Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Paul Busch, Pekka Lahti, and Reinhard F. Werner
Rev. Mod. Phys. 86, 1261 – Published 18 December 2014

Abstract

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.

  • Figure
  • Received 17 December 2013

DOI:https://doi.org/10.1103/RevModPhys.86.1261

© 2014 American Physical Society

Authors & Affiliations

Paul Busch*

  • Department of Mathematics, University of York, York, YO10 5DD, United Kingdom

Pekka Lahti

  • Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland

Reinhard F. Werner

  • Institut für Theoretische Physik, Leibniz Universität, D-30167 Hannover, Germany

  • *paul.busch@york.ac.uk
  • pekka.lahti@utu.fi
  • reinhard.werner@itp.uni-hannover.de

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Issue

Vol. 86, Iss. 4 — October - December 2014

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