International Science Index


2946

Impact of the Existence of One-Way Functionson the Conceptual Difficulties of Quantum Measurements

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.
References:
[1] Goldreich, O. Modern Cryptography, Probabilistic Proofs, and Pseudorandomness. Springer, 1999.
[2] Sipser M. Introduction to the Theory of Computation. PWS Publishing, Section 10.6.3: One-way functions, 1997, pp. 374-376.
[3] Papadimitriou C. Computational Complexity. 1st edition, Addison Wesley, Section 12.1: One-way functions, 1993, pp.279-298.
[4] Greenlaw, R., Hoover, H. J., and Ruzzo, W. L. Limits to Parallel Computation: P-Completeness Theory. Oxford, England: Oxford University Press, 1995.
[5] Cook, S. The P versus NP Problem http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Descr iption.pdf.
[6] 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.
[7] Laloë F. Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems. Am. J. Phys., Vol. 69, No. 6, 2001, pp. 655-701.
[8] Griffiths D. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall, 2004.
[9] Schrödinger E. Proc. Cambridge Philos. Soc. 31, 1935, 555; 32, 1936, 446.
[10] Bassi, A., and Ghirardi, G.C. A general argument against the universal validity of the superposition principle. Phys. papers A, 275, 2000, 373- 381.
[11] Lui, Y.-K., Christiandl, M., Verstraete, F. Quantum Computational Complexity of the N-Representability Problem: QMA Complete. Phys. Rev. Letters 98, 2007, 110503(4).
[12] Wheeler J. and Zurek W. (eds). Quantum Theory and Measurement. Princeton University Press, 1983.
[13] 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.
[14] Wigner E. P. On hidden variables and quantum mechanical probabilities. Am. J. Phys. 38, 1970, 1005-1009.
[15] Wiener N. and Siegel A. A new form for the statistical postulate of quantum mechanics. Phys. Rev. 91, 1953, 1551-1560.
[16] Siegel A. and Wiener N. Theory of measurement in differential space quantum theory. Phys. Rev. 101, 1956, 429-432.
[17] 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.
[18] Pearle P. Reduction of the state vector by a non-linear Schrödinger equation. Phys. Rev. D 13, 1976, 857-868.
[19] Ghirardi G. C., Rimini A., and Weber T. Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 1986, 470-491.
[20] Diosi L. Quantum stochastic processes as models for state vector reduction. J. Phys. A 21, 1988, 2885-2898.
[21] Griffiths R. B. Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36, 1984, 219-272.
[22] Gell-Mann M. and Hartle J. Classical equations for quantum systems. Phys. Rev. D 47, 1993, 3345-3382.
[23] Omnés R. Understanding Quantum Mechanics. Princeton U.P., Princeton, 1999.
[24] Everett III H. Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29, 1957, 454-462.