Classification of Bipartite and Tripartite Qutrit Entanglement under SLOCC

It is well known that quantum entanglement plays very important roles in quantum information theory. One of the main tasks in quantum information theory is to find out how many different ways multipartite pure states can be entangled. It has been shown that entangled qudits are less affected by noise than entangled qubits [1, 2]. In quantum cryptography it is more secure against evesdropping attacks using entangled qutrits or qudits than using qubits [3, 4, 5, 6]. These facts motivate our interest in multi-dimensional entangled states. As the concept of entanglement is related to the nonlocal properties of a state, local quantum operations can not affect the intrinsic nature of entanglement[7]. It is natural and meaningful to classify pure states in terms of stochastic local operations and classical communication (SLOCC). In [8] it was shown that SLOCC equivalent pure states can carry out the same quantum-informational tasks with non-null possibly different possibilities, and two N -partite states Ψ and Φ are equivalent under SLOCC if and only if there exist invertible local operations (ILOs) F , · · · , F [N ] such that Ψ = F [1]⊗F [2]⊗· · ·⊗F [N Φ. In recent years, a lot of efforts have been made on classification of multipartite entanglement under SLOCC [3, 9-18]. In [15] an inductive method of classifying n-qubit entanglement under SLOCC has been presented, from which the entanglement classifica-