Abstract
In this chapter, we provide a brief overview of various uncertainty relations. First, we review historical uncertainty relations: Heisenberg’s gamma-ray microscope and von-Neumann’s Doppler speed meter. These uncertainty relations epitomize trade-off relation between error and disturbance in quantum measurement process. Next, we review a different type of uncertainty relations: Kennard-Robertson’s inequality and Schrödinger’s inequality. These characterize trade-off relations of inherent fluctuations of observables. Finally, we review Arthurs-Goodman’s inequality and Ozawa’s inequality that based on modern quantum measurement theory.
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W. Heisenberg, Zeitschrift fr Physik A Hadrons Nuclei 43, 172 (1927)
J.A. Wheeler, W.H. Zurek, Quantum Theory and Measurement (Princeton Univ Press, New Jersey, 1983)
E.B. Davies, J.T. Lewis, Commun. Math. Phys. 17, 239 (1970)
M. Ozawa, J. Math. Phys. 25, 79 (1984)
J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ Pr, 1955)
V. Braginsky, F. Khalili, K. Thorne, Quantum Measurement (Cambridge Univ Pr, Cambridge, 1992)
E.H. Kennard, Zeitschrift fr Physik A Hadrons Nuclei 44, 326 (1927)
H.P. Robertson, Phys. Rev. 34, 163 (1929)
E. Schrödinger, Proc. Prussian Acad. Sci. Phys. Math. Sect. XIX, 293 (1930)
E. Arthurs, J.L. Kelly, Bell Syst. Tech. J 44, 725 (1965)
E. Arthurs, M.S. Goodman, Phys. Rev. Lett. 60, 2447 (1988)
M. Ozawa, Phys. Rev. A 67, 042105 (2003)
M. Ozawa, Phys. Lett. A 320, 367 (2004)
M. Ozawa, Ann. Phys. 311, 350 (2004)
C.W. Helstrom, J. Stat. Phys. 1, 231 (1969)
A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982)
M. Hayashi, Quantum Information: An Introduction (Springer Verlag, Berlin, 2006)
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Watanabe, Y. (2014). Reviews of Uncertainty Relations. In: Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54493-7_2
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DOI: https://doi.org/10.1007/978-4-431-54493-7_2
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