Abstract
The recently established universal uncertainty principle revealed that two nowhere commuting observables can be measured simultaneously in some state, whereas they have no joint probability distribution in any state. Thus, one measuring apparatus can simultaneously measure two observables that have no simultaneous reality. In order to reconcile this discrepancy, an approach based on quantum logic is proposed to establish the relation between quantum reality and measurement. We provide a language speaking of values of observables independent of measurement based on quantum logic and we construct in this language the state-dependent notions of joint determinateness, value identity, and simultaneous measurability. This naturally provides a contextual interpretation, in which we can safely claim such a statement that one measuring apparatus measures one observable in one context and simultaneously it measures another nowhere commuting observable in another incompatible context.
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References
Ballentine, L.E.: The statistical interpretation of quantum mechanics. Rev. Mod. Phys. 42, 358–381 (1970)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–845 (1936)
Bohr, N.: The quantum postulate and the recent development of atomic theory. Nature (London) 121, 580–590 (1928)
Davies, E.B.: Quantum Theory of Open Systems. Academic, London (1976)
Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)
Gudder, S.: Joint distributions of observables. J. Math. Mech. 18, 325–335 (1968)
Heisenberg, W.: The Physical Principles of the Quantum Theory. University of Chicago Press, Chicago (1930). [Reprinted by Dover, New York (1949, 1967)]
Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Ozawa, M.: Quantum measuring processes of continuous observables. J. Math. Phys. 25, 79–87 (1984)
Ozawa, M.: Physical content of Heisenberg’s uncertainty relation: Limitation and reformulation. Phys. Lett. A 318, 21–29 (2003)
Ozawa, M.: Uncertainty principle for quantum instruments and computing. Int. J. Quantum Inf. 1, 569–588 (2003)
Ozawa, M.: Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A 67, 042105 (2003) (6 pages)
Ozawa, M.: Uncertainty relations for noise and disturbance in generalized quantum measurements. Ann. Phys. (N.Y.) 311, 350–416 (2004)
Ozawa, M.: Perfect correlations between noncommuting observables. Phys. Lett. A 335, 11–19 (2005)
Ozawa, M.: Quantum perfect correlations. Ann. Phys. (N.Y.) 321, 744–769 (2006)
Ozawa, M.: Simultaneous measurability of non-commuting observables and the universal uncertainty principle. In: Hirota, O., Shapiro, J., Sasaki, M. (eds.) Proc. 8th Int. Conf. on Quantum Communication, Measurement and Computing, pp. 363–368. NICT Press, Tokyo (2007)
Ozawa, M.: Transfer principle in quantum set theory. J. Symb. Log. 72, 625–648 (2007)
Takeuti, G.: Quantum set theory. In: Beltrametti, E.G., van Fraassen, B.C. (eds.) Current Issues in Quantum Logic, pp. 303–322. Plenum, New York (1981)
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Ozawa, M. Quantum Reality and Measurement: A Quantum Logical Approach. Found Phys 41, 592–607 (2011). https://doi.org/10.1007/s10701-010-9462-y
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DOI: https://doi.org/10.1007/s10701-010-9462-y