  We characterize the boundary of the convex compact set of absolutely
separable states, referred as {\bf AS}, that cannot be transformed to entangled
states by global unitary operators, in $2\otimes d$ Hilbert space. However, we
show that the absolutely separable states of rank-$(2d-1)$ are extreme points
of such sets. We then discuss conditions to examine if a given full-rank
absolutely separable state is an interior point or a boundary point of {\bf
AS}. Moreover, we construct two-qubit absolutely separable states which are
boundary points but not extreme points of {\bf AS} and prove the existence of
full-rank extreme points of {\bf AS}. Properties of certain interior points are
also explored. We further show that by examining the boundary of the above set,
it is possible to develop an algorithm to generate the absolutely separable
states which stay outside the maximal ball. By considering paradigmatic noise
models, we find the amount of local noise which the input entangled states can
sustain, so that the output states do not become absolutely separable.
Interestingly, we report that with the decrease of entanglement of the pure
input state, critical depolarizing noise value, transferring an entangled state
to an absolutely separable one, increases, thereby showing advantages of
sharing nonmaximally entangled states. Furthermore, when the input two-qubit
states are Haar uniformly generated, we report a hierarchy among quantum
channels according to the generation of absolutely separable states.
