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

Non-commuting Observables are Jointly Measureable under Disturbance Correction Strategy