Fig. 3: Normalized quantum Fisher information \({{{{{{{\mathcal{Q}}}}}}}}\).

a The Bloch-sphere illustration of the quantum jump tracking (QJT). For a two-component Fock state \(|{\psi }_{m,n}\rangle\), although it could be confined in the code space \({{{{{{{\rm{span}}}}}}}}\left\{|m\rangle ,|n\rangle \right\}\) via QEC, the amplitudes vary depending on the number (j) of the single-photon-loss errors occurring, i.e., \(|{\psi }_{m,n}\rangle \mapsto |{\psi }_{m,n}^{(j)}\rangle\). The output would be a mixed state \({\rho }_{m,n}^{{{{{{{{\rm{(tot)}}}}}}}}}\), if different evolution trajectories could not be distinguished. b–d Quantitative performance of different quantum sensing strategies characterized by \({{{{{{{\mathcal{Q}}}}}}}}\) for the probe states \(|{\psi }_{1,3}\rangle\), \(|{\psi }_{1,5}\rangle\), and \(|{\psi }_{1,7}\rangle\), respectively. TLS: the two-level system encoding with the two lowest Fock states. QEC and No QEC: results with and without QEC, respectively. QEC+QJT: the strategy that combines QEC and QJT. The error bars are obtained through error propagation of the fit parameter uncertainties. Inset: Wigner functions of the corresponding probe states, with the same axes and color scale bar as in Fig. 1b.