Sufficient statistic and recoverability via Quantum Fisher Information metrics

We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output, if and only if the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger-Yanase-Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analog of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying Riemannian monotone metrics on the quantum state space, i.e. metrics monotone decreasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum $\chi^2$ divergence. We obtain an approximate recovery result in the sense that, if the quantum $\chi^2$ divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the $\chi_{\frac{1}{2}}$ divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.