Structural physical approximation of partial transposition makes possible to distinguish SLOCC inequivalent classes of three-qubit system

Detection and classification of entanglement properties of a multi-qubit system is a topic of great interest. This topic has been studied extensively, and thus we found different approaches for the detection and classification of multi-qubit entangled states. We have applied partial transposition operation on one of the qubits of the three-qubit system and then studied the entanglement properties of the three-qubit system, which is under investigation. Since the partial transposition operation is not a quantum operation, we have approximated partial transposition operation in such a way that it represents a completely positive map. The approximated partial transposition operation is also known as structural physical approximation of partial transposition (SPA-PT). We have studied in detail the application of SPA-PT on a three-qubit system and provided explicitly the matrix elements of the density matrix describing SPA-PT of a three-qubit system. Moreover, we propose a criterion to classify all possible stochastic local operations and classical communication inequivalent classes of a pure as well as mixed three-qubit state through SPA-PT map, which makes our criterion experimentally realizable. We have illustrated our criterion for detection and classification of three-qubit entangled states by considering few examples