  In this paper, we study the separability of quantum states in bosonic system.
Our main tool here is the "separability witnesses", and a connection between
"separability witnesses" and a new kind of positivity of matrices--- "Power
Positive Matrices" is drawn. Such connection is employed to demonstrate that
multi-qubit quantum states with Dicke states being its eigenvectors is
separable if and only if two related Hankel matrices are positive semidefinite.
By employing this criterion, we are able to show that such state is separable
if and only if it's partial transpose is non-negative, which confirms the
conjecture in [Wolfe, Yelin, Phys. Rev. Lett. (2014)]. Then, we present a class
of bosonic states in $d\otimes d$ system such that for general $d$, determine
its separability is NP-hard although verifiable conditions for separability is
easily derived in case $d=3,4$.
