Discontinuities of the quantum Fisher information and the Bures metric

TL;DR

This paper clarifies the relationship between the quantum Fisher information matrix and the Bures metric, showing they differ at points where the density matrix rank changes, but the Bures metric provides a continuous alternative.

Contribution

It demonstrates the discontinuities of the quantum Fisher information at rank-changing points and derives an explicit formula for the Bures metric applicable to all density matrices.

Findings

Quantum Fisher information is discontinuous at rank-changing points.

The Bures metric provides a continuous alternative to quantum Fisher information.

Explicit formulas for the Bures metric are derived for all density matrices.

Abstract

We show that two quantities in quantum metrology that were thought to be the same, the quantum Fisher information matrix and the Bures metric, are not the same. They differ at points at which the rank of the density matrix changes. The quantum Fisher information matrix is discontinuous at these points. However, these discontinuities are removable in some sense. We show that the expression given by the Bures metric represents the continuous version of the quantum Fisher information matrix. We also derive an explicit formula for the Bures metric for both singular and non-singular density matrices.