Three-Qubit Quantum System Classification Method Using Special Linear Group Under Stochastic Local Operations and Classical Communication ( SLOCC) Protocol

Quantum technology has been instigated in Industrial Revolution 4.0, transforming the
future with advanced technology. Therefore, understanding one of the key resources of
quantum technology foundations, which is quantum entanglement, is crucial and
valuable. An exponentially growing interest by many entities in quantum technology
has propelled comprehensive studies of quantum entanglement, especially on the
entanglement classification protocol. The three main protocols of entanglement
classification are local unitary (LU), local operations and classical communication
(LOCC), and stochastic local operations and classical communication (SLOCC). The
special linear group, ( ) SL n of multipartite entanglement classification under SLOCC
protocol has not been extensively studied due to its complex structure; thus, it is
challenging to develop its classification method. Therefore, the main objective of this
research is to develop the classification method of multipartite entanglement state using
a combination of the special linear group, ( ) 2SL into an (2) (2) (2) SL SL SL   model
operator under SLOCC protocol. The model operator was used to classify the
entanglement of the pure state with selected parameters. An in-depth analysis was done
to determine the subgroups of the special linear group, ( ) SL n and to define the
characteristics and mathematical details of its theory and SLOCC transformation. This
model was developed using Mathematica version 13.2 software with three parameters.
In conclusion, the SL(2) SL(2) SL(2) model operator with three selected parameters
changed the type of entanglement state under SLOCC protocol