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Quantum Cramér-Rao bound for quantum statistical models with parameter-dependent rank

Recently, a widely-used computation expression for quantum Fisher information was shown to be discontinuous at the parameter points where the rank of the parametric density operator changes. The quantum Cramér-Rao bound can be violated on such singular parameter points if one uses this computation expression for quantum Fisher information. We point out that the discontinuity of the computation expression of quantum Fisher information is accompanied with the unboundedness of the symmetric logarithmic derivation operators, based on which the quantum Fisher information is formally defined and the quantum Cramér-Rao bound is originally proved. We argue that the limiting version of quantum Cramér-Rao bound still holds when the parametric density operator changes its rank by closing the potential loophole of involving an unbounded SLD operator in the proof of the bound. Moreover, we analyze a typical example of the quantum statistical models with parameter-dependent rank.