Construction of PPT entangled state and its detection by using second-order moment of the partial transposition

We adopt a formalism by which we construct and detect a new family of positive partial transpose entangled states in $d_1\otimes d_2$ dimensional system. Our detection method is based on the second order moment $p_2(\rho^{T_B})$ as it is very easy to calculate and may be realizable in laboratory. We show that if the second order moment $p_2(\rho^{T_B})$ in $d_1\otimes d_2$ dimensional system satisfy $p_2(\rho^{T_B})\leq\frac{1}{d_1 d_2-1}$, then the state is a PPT state. We also derive an equivalent condition on the bloch vector. Then, we construct a quantum state by considering the mixture of a separable and an entangled state and obtain a condition on the mixing parameter for which the mixture represents a PPTES. Finally, applying our results, we have shown that the distillable key rate of the private state, prepared through our prescription, is positive. It suggests that our result also has potential applications in quantum cryptography.