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Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems

by 1,*,† and 2,†
1
Institute of Industrial Science, The University of Tokyo, Chiba 277-8574, Japan
2
Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan
*
Author to whom correspondence should be addressed.
J.L. conceptualized the research, formulated the problem, conducted the analyses, obtained all the results, and created the manuscript. Historical remarks have been enriched and the presentation of the main result has been refined by comments of I.T. in communications and discussions.
Entropy 2020, 22(11), 1222; https://doi.org/10.3390/e22111222
Received: 4 September 2020 / Revised: 4 October 2020 / Accepted: 19 October 2020 / Published: 27 October 2020
(This article belongs to the Special Issue Quantum Probability, Statistics and Control)
A novel uncertainty relation for errors of general quantum measurement is presented. The new relation, which is presented in geometric terms for maps representing measurement, is completely operational and can be related directly to tangible measurement outcomes. The relation violates the naïve bound /2 for the position-momentum measurement, whilst nevertheless respecting Heisenberg’s philosophy of the uncertainty principle. The standard Kennard–Robertson uncertainty relation for state preparations expressed by standard deviations arises as a corollary to its special non-informative case. For the measurement on two-state quantum systems, the relation is found to offer virtually the tightest bound possible; the equality of the relation holds for the measurement performed over every pure state. The Ozawa relation for errors of quantum measurements will also be examined in this regard. In this paper, the Kolmogorovian measure-theoretic formalism of probability—which allows for the representation of quantum measurements by positive-operator valued measures (POVMs)—is given special attention, in regard to which some of the measure-theory specific facts are remarked along the exposition as appropriate. View Full-Text
Keywords: quantum foundations; quantum measurement; uncertainty relation quantum foundations; quantum measurement; uncertainty relation
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MDPI and ACS Style

Lee, J.; Tsutsui, I. Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems. Entropy 2020, 22, 1222. https://doi.org/10.3390/e22111222

AMA Style

Lee J, Tsutsui I. Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems. Entropy. 2020; 22(11):1222. https://doi.org/10.3390/e22111222

Chicago/Turabian Style

Lee, Jaeha, and Izumi Tsutsui. 2020. "Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems" Entropy 22, no. 11: 1222. https://doi.org/10.3390/e22111222

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