Non-commuting Observables are Jointly Measureable under Disturbance Correction Strategy

In the study of Heisenberg's error-disturbance relation, it is commonly believed that the non-unitary change of states hinders us from deducing the information encoded in original states about subsequently measured observable. However, we find that the disturbance can be corrected iff the pre-measurement is non-projective. In this work, by analysing the effect of decoherence on statistics of the subsequential measurement, we find the acquired information from pre-measurement can be used to developed a correction strategy, and then the information about post-measured observable can be recovered. In viewpoint of estimation theory, this result is the unbiasedness condition, which enable us to define precisions of measurement directly in terms of Fisher information. Moreover, we study the precisions trade-off relations in the information theoretic viewpoint.