Von Neumann’s Theory, Projective Measurement, and Quantum Computation


We discuss the fact that there is a crucial contradiction within Von Neumann’s theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1 ≤ N < +∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of Von Neumann’s projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. We have to give up either the existence of the directions or the formalism of Von Neumann’s projective measurement. Therefore, Von Neumann’s theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on Von Neumann’s theory. We investigate whether Von Neumann’s theory meets the Deutsch-Jozsa algorithm. We discuss the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we discuss the fact that projective measurement theory does not meet an easy detector model for a single Pauli observable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. We propose a solution of the problem. Our solution is equivalent to changing Planck’s constant to a new constant . It may be said that a new type of the quantum theory early approaches Newton’s theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch’s algorithm.

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Nagata, K. and Nakamura, T. (2015) Von Neumann’s Theory, Projective Measurement, and Quantum Computation. Journal of Applied Mathematics and Physics, 3, 874-897. doi: 10.4236/jamp.2015.37108.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] von Neumann, J. (1955) Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton.
[2] Nagata, K., Ren, C.-L. and Nakamura, T. (2011) Whether Quantum Computation Can Be Almighty. Advanced Studies in Theoretical Physics, 5, 1-14.
[3] Gershenfeld, N. and Chuang, I.L. (1998) Quantum Computing with Molecules. Scientific American.
[4] Deutsch, D. (1992) Quantum Computation. Physics World, 1/6/92.
[5] Quantum Computer. Wikipedia, the Free Encyclopedia.
[6] Feynman, R.P., Leighton, R.B. and Sands, M. (1965) Lectures on Physics, Volume III, Quantum Mechanics. Addison-Wesley Publishing Company, Reading.
[7] Redhead, M. (1989) Incompleteness, Nonlocality, and Realism. 2nd Edition, Clarendon Press, Oxford.
[8] Peres, A. (1993) Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht.
[9] Sakurai, J.J. (1995) Modern Quantum Mechanics. Revised Edition, Addison-Wesley Publishing Company, Reading.
[10] Nielsen, M.A. and Chuang, I.L. (2000) Quantum Computation and Quantum Information. Cambridge University Press, Cambridge.
[11] Leggett, A.J. (2003) Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem. Foundations of Physics, 33, 1469-1493. http://dx.doi.org/10.1023/A:1026096313729
[12] Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, C., Zukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) An Experimental Test of Non-Local Realism. Nature, 446, 871-875.
[13] Paterek, T., Fedrizzi, A., Gröblacher, S., Jennewein, T., Zukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) Experimental Test of Nonlocal Realistic Theories without the Rotational Symmetry Assumption. Physical Review Letters, 99, Article ID: 210406. http://dx.doi.org/10.1103/PhysRevLett.99.210406
[14] Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A. and Scarani, V. (2007) Experimental Falsification of Leggett’s Nonlocal Variable Model. Physical Review Letters, 99, Article ID: 210407.
[15] Deutsch, D. (1985) Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 400, 97-117.
[16] Jones, J.A. and Mosca, M. (1998) Implementation of a Quantum Algorithm on a Nuclear Magnetic Resonance Quantum Computer. The Journal of Chemical Physics, 109, 1648.
[17] Gulde, S., Riebe, M., Lancaster, G.P.T., Becher, C., Eschner, J., Häffner, H., Schmidt-Kaler, F., Chuang, I.L. and Blatt, R. (2003) Implementation of the Deutsch-Jozsa Algorithm on an Ion-Trap Quantum Computer. Nature, 421, 48-50. http://dx.doi.org/10.1038/nature01336
[18] de Oliveira, A.N., Walborn, S.P. and Monken, C.H. (2005) Implementing the Deutsch Algorithm with Polarization and Transverse Spatial Modes. Journal of Optics B: Quantum and Semiclassical Optics, 7, 288-292. http://dx.doi.org/10.1088/1464-4266/7/9/009
[19] Kim, Y.-H. (2003) Single-Photon Two-Qubit Entangled States: Preparation and Measurement. Physical Review A, 67, Article ID: 040301(R).
[20] Mohseni, M., Lundeen, J.S., Resch, K.J. and Steinberg, A.M. (2003) Experimental Application of Decoherence-Free Subspaces in an Optical Quantum-Computing Algorithm. Physical Review Letters, 91, Article ID: 187903. http://dx.doi.org/10.1103/PhysRevLett.91.187903
[21] Tame, M.S., Prevedel, R., Paternostro, M., Böhi, P., Kim, M.S. and Zeilinger, A. (2007) Experimental Realization of Deutsch’s Algorithm in a One-Way Quantum Computer. Physical Review Letters, 98, Article ID: 140501. http://dx.doi.org/10.1103/PhysRevLett.98.140501
[22] Deutsch, D. and Jozsa, R. (1992) Rapid Solution of Problems by Quantum Computation. Proceedings of the Royal Society A, 439, 553-558. http://dx.doi.org/10.1098/rspa.1992.0167
[23] Nagata, K. and Nakamura, T. (2010) Can von Neumann’s Theory Meet the Deutsch-Jozsa Algorithm? International Journal of Theoretical Physics, 49, 162-170. http://dx.doi.org/10.1007/s10773-009-0189-5
[24] Nagata, K. (2009) There Is No Axiomatic System for the Quantum Theory. International Journal of Theoretical Physics, 48, 3532-3536. http://dx.doi.org/10.1007/s10773-009-0158-z
[25] Nagata, K. and Nakamura, T. (2013) Von Neumann’s Theory Does Not Meet Deutsch’s Algorithm. Precision Instrument and Mechanology, 2, 104.
[26] Nagata, K. and Nakamura, T. (2013) An Additional Condition for Bell Experiments for Accepting Local Realistic Theories. Quantum Information Processing, 12, 3785-3789.
[27] Nagata, K. (2014) Reply to “Comments on ‘There Is No Axiomatic System for the Quantum Theory’”. Journal of Quantum Information Science, 4, 195-200. http://dx.doi.org/10.4236/jqis.2014.44018
[28] De Broglie-Bohm Theory—Wikipedia, the Free Encyclopedia.
[29] Schon, C. and Beige, A. (2001) Analysis of a Two-Atom Double-Slit Experiment Based on Environment-Induced Measurements. Physical Review A, 64, Article ID: 023806.

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