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 rank of parametric density operator changes. The Cram\'er-Rao bound can violated on such singular if one uses this information. We point out that discontinuity is accompanied with unboundedness symmetric logarithmic derivation operators, based which formally defined and originally proved. argue limiting version still holds when changes its by closing potential loophole involving an unbounded SLD in proof bound. Moreover, we analyze typical example statistical models parameter-dependent rank.