Finite-Syndrome Compression of Quantum Fisher Information

Readable error records can protect quantum sensing because they prevent physically distinct noise trajectories from being irreversibly mixed. A finite detector or ancilla, however, can retain only finitely many syndrome values, and a general criterion for deciding which records may be merged without losing metrological information is absent. We formulate the problem for a fixed fine-grained classical-quantum record and parameter-independent compression into at most $M$ flags. We prove an exact identity expressing the lost symmetric-logarithmic-derivative quantum Fisher information (SLD QFI) as a sum of state-weighted squared distances between fine and coarse SLD scores. Consequently, optimal finite-syndrome design is exactly an operator-valued clustering problem, and zero loss is characterized by a support-resolved common-SLD condition. We extend the identity to the full multiparameter SLD QFI matrix and distinguish local QFI preservation from recovery of an entire statistical model. For exact recovery of a quantum code, we separately show that, when each fine error is individually correctable, the minimum number of readable syndromes is the chromatic number of a Knill-Laflamme incompatibility graph. For qubit random-unitary noise we obtain a finite partition formula. A planar random-Pauli model admits a conditional-variance representation and, for a uniform error axis, the exact optimum $F_M^{\star}=[M\sin(\pi/M)/\pi]^2$, with deficit $\pi^2/(3M^2)+O(M^{-4})$. These results identify the information-theoretic cost of finite syndrome resolution while making explicit the side-information assumptions required for any passive noise-to-erasure interpretation.