  The work is intended to represent some interesting and apparently peculiar
features of entangled system in both pure as well as mixed states level. In the
pure state level, we are largely concerned about the existence and
characteristics of bipartite incomparable pair of states. We have found some
schemes for local transformation of one pure bipartite state to another,
incomparable in nature, by exploiting free unpreserved entanglement. Also, we
observed that different incomparable pair of states in $3\times 3$ may
co-operate with each other to cross the barrier of incomparability. Our next
goal was to probe critically the roots of the existence of incomparable states.
We have succeeded in connecting incomparability with some famous no-go theorems
of quantum mechanics. We have established it with no-flipping principle,
general anti-unitary operators and lastly with inner product preserving
operations. Thus we are able to pose incomparability as a good detector of some
non-physical operations. In the next part, we provide a new class of
multipartite bound entanglement in mixed state level. The class of states are
activable bound entangled. There are exactly four orthogonal activable bound
entangled states in any even number of qubit system. They have some interesting
features like Bell correlation, permutation symmetry, local Pauli connection,
non-compatibility, etc. Apart from those properties, the class of bound
entangled states have the remarkable feature of local indistinguishability in
quantum systems.
