Commenced in January 2007
Paper Count: 31242
Impact of the Existence of One-Way Functionson the Conceptual Difficulties of Quantum Measurements
Authors: Arkady Bolotin
Abstract:One-way functions are functions that are easy to compute but hard to invert. Their existence is an open conjecture; it would imply the existence of intractable problems (i.e. NP-problems which are not in the P complexity class). If true, the existence of one-way functions would have an impact on the theoretical framework of physics, in particularly, quantum mechanics. Such aspect of one-way functions has never been shown before. In the present work, we put forward the following. We can calculate the microscopic state (say, the particle spin in the z direction) of a macroscopic system (a measuring apparatus registering the particle z-spin) by the system macroscopic state (the apparatus output); let us call this association the function F. The question is: can we compute the function F in the inverse direction? In other words, can we compute the macroscopic state of the system through its microscopic state (the preimage F -1)? In the paper, we assume that the function F is a one-way function. The assumption implies that at the macroscopic level the Schrödinger equation becomes unfeasible to compute. This unfeasibility plays a role of limit of the validity of the linear Schrödinger equation.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1058205Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1033
 Goldreich, O. Modern Cryptography, Probabilistic Proofs, and Pseudorandomness. Springer, 1999.
 Sipser M. Introduction to the Theory of Computation. PWS Publishing, Section 10.6.3: One-way functions, 1997, pp. 374-376.
 Papadimitriou C. Computational Complexity. 1st edition, Addison Wesley, Section 12.1: One-way functions, 1993, pp.279-298.
 Greenlaw, R., Hoover, H. J., and Ruzzo, W. L. Limits to Parallel Computation: P-Completeness Theory. Oxford, England: Oxford University Press, 1995.
 Cook, S. The P versus NP Problem http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Descr iption.pdf.
 Schrödinger, E. Die gegenwartige Situation in der Quantenmechanik, Naturwissenschaftern. 23: 1935, pp. 807-812; 823-823, 844-849. English translation: John D. Trimmer, Proceedings of the American Philosophical Society, 124, 1980, pp. 323-38.
 Lalo├½ F. Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems. Am. J. Phys., Vol. 69, No. 6, 2001, pp. 655-701.
 Griffiths D. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall, 2004.
 Schrödinger E. Proc. Cambridge Philos. Soc. 31, 1935, 555; 32, 1936, 446.
 Bassi, A., and Ghirardi, G.C. A general argument against the universal validity of the superposition principle. Phys. papers A, 275, 2000, 373- 381.
 Lui, Y.-K., Christiandl, M., Verstraete, F. Quantum Computational Complexity of the N-Representability Problem: QMA Complete. Phys. Rev. Letters 98, 2007, 110503(4).
 Wheeler J. and Zurek W. (eds). Quantum Theory and Measurement. Princeton University Press, 1983.
 Krips H. Measurement in Quantum Mechanics. In Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/qt measurement/. First published Tue Oct 12, 1999; substantive revision Wed Aug 22, 2007.
 Wigner E. P. On hidden variables and quantum mechanical probabilities. Am. J. Phys. 38, 1970, 1005-1009.
 Wiener N. and Siegel A. A new form for the statistical postulate of quantum mechanics. Phys. Rev. 91, 1953, 1551-1560.
 Siegel A. and Wiener N. Theory of measurement in differential space quantum theory. Phys. Rev. 101, 1956, 429-432.
 Bohm D. and Bub J. A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory. Rev. Mod. Phys. 38, 1966, 453-469.
 Pearle P. Reduction of the state vector by a non-linear Schrödinger equation. Phys. Rev. D 13, 1976, 857-868.
 Ghirardi G. C., Rimini A., and Weber T. Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 1986, 470-491.
 Diosi L. Quantum stochastic processes as models for state vector reduction. J. Phys. A 21, 1988, 2885-2898.
 Griffiths R. B. Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36, 1984, 219-272.
 Gell-Mann M. and Hartle J. Classical equations for quantum systems. Phys. Rev. D 47, 1993, 3345-3382.
 Omnés R. Understanding Quantum Mechanics. Princeton U.P., Princeton, 1999.
 Everett III H. Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29, 1957, 454-462.