  Quantum states of a few-particle system capacitively coupled to a metal gate
can be discriminated by measuring the quantum capacitance, which can be
identified with the second derivative of the system energy with respect to the
gate voltage. This approach is here generalized to the multi-voltage case,
through the introduction of the quantum capacitance matrix. The matrix
formalism allows us to determine the dependence of the quantum capacitance on
the direction of the voltage oscillations in the parameter space, and to
identify the optimal combination of gate voltages. As a representative example,
this approach is applied to the case of a quantum-dot array, described in terms
of a Hubbard model. Here, we first identify the potentially relevant regions in
the multi-dimensional voltage space with the boundaries between charge
stability regions, determined within a semiclassical approach. Then, we
quantitatively characterize such boundaries by means of the quantum capacitance
matrix. Altogether, this provides a procedure for optimizing the discrimination
between states with different particle numbers and/or total spins.
