  Given an ensemble of orthogonal quantum states, we consider that the states
cannot be perfectly distinguished by local operations and classical
communication (LOCC). If such states remain indistinguishable under LOCC when
multiple (finite) identical copies of the states are available, then in brief,
we say that the ensemble is many-copy indistinguishable. Here we explore the
notions -- unextendibility, uncompletability and analyze their connections to
many-copy indistinguishable ensembles. We show that the orthogonal complement
of any bipartite pure entangled state is spanned by product states which form a
nonorthogonal unextendible product basis (nUPB) of maximum cardinality. This
subspace, which is spanned by the states of an nUPB of maximum cardinality, has
one to one correspondence with the maximum dimensional subspace where there is
no orthonormal product basis. Explicit construction of the product states,
forming any such nUPB is shown. Due to these, the proof of indistinguishability
of a class of ensembles under LOCC in many-copy scenario becomes simpler.
Furthermore, it is now clear that there are several many-copy indistinguishable
ensembles, construction-wise they are different but if we consider only
indistinguishability property under LOCC, then, many of them are connected
through the general notion of unextendible product basis. Thereafter, we
introduce the notion of positive partial transpose uncompletability to unify
different many-copy indistinguishable ensembles. We also report a class of
multipartite many-copy indistinguishable ensembles for which local
indistinguishability property increases with decreasing mixedness. This is not
usual as mixedness usually introduce difficulty in discrimination. Finally, we
discuss the role of bound entanglement in this context.
