Quantum Hair on D-branes and Black Hole Information in String Theory

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'' .