Symmetry-Projected Compatible Multiparameter Quantum Sensing

We establish a general symmetry-projection framework for multiparameter quantum sensing. Decomposing encoding generators into subspace-preserving and subspace-changing components relative to a symmetry sector identically eliminates all cross-sector elements of the quantum Fisher information matrix (QFIM) and the mean symmetric logarithmic derivative (SLD) commutator matrix. When projected subspace-changing generators act as a scalar within the occupied subspace, the corresponding QFIM block reduces to four times the symmetrized covariance matrix, regardless of probe state purity. For parity-protected collective $\mathrm{SU}(2)$ spin systems, this renders the transverse QFIM directly certifiable via spin fluctuations, with the optimal axis aligned with the anti-squeezed quadrature. Applied to a dissipative one-axis-twisting system, our framework reveals that highly mixed transient states can exhibit nearly balanced, Heisenberg-scaled QFIM components for transverse--longitudinal parameter pairs $(\theta_y,\theta_z)$ over a broad time window. Furthermore, while the steady state retains an isotropic transverse QFIM scaling as $N^2/3$, weak compatibility for transverse parameter pairs $(\theta_x,\theta_y)$ exhibits a sharp parity dependence---failing for odd $N$ but restored for even $N$. The resulting symmetry protection eliminates the Uhlmann curvature for transverse--longitudinal pairs, enabling simultaneous saturation of the multi-parameter quantum Cram\'{e}r-Rao bound in the asymptotic limit.