On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank

We address the discontinuities of the quantum Fisher information (QFI) that may arise when the parameter of interest takes values that change the rank of the quantum statistical model. We revisit the classical and the quantum Cramér-Rao theorems, show that they do not hold in these limiting cases, and discuss how this impacts on the relationship between the QFI and the Bures metric. In order to illustrate the metrological implications of our findings, we present two paradigmatic examples, where we discuss in detail the role of the discontinuities. We show that the usual equivalence between the variance of the maximum likelihood estimator and the inverse of the QFI breaks down.