  A multipartite quantum state is entangled if it is not separable. Quantum
entanglement plays a fundamental role in many applications of quantum
information theory, such as quantum teleportation. Stochastic local quantum
operations and classical communication (SLOCC) cannot essentially change
quantum entanglement without destroying it. Therefore, entanglement can be
classified by dividing quantum states into equivalence classes, where two
states are equivalent if each can be converted into the other by SLOCC.
Properties of this classification, especially in the case of non
two-dimensional quantum systems, have not been well studied. Graphical
representation is sometimes used to clarify the nature and structural features
of entangled states. SLOCC equivalence of quantum bits (qubits) has been
described graphically via a connection between tripartite entangled qubit
states and commutative Frobenius algebras (CFAs) in monoidal categories. In
this paper, we extend this method to qutrits, i.e., systems that have three
basis states. We examine the correspondence between CFAs and tripartite
entangled qutrits. Using the symmetry property, which is required by the
definition of a CFA, we find that there are only three equivalence classes that
correspond to CFAs. We represent qutrits graphically, using the connection to
CFAs. We derive equations that characterize the three equivalence classes.
Moreover, we show that any qutrit can be represented as a composite of three
graphs that correspond to the three classes.
