrit Entanglement

TL;DR

The paper addresses entanglement in bipartite two-qutrit pure states by separating pairwise from genuinely three-level correlations using symmetric polynomials of the reduced-density eigenvalues. It introduces a rank-sensitive determinant-based invariant $G = 3\sqrt{3}\sqrt{s_3}$ and an analytic constraint coupling $G$ to the I-concurrence $C_I$, delineating the physically allowed region in $(C_I,G)$ and clarifying spectral-degeneracy boundaries. An operational interpretation is provided via a three-path interferometry mapping, deriving conditional visibility $V_{\mathrm{cond}}$, conditional predictability $P_{\mathrm{cond}}$, and a three-way complementarity relation that accommodates unequal path transmittances through $G_T$. Numerical simulations validate the analytic bounds and complementarity relations, illustrating the geometric-content of two-qutrit entanglement and its interferometric manifestations. The work offers a cohesive geometric-operational framework with potential extensions to mixed states, higher-dimensional systems, and experimental implementations.

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.

Figures (2)

• Figure 1: Scatter plot of randomly generated two-qutrit pure states in the $(C_I, G)$ plane. Rank-2 states lie on the line $G = 0$, while rank-3 states populate the interior of the physically allowed region. The solid curve denotes the analytic boundary obtained by saturating Eq. (16).
• Figure 2: Conditional visibility $V_{\mathrm{cond}}$ and conditional predictability $P_{\mathrm{cond}}$ as functions of the erasure parameter $\tau$. The tradeoff between the two quantities demonstrates wave--particle complementarity in the qutrit quantum erasure scheme.