An Entanglement-Assisted Stabilizer Framework for Distributed Sensing of Local Phases
Distributed quantum sensing requires spatially separated probes to acquire local parameters while maintaining compatibility with network-level quantum information processing. We develop an entanglement-assisted stabilizer framework based on the extended structure of entanglement-assisted quantum error-correcting (EAQEC) codes, in which the remote halves of pre-shared ebits are used directly as local phase probes while the joint state simultaneously carries an encoded logical subsystem. Each remote probe acquires a local Z-axis phase and subsequently returns through an X-type noise channel. Within the extended EAQEC stabilizer structure, the stabilizer containing $X_{B_j}$ provides phase-dependent measurement statistics, whereas its partner containing $Z_{B_j}$ records the corresponding return-error syndrome. A graph-code formulation is introduced to make this structure explicit, together with an illustrative [[5,1,3;2]] construction. We further show that, conditioned on the joint sensing-and-syndrome measurement record, the post-sensing state differs from the original encoded state only by a known Pauli transformation, so that the logical information remains available for subsequent encoded operations. For the local-phase model considered here, the stabilizer readout attains the available quantum Fisher information, while the finite-shot estimator approaches the corresponding $1/\sqrt{M}$ scaling as the number of repetitions increases. The framework therefore provides a common entanglement-assisted stabilizer structure for distributed local-phase sensing, restricted return-error identification, and post-sensing logical-state retention, without relying on an intrinsic metrological enhancement from EAQEC itself.