  We present a theoretical analysis of the capacitance of a double quantum dot
in the charge and spin qubit configurations probed at high-frequencies. We find
that in general the total capacitance of the system consists of two
state-dependent terms: The quantum capacitance arising from adiabatic charge
motion and the tunnelling capacitance that appears when repopulation occurs at
a rate comparable or faster than the probing frequency. The analysis of the
capacitance lineshape as a function of externally controllable variables offers
a way to characterize the qubits' charge and spin state as well as relevant
system parameters such as charge and spin relaxation times, tunnel coupling,
electron temperature and electron g-factor. Overall, our analysis provides a
formalism to understand dispersive qubit-resonator interactions which can be
applied to high-sensitivity and non-invasive quantum-state readout.
