Entanglement Distillation of some Rank-Five Symmetric NPT States in Two-Qutrit Systems

Entanglement distillation is the process of converting multiple copies of mixed entangled states into pure entangled states using local operations and classical communication (LOCC). While it is known that all two-qubit NPT states are distillable, the distillability of higher-dimensional states—particularly those of rank five—remains a significant open problem. This paper investigates the distillability of a specific family of rank-five symmetric two-qutrit states, aiming to resolve the status of a previously undetermined parameter interval.

The authors analyze a family of symmetric two-qutrit states $\rho$ defined by five orthonormal pure states. They focus on the interval of the eigenvalue parameter $\lambda_5 \in [\frac{24\sqrt{2}-33}{7}, \frac{33-12\sqrt{6}}{25})$, where the distillability was previously unknown. To test for 1-distillability, they apply the principal minors criterion to projected matrices derived from the state's partial transpose. For 2-distillability, they examine the negative subspace of the two-copy state $\sigma = (\rho^\Gamma)^{\otimes 2}$ and employ a quadratic form decomposition to search for Schmidt-rank-two vectors that could yield a negative expectation value.

The study establishes that the states in the specified parameter interval are 1-undistillable, as all principal minors of the projected matrices are non-negative. Regarding 2-distillability, the authors uncover a structural obstruction: no Schmidt-rank-two vector within a 17-dimensional subspace of the negative eigenspace of $\sigma$ can produce a negative expectation value. This effectively narrows the search space for potential 2-distillability and provides a rigorous framework for testing higher-order distillability in high-dimensional systems.