Simple expression for the quantum Fisher information matrix

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