Geometric obstructions to quadratic time scaling in multiparameter quantum estimation

Abstract

Unitary encoding of a single parameter provides quadratic enhancement in precision, with the quantum Fisher information scaling quadratically with the encoding time. However, when estimating multiple parameters simultaneously, this fundamental scaling is not guaranteed. Here, we establish a universal geometric obstruction that dictates when multiparameter quantum metrology fails to achieve simultaneous $t^{-2}$ scaling. By decomposing the Hamiltonian derivatives into components that commute and do not commute with the system Hamiltonian, we prove that linear dependence among the commuting components inevitably generates a slow parameter direction whose Fisher information remains bounded as O$(t^0)$, limiting the overall estimation precision. We demonstrate this mechanism in both discrete- and continuous-variable setups, including collective spin magnetometry and a generalized quantum harmonic oscillator, and contrast it with the Lipkin--Meshkov--Glick model where $t^{-2}$ decay is preserved. Remarkably, while the slow direction fundamentally limits the achievable precision, the measurement incompatibility between fast and slow directions decays as $1/t$, rendering the symmetric logarithmic derivative bound asymptotically saturable. Our framework provides a readily computable diagnostic, given by the Gram matrix of the diagonal generators, for identifying such obstructions in arbitrary multiparameter estimation problems. We further show that the bottleneck can be circumvented by relegating slow directions to nuisance parameters or by employing adaptive quantum control.