  In this work, we present new connections between three types of quantum
states: positive under partial transpose states, symmetric with positive
coefficients states and invariant under realignment states. First, we obtain a
common upper bound for their spectral radii and a result on their filter normal
forms. Then we prove the existence of a lower bound for their ranks and the
fact that whenever this bound is attained the states are separable. These
connections add new evidence to the pattern that for every proven result for
one of these types, there are counterparts for the other two, which is a
potential source of information for entanglement theory.
