  Ensembles of composite quantum states can exhibit nonlocal behaviour in the
sense that their optimal discrimination may require global operations. Such an
ensemble containing N pairwise orthogonal pure states, however, can always be
perfectly distinguished under adaptive local scheme if (N-1) copies of the
state are available. In this letter, we provide examples of orthonormal bases
in two-qubit Hilbert space whose adaptive discrimination require 3 copies of
the state. For this composite system we analyze multi-copy adaptive local
distinguishability of orthogonal ensembles in full generality which in turn
assigns varying nonlocal strength to different such ensembles. We also come up
with ensembles whose discrimination under adaptive separable scheme require
less number of copies than adaptive local schemes. Our construction finds
important application in multipartite secret sharing tasks and indicates
towards an intriguing super-additivity phenomenon for locally accessible
information.
