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

## TL;DR

This paper investigates how the quantum Fisher information can become discontinuous when the rank of quantum models changes with parameters, affecting fundamental estimation bounds and the relationship with the Bures metric.

## Contribution

It reveals the limitations of classical and quantum Cramér-Rao theorems in cases of rank change and discusses the implications for quantum metrology.

## Key findings

- Quantum Fisher information can be discontinuous at rank-changing points.
- Classical and quantum Cramér-Rao bounds may be violated in these cases.
- Discontinuities impact the relationship between QFI and the Bures metric.

## Abstract

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 Cram\'er-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 and show that the Cram\'er-Rao may be easily violated.