Simple expression for the quantum Fisher information matrix

TL;DR

This paper presents a simple, efficient formula for the quantum Fisher information matrix that avoids diagonalization, facilitating easier computation and deeper understanding in quantum metrology and information geometry.

Contribution

The authors derive a new, straightforward formula for QFIM that is computationally more efficient and broadly applicable to finite-dimensional density matrices.

Findings

The formula does not require diagonalization of the density matrix.

It is at least as efficient as previous methods.

Applicable to any finite-dimensional density matrix.

Abstract

Quantum Fisher information matrix (QFIM) is a cornerstone of modern quantum metrology and quantum information geometry. Apart from optimal estimation, it finds applications in description of quantum speed limits, quantum criticality, quantum phase transitions, coherence, entanglement, and irreversibility. We derive a surprisingly simple formula for this quantity, which, unlike previously known general expression, does not require diagonalization of the density matrix, and is provably at least as efficient. With a minor modification, this formula can be used to compute QFIM for any finite-dimensional density matrix. Because of its simplicity, it could also shed more light on the quantum information geometry in general.