A New Method of Classification of Pure Tripartite Quantum States

The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states $\sigma_{AB}= p|\chi_{1}\rangle\langle\chi_{1}|\otimes\rho_{1}+(1-p)|\chi_{2}\rangle\langle\chi_{2}|\otimes\rho_{2}$, where $|\chi_{1}\rangle=\alpha|0\rangle+\beta|1\rangle$, $|\chi_{2}\rangle=\beta|0\rangle+(-1)^{n}\alpha|1\rangle$. We have shown that the state $\sigma_{AB}$ represent a classical state when $n$ is odd while it represent a non-classical state when $n$ is even. The purification of the state $\sigma_{AB}$ is studied and found that the purification is possible if the spectral decomposition of the density matrices $\rho_{1}$ and $\rho_{2}$ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity $\langle\chi_{1}|\chi_{2}\rangle$, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.