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Given a quantum state $\rho$ and a measurement operator $m$ one can define the classical and quantum Fisher information indices (CFI/QFI), the former depending on both $\rho$ and $m$, while the latter being an intrinsic property of the quantum state.

Shortly after their introduction, it was observed how the CFI is bounded by the QFI, allowing one to ask what optimal measurements can attain the bound.
However, the problem of actually computing (and defining) the QFI is an obstruction that kept researchers from addressing such an optimisation, except for simple cases.

Rephrasing (finite dimensional) quantum mechanics in the geometric framework of co-adjoint orbits of the unitary group has lead to the solution of the computation problem of the quantum Fisher information, reinterpreted as a natural object on such symplectic manifolds.

In this talk I will introduce the Fisher information optimisation problem, highlighting the parts where geometry has shown to be crucial, and I will describe the main construction of the Fisher information tensor and related quantities on the spaces of quantum states.