Current Science

Open Access Open Access  Restricted Access Subscription Access

Heisenberg's Original Derivation of the Uncertainty Principle and its Universally Valid Reformulations


Affiliations
1 Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan
 

Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement, but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the present author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.

Keywords

Error-Disturbance Relations, Quantum Root Mean Square, Quantum Measurement, Repeatability Hypothesis, Uncertainty Principle.
User
Notifications
Font Size

  • Heisenberg, W., The physical content of quantum kinematics and mechanics. In Quantum Theory and Measurement (eds Wheeler, J. A. and Zurek, W. H.), Princeton University Press, Princeton, NJ, USA, 1983, pp. 62–84 [originally published: Z. Phys., 1927, 43(3–4), 172–198].

  • Kennard, E. H., Zur quantenmechanik einfacher bewegungstypen. Z. Phys., 1927, 44(4–5), 326–352.

  • von Neumann, J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955 [Originally published: Mathematische Grundlagen der Quantenmechanik, Springer, 1932].

  • Bohm, D., Quantum Theory, Prentice-Hall, New York, 1951.

  • Messiah, A., M´ecanique Quantique, Dunod, Paris, 1959, vol. I [Quantum Mechanics, North-Holland, Amsterdam, 1959, vol. I].

  • Schiff, L. I., Quantum Mechanics, McGraw-Hill, New York, 1968.

  • Ballentine, L. E., The statistical interpretation of quantum mechanics. Rev. Mod. Phys., 1970, 42(4), 358–381.

  • Schrodinger, E., Die gegenwärtige situation in der quantenmechanik. Naturwissenshaften, 1935, 23(48, 49, 50), 807–812, 823–828, 844–849 [English translation by Trimmer, J. D., Proc. Am. Philos. Soc., 1980, 124(5), 323–338].

  • Braginsky, V. B. and Khalili, F. Y., Quantum Measurement, Cambridge University Press, Cambridge, 1992.

  • Giovannetti, V., Lloyd, S. and Maccone, L., Quantum-enhanced measurements: Beating the standard quantum limit. Science, 2004, 306(5700), 1330–1336.

  • Ozawa, M., Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A, 2003, 67(4), 042105-1 to 042105-6.

  • Ozawa, M., Uncertainty principle for quantum instruments and computing. Int. J. Quant. Inf., 2003, 1(4), 569–588.

  • Ozawa, M., Uncertainty relations for noise and disturbance in generalized quantum measurements. Ann. Phys. (N.Y.), 2004, 311(2), 350–416.

  • Halmos, P. R., Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1951.

  • Imoto, N., Ueda, M. and Ogawa, T., Microscopic theory of the continuous measurement of photon number. Phys. Rev. A, 1990, 41(7), 4127–4130.

  • Ozawa, M., Quantum measuring processes of continuous observables. J. Math. Phys., 1984, 25(1), 79–87.

  • Ozawa, M., Conditional probability and a posteriori states in quantum mechanics. Publ. Res. Inst. Math. Sci., Kyoto Univ., 1985, 21(2), 279–295.

  • Srinivas, M. D., Collapse postulate for observables with continuous spectra. Commun. Math. Phys., 1980, 71(2), 131–158.

  • Ozawa, M., Measuring processes and repeatability hypothesis. In Probability Theory and Mathematical Statistics, Lecture Notes in Math. 1299 (eds Watanabe, S. and Prohorov, Y. V.), Springer, Berlin, 1988, pp. 412–421.

  • Davies, E. B. and Lewis, J. T., An operational approach to quantum probability. Commun. Math. Phys., 1970, 17(3), 239–260.

  • Heistrom, C. W., Quantum Detection and Estimation Theory, Academic Press, New York, 1976.

  • Ozawa, M., Conditional expectation and repeated measurements of continuous quantum observables. In Probability Theory and Mathematical Statistics, Lecture Notes in Math. 1021 (eds Itô, K. and Prohorov, J. V.), Springer, Berlin, 1983, pp. 518–525.

  • Reed, M. and Simon, B., Methods of Modern Mathematical Physics, I: Functional Analysis (Revised and Enlarged Edition), Academic Press, 1980.

  • Gauss, C. F., Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement, SIAM, Philadelphia, USA, 1995 [originally published: Theoria Combinationis Observationum Erroribus Miinimis Obnoxiae, Pars Prior, Pars Posterior, Supplementum, Societati Regiae Scientiarum Exhibita, 15 February 1821].

  • Braginsky, V. B., Vorontsov, Y. I. and Thorne, K. S., Quantum nondemolition measurements. Science, 1980, 209(4456), 547–557.

  • Caves, C. M., Thorne, K. S., Drever, R. W. P., Sandberg, V. D. and Zimmermann, M., On the measurement of a weak classical force coupled to a quantum mechanical oscillator, I. Issues of principle. Rev. Mod. Phys., 1980, 52(2), 341–392.

  • Caves, C. M., Defense of the standard quantum limit for free-mass position. Phys. Rev. Lett., 1985, 54(23), 2465–2468.

  • Yuen, H. P., Contractive states and the standard quantum limit for monitoring free-mass positions. Phys. Rev. Lett., 1983, 51(9), 719–722.

  • Ozawa, M., Measurement breaking the standard quantum limit for free-mass position. Phys. Rev. Lett., 1988, 60(5), 385–388.

  • Ozawa, M., Realization of measurement and the standard quantum limit. In Squeezed and Nonclassical Light (eds Tombesi, P. and Pike, E. R.), Plenum Press, New York, 1989, pp. 263–286.

  • Ozawa, M., Position measuring interactions and the Heisenberg uncertainty principle. Phys. Lett. A, 2002, 299(1), 1–7.

  • Gudder, S., Joint distributions of observables. Indiana Univ. Math. J., 1969, 18(4), 325–335.

  • Ozawa, M., Perfect correlations between noncommuting observables. Phys. Lett. A, 2005, 335(1), 11–19.

  • Ozawa, M., Quantum perfect correlations. Ann. Phys. (N.Y.), 2006, 321(3), 744–769.

  • Ozawa, M., Physical content of Heisenberg’s uncertainty relation: limitation and reformulation. Phys. Lett. A, 2003, 318(1–2), 21– 29.

  • Arthurs, E. and Goodman, M. S., Quantum correlations: a generalized Heisenberg uncertainty relation. Phys. Rev. Lett., 1988, 60(24), 2447–2449.

  • Raymer, M. G., Uncertainty principle for joint measurement of noncommuting variables. Am. J. Phys., 1994, 62(11), 986–993.

  • Ozawa, M., Quantum limits of measurements and uncertainty principle. In Quantum Aspects of Optical Communications (eds Bendjaballah, C., Hirota, O. and Reynaud, S.), Springer, Berlin, 1991, pp. 3–17.

  • Ishikawa, S., Uncertainty relations in simultaneous measurements for arbitrary observables. Rep. Math. Phys., 1991, 29(3), 257–273.

  • Ozawa, M., Uncertainty relations for joint measurements of noncommuting observables. Phys. Lett. A, 2004, 320(5–6), 367–374.

  • Ozawa, M., Universal uncertainty principle in measurement operator formalism. J. Opt. B: Quantum Semiclass., 2005, 7(12), S672– S681.

  • Ozawa, M., Conservation laws, uncertainty relations, and quantum limits of measurements. Phys. Rev. Lett., 2002, 88(5), 050402-1 to 050402-4.

  • Ozawa, M., Universal uncertainty principle and quantum state control under conservation laws. AIP Conf. Proc., 2004, 734, 95– 98.

  • Busch, P. and Loveridge, L., Position measurements obeying momentum conservation. Phys. Rev. Lett., 2011, 106(11), 1104061 to 110406-4.

  • Wigner, E. P., Die messung quntenmechanischer operatoren. Z. Phys., 1952, 133(1–2), 101–108.

  • Araki, H. and Yanase, M. M., Measurement of quantum mechanical operators. Phys. Rev., 1960, 120(2), 622–626.

  • Yanase, M. M., Optimal measuring apparatus. Phys. Rev., 1961, 123(2), 666–668.

  • Ozawa, M., Does a conservation law limit position measurements? Phys. Rev. Lett., 1991, 67(15), 1956–1959.

  • Busch, P., Heinonen, T. and Lahti, P., Noise and disturbance in quantum measurement. Phys. Lett. A, 2004, 320(4), 261–270.

  • Dressel, J. and Nori, F., Certainty in Heisenberg’s uncertainty principle: revisiting definitions for estimation errors and disturbance. Phys. Rev. A, 2014, 89(2), 022106-1 to 022106-14.

  • Korzekwa, K., Jennings, D. and Rudolph, T., Operational constraints on state-dependent formulations of quantum error-disturbance tradeoff relations. Phys. Rev. A, 2014, 89(5), 052108-1 to 052108-6.

  • Busch, P., Lahti, P. and Werner, R. F., Colloquium: Quantum ischolar_main-mean-square error and measurement uncertainty relations. Rev. Mod. Phys., 2014, 86(4), 1261–1281.

  • Lund, A. P. and Wiseman, H. M., Measuring measurement– disturbance relationships with weak values. New J. Phys., 2010, 12, 093011-1 to 093011-11.

  • Garretson, J. L., Wiseman, H. M., Pope, D. T. and Pegg, D. T., The uncertainty relation in ‘which-way’ experiments: how to observe directly the momentum transfer using weak values. J. Opt. B, 2004, 6(6), S506–S517.

  • Mir, R., Lundeen, J. S., Mitchell, M. W., Steinberg, A. M., Garretson, J. L. and Wiseman, H. M., A double-slit ‘which-way’ experiment on the complementarity-uncertainty debate. New J. Phys., 2007, 9, 287-1 to 287-11.

  • Ozawa, M., Noise and disturbance in quantum measurements and operations. Proc. SPIE, 2006, 6244, 62440Q-1 to 62440Q-9.

  • Ozawa, M., Heisenberg’s uncertainty relation: violation and reformulation. J. Phys. Conf. Ser., 2014, 504, 012024-1 to 012024-12.


Abstract Views: 410

PDF Views: 261




  • Heisenberg's Original Derivation of the Uncertainty Principle and its Universally Valid Reformulations

Abstract Views: 410  |  PDF Views: 261

Authors

Masanao Ozawa
Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan

Abstract


Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement, but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the present author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.

Keywords


Error-Disturbance Relations, Quantum Root Mean Square, Quantum Measurement, Repeatability Hypothesis, Uncertainty Principle.

References





DOI: https://doi.org/10.18520/cs%2Fv109%2Fi11%2F2006-2016