Quantum root-mean-square error and measurement uncertainty relations

Paul Busch, Pekka Lahti, Reinhard F Werner

Research output: Contribution to journalArticlepeer-review


Recent years have witnessed a controversy over Heisenberg’s famous error-disturbance relation. Here the conflict is resolved by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. Two approaches to adapting the classic notion of root-mean-square error to quantum measurements are discussed. One is based on the concept of a noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.
Original languageEnglish
Pages (from-to)1261-1281
Number of pages21
JournalReviews of Modern Physics
Issue number4
Publication statusPublished - 18 Dec 2014


  • Quantum measurement
  • Uncertainty principle
  • Measurement error
  • Heisenberg effect
  • Disturbance

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