Root-system structure of sloppiness in passive Gaussian metrology

Passive transformations of an $n$-mode squeezed probe become locally unidentifiable when the quantum Fisher matrix loses rank. For pure zero-mean Gaussian probes, we show that the $C_{n}$ restricted-root system of $\mathrm{Sp}(2n,\mathbb{R})/\mathrm{U}(n)$ governs this exact sloppiness. For squeezin