  Absolute separable states is a kind of separable state that remain separable
under the action of any global unitary transformation. These states may or may
not have quantum correlation and these correlations can be measured by quantum
discord. We find that the absolute separable states are useful in quantum
computation even if it contains infinitesimal quantum correlation in it. Thus
to search for the class of two-qubit absolute separable states with zero
discord, we have derived an upper bound for $Tr(\varrho^{2})$, where $\varrho$
denoting all zero discord states. In general, the upper bound depends on the
state under consideration but if the state belong to some particular class of
zero discord states then we found that the upper bound is state independent.
Later, it is shown that among these particular classes of zero discord states,
there exist sub-classes which are absolutely separable. Furthermore, we have
derived necessary conditions for the separability of a given qubit-qudit
states. Then we used the derived conditions to construct a ball for $2\otimes
d$ quantum system described by $Tr(\rho^{2})\leq Tr(X^{2})+2Tr(XZ)+Tr(Z^{2})$,
where the $2\otimes d$ quantum system is described by the density operator
$\rho$ which can be expressed by block matrices $X,Y$ and $Z$ with $X,Z\geq 0$.
In particular, for qubit-qubit system, we show that the newly constructed ball
contain larger class of absolute separable states compared to the ball
described by $Tr(\rho^{2})\leq \frac{1}{3}$. Lastly, we have derived the
necessary condition in terms of purity for the absolute separability of a
qubit-qudit system under investigation.
