  We revisit the problem of determining conditions under which a pure state,
that evolves under an arbitrary unitary transformation, reaches an orthogonal
state in a finite amount of the transformation parameter. Simple geometric
considerations disclose the existence of a fundamental limit for the minimal
amount required, providing, in particular, an intuitive hint of the
Mandelstam-Tamm bound. The geometric considerations leads us to focus on a
particular, yet relevant, family of states that evolve towards orthogonality.
Several dynamical features are discussed, which include the (relative) entropy
production during transformation, and special attention is paid to multipartite
systems of $N$ bosons that are allowed to tunnel between two sites. The effects
of the tunneling in the amount of transformation required for the system to
attain an orthogonal state are revealed, and the relation between the latter,
the tunneling intensity and the mode-entanglement is explored.
