We revisit the definitions of error and disturbance recently used inerror-disturbance inequalities derived by Ozawa and others by expressing themin the reduced system space. The interpretation of the definitions asmean-squared deviations relies on an implicit assumption that is generallyincompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, andwhich results in averaging the deviations over a non-positive-semidefinitejoint quasiprobability distribution. For unbiased measurements, the erroradmits a concrete interpretation as the dispersion in the estimation of themean induced by the measurement ambiguity. We demonstrate how to directlymeasure not only this dispersion but also every observable moment with the sameexperimental data, and thus demonstrate that perfect distributional estimationscan have nonzero error according to this measure. We conclude that theinequalities using these definitions do not capture the spirit of Heisenberg'seponymous inequality, but do indicate a qualitatively different relationshipbetween dispersion and disturbance that is appropriate for ensembles beingprobed by all outcomes of an apparatus. To reconnect with the discussion ofHeisenberg, we suggest alternative definitions of error and disturbance thatare intrinsic to a single apparatus outcome. These definitions naturallyinvolve the retrodictive and interdictive states for that outcome, and producecomplementarity and error-disturbance inequalities that have the same form asthe traditional Heisenberg relation.
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