  Quantum Fisher information matrices (QFIMs) are fundamental to estimation
theory: they encode the ultimate limit for the sensitivity with which a set of
parameters can be estimated using a given probe. Since the limit invokes the
inverse of a QFIM, an immediate question is what to do with singular QFIMs.
Moreover, the QFIM may be discontinuous, forcing one away from the paradigm of
regular statistical models. These questions of nonregular quantum statistical
models are present in both single- and multiparameter estimation.
Geometrically, singular QFIMs occur when the curvature of the metric vanishes
in one or more directions in the space of probability distributions, while
QFIMs have discontinuities when the density matrix has parameter-dependent
rank. We present a nuanced discussion of how to deal with each of these
scenarios, stressing the physical implications of singular QFIMs and the
ensuing ramifications for quantum metrology.
