Root-system structure of sloppiness in passive Gaussian metrology

Abstract: 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 squeezing magnitudes $r_{1},\ldots,r_{n}$ in the canonical basis, the roots $2r_{j}$ and $r_{j}\pm r_{k}$ label the local phase rotations and two beam-splitter quadratures, and their vanishing identifies every Fisher-null direction. The associated Fisher weights quantify the approach to each singular wall. At fixed positive mean photon number, we solve the E-optimal probe-design problem of maximizing the smallest passive Fisher eigenvalue in two explicit generator normalizations. In the canonical phase and beam-splitter angle convention, the unique optimum in the fundamental Weyl chamber is an arithmetic progression approaching the consecutive-odd-integer dual-Weyl direction at large resource. With an invariant generator norm, the smallest eigenvalue is independent of the passive frame and the optimum lies exactly on that ray at every resource. Uniform squeezing instead minimizes the local-phase A-optimal cost while leaving one beam-splitter quadrature unidentifiable for every mode pair. Finally, the two quadratures of each identifiable mode pair saturate the quantum-geometric incompatibility bound, and the largest compatible passive submodel has $n(n+1)/2$ parameters at a regular spectrum. The resulting classification separates local identifiability, sensitivity optimization, and simultaneous attainability.