  We introduce a notion of quantum hair which completely characterizes the
state of a D-brane in perturbative string theory. The hair manifests itself as
a phase (more generally a unitary matrix in subspaces of degenerate string
eigenstates) in the scattering amplitudes of elementary strings on the D brane.
As the separation of the D-brane and the string center of mass becomes large,
the phase goes to zero if we keep the string excitation level fixed. However,
by letting the level number increase with the distance, we can keep the phase
constant. We argue that this implies that scattering experiments with highly
excited strings can detect the state of a D-brane long after it has ``fallen
into a black hole'' .
