  We consider localized qubits evolving around a black hole following a quantum
adiabatic dynamics. We develop a geometric structure (based on fibre bundles)
permitting to describe the quantum states of a qubit and the spacetime geometry
in a single framework. The quantum decoherence induced by the black hole on the
qubit is analysed in this framework (the role of the dynamical and geometric
phases in this decoherence is treated), especially for the quantum
teleportation protocol when one qubit falls to the event horizon. A simple
formula to compute the fidelity of the teleportation is derived. The case of a
Schwarzschild black hole is analysed.
