  Braunstein and Caves (1994) proposed to use Helstrom's {\em quantum
information} number to define, meaningfully, a metric on the set of all
possible states of a given quantum system. They showed that the quantum
information is nothing else than the maximal Fisher information in a
measurement of the quantum system, maximized over all possible measurements.
Combining this fact with classical statistical results, they argued that the
quantum information determines the asymptotically optimal rate at which
neighbouring states on some smooth curve can be distinguished, based on
arbitrary measurements on $n$ identical copies of the given quantum system.
  We show that the measurement which maximizes the Fisher information typically
depends on the true, unknown, state of the quantum system. We close the
resulting loophole in the argument by showing that one can still achieve the
same, optimal, rate of distinguishability, by a two stage adaptive measurement
procedure.
  When we consider states lying not on a smooth curve, but on a manifold of
higher dimension, the situation becomes much more complex. We show that the
notion of ``distinguishability of close-by states'' depends strongly on the
measurement resources one allows oneself, and on a further specification of the
task at hand. The quantum information matrix no longer seems to play a central
role.
