Fisher Information and BKM Metric

This section is under development.

The Bogoliubov-Kubo-Mori Metric

The quantum Fisher information is defined using the Bogoliubov-Kubo-Mori (BKM) inner product:

\[G_{ab}(\theta) = \int_0^1 \text{tr}\left[\rho^s F_a \rho^{1-s} F_b\right] ds\]

Properties:

  • Symmetric: \(G_{ab} = G_{ba}\)

  • Positive semidefinite

  • Reduces to classical Fisher information for commuting operators

Quantum Covariance

The BKM metric can be expressed as a quantum covariance:

\[G_{ab} = \text{cov}_{BKM}(F_a, F_b)\]

This generalizes the classical covariance to non-commuting operators.

See Also