Taming singularities of the quantum Fisher information

TL;DR

This paper investigates the challenges posed by singular and discontinuous quantum Fisher information matrices in quantum estimation, providing insights into their physical implications and methods to address these issues in quantum metrology.

Contribution

It offers a detailed analysis of singular and discontinuous QFIMs, highlighting their geometric and physical significance, and discusses strategies to handle nonregular quantum statistical models.

Findings

Singular QFIMs occur when the metric curvature vanishes in certain directions.

Discontinuities in QFIMs relate to parameter-dependent rank changes in density matrices.

Addressing these issues is crucial for accurate quantum parameter estimation.

Abstract

Quantum Fisher information matrices (QFIMs) are fundamental to estimation theory: they encode the ultimate limit for the sensitivity with which a set of parameters can be estimated using a given probe. Since the limit invokes the inverse of a QFIM, an immediate question is what to do with singular QFIMs. Moreover, the QFIM may be discontinuous, forcing one away from the paradigm of regular statistical models. These questions of nonregular quantum statistical models are present in both single- and multiparameter estimation. Geometrically, singular QFIMs occur when the curvature of the metric vanishes in one or more directions in the space of probability distributions, while QFIMs have discontinuities when the density matrix has parameter-dependent rank. We present a nuanced discussion of how to deal with each of these scenarios, stressing the physical implications of singular QFIMs and the ensuing ramifications for quantum metrology.