Two-qutrit Entanglement Witnesses and Gell-Mann Matrices

TL;DR

This paper constructs entanglement witnesses for two-qutrit systems using Gell-Mann matrices and linear programming, analyzing their decomposability and providing new tools for quantum entanglement detection.

Contribution

It introduces a method to construct two-qutrit entanglement witnesses based on Gell-Mann matrices, including analysis of their decomposability properties.

Findings

All λ-diagonal EWs are decomposable.

Existence of non-decomposable λ-non-diagonal EWs.

Construction methods include exact and approximate linear programming.

Abstract

The Gell-Mann $\lambda$ matrices for Lie algebra su(3) are the natural basis for the Hilbert space of Hermitian operators acting on the states of a three-level system(qutrit). So the construction of EWs for two-qutrit states by using these matrices may be an interesting problem. In this paper, several two-qutrit EWs are constructed based on the Gell-Mann matrices by using the linear programming (LP) method exactly or approximately. The decomposability and non-decomposability of constructed EWs are also discussed and it is shown that the $\lambda$-diagonal EWs presented in this paper are all decomposable but there exist non-decomposable ones among $\lambda$-non-diagonal EWs.