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The quantum Cram\'er-Rao bound is a cornerstone of modern quantum metrology,as it provides the ultimate precision in parameter estimation. In themultiparameter scenario, this bound becomes a matrix inequality, which can becast to a scalar form with a properly chosen weight matrix. Multiparameterestimation thus elicits tradeoffs in the precision with which each parametercan be estimated. We show that, if the information is encoded in a unitarytransformation, we can naturally choose the weight matrix as the metric tensorlinked to the geometry of the underlying algebra $\mathfrak{su}(n)$, withapplications in numerous fields. This ensures an intrinsic bound that isindependent of the choice of parametrization.

Intrinsic Sensitivity Limits for Multiparameter Quantum Metrology