  We show that localisation microscopy of multiple weak, incoherent point
sources with possibly different intensities in one spatial dimension is
equivalent to estimating the amplitudes of a classical mixture of coherent
states of a simple harmonic oscillator. This enables us to bound the
multi-parameter covariance matrix for an unbiased estimator for the locations
in terms of the quantum Fisher information matrix, which we obtained
analytically. In the regime of arbitrarily small separations we find it to be
no more than rank two -- implying that no more than two independent parameters
can be estimated irrespective of the number of point sources. We use the
eigenvalues of the classical and quantum Fisher information matrices to compare
the performance of spatial-mode demultiplexing and direct imaging in
localisation microscopy with respect to the quantum limits.
