Quantum Cram\'er-Rao bound for quantum statistical models with parameter-dependent rank

TL;DR

This paper investigates the quantum Cramér-Rao bound's validity at points where the quantum state rank changes, showing that the bound still holds despite discontinuities in quantum Fisher information.

Contribution

It clarifies the conditions under which the quantum Cramér-Rao bound remains valid for models with parameter-dependent rank, addressing issues caused by unbounded operators.

Findings

The quantum Cramér-Rao bound holds in the limit even at singular points.

Discontinuities in quantum Fisher information are linked to unbounded symmetric logarithmic derivatives.

The analysis includes a typical example of models with parameter-dependent rank.

Abstract

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 Cram\'er-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 Cram\'er-Rao bound is originally proved. We argue that the limiting version of quantum Cram\'er-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.