  The quantum Fisher information matrix is a central object in multiparameter
quantum estimation theory. It is usually challenging to obtain analytical
expressions for it because most calculation methods rely on the diagonalization
of the density matrix. In this paper, we derive general expressions for the
quantum Fisher information matrix which bypass matrix diagonalization and do
not require the expansion of operators on an orthonormal set of states.
Additionally, we can tackle density matrices of arbitrary rank. The methods
presented here simplify analytical calculations considerably when, for example,
the density matrix is more naturally expressed in terms of non-orthogonal
states, such as coherent states. Our derivation relies on two matrix inverses
which, in principle, can be evaluated analytically even when the density matrix
is not diagonalizable in closed form. We demonstrate the power of our approach
by deriving novel results in the timely field of discrete quantum imaging: the
estimation of positions and intensities of incoherent point sources. We find
analytical expressions for the full estimation problem of two point sources
with different intensities, and for specific examples with three point sources.
We expect that our method will become standard in quantum metrology.
