Symmetry-Projected Weakly Compatible Multiparameter Quantum Sensing
Achieving joint quantum-enhanced precision in multiparameter sensing requires both high sensitivity and measurement compatibility. These two aspects are characterized by the quantum Fisher information matrix (QFIM) and the Uhlmann curvature matrix (UCM), respectively, with weak compatibility corresponding to the vanishing of the relevant UCM elements. Here, we develop a symmetry-projection framework that classifies phase generators into subspace-preserving and subspace-changing sectors. For probe states confined to a symmetry subspace, symmetry projection imposes a common block-diagonal structure on the QFIM and UCM, rendering cross-sector parameters simultaneously free from information cross-talk and measurement incompatibility. When the subspace-changing generators act as scalars within the occupied subspace, the corresponding QFIM block reduces to four times the symmetrized covariance matrix, even for mixed probe states. For parity-protected collective $\mathrm{SU}(2)$ systems, this structure singles out the transverse anti-squeezed quadrature and the longitudinal mean-spin direction as natural optimal sensing axes. Applied to a dissipative one-axis twisting model, the dynamically generated probe state exhibits identically vanishing UCM elements for transverse--longitudinal parameter pairs, while maintaining nearly balanced, Heisenberg-scaled QFIM components over a broad transient window. Our work opens a route to symmetry-protected, weakly compatible multiparameter sensing in interacting quantum many-body systems.