Simple expression for the quantum Fisher information matrix

Understand

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.