  We develop a theory of adiabatic response for open systems governed by
Lindblad evolutions. The theory determines the dependence of the response
coefficients on the dephasing rates and allows for residual dissipation even
when the ground state is protected by a spectral gap. We give quantum response
a geometric interpretation in terms of Hilbert space projections: For a two
level system and, more generally, for systems with suitable functional form of
the dephasing, the dissipative and non-dissipative parts of the response are
linked to a metric and to a symplectic form. The metric is the Fubini-Study
metric and the symplectic form is the adiabatic curvature. When the metric and
symplectic structures are compatible the non-dissipative part of the inverse
matrix of response coefficients turns out to be immune to dephasing. We give
three examples of physical systems whose quantum states induce compatible
metric and symplectic structures on control space: The qubit, coherent states
and a model of the integer quantum Hall effect.
