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We address the discontinuities of the quantum Fisher information (QFI) thatmay arise when the parameter of interest takes values that change the rank ofthe quantum statistical model. We revisit the classical and the quantumCram\'er-Rao theorems, show that they do not hold in these limiting cases, anddiscuss how this impacts on the relationship between the QFI and the Buresmetric. In order to illustrate the metrological implications of our findings,we present two paradigmatic examples, where we discuss in detail the role ofthe discontinuities and show that the Cram\'er-Rao may be easily violated.

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