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Computer Tomography and Quantum Mechanics

Published online by Cambridge University Press:  01 July 2016

Lev B. Klebanov*
Affiliation:
Russian Academy of Natural Sciences
Svetlozar T. Rachev*
Affiliation:
University of California, Santa Barbara
*
Postal address: Institute of Mathematical Geology, Russian Academy of Natural Science, 12 Shpalemaya, 191187, Russia.
∗∗ Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara CA 93106-3110, USA.

Abstract

In this paper we study some topics of interest to specialists in computer tomography. These are the following. (a) The Radon transform and its applications to computer tomography. (b) Problems of computer tomography with partially known data. Estimates of stability will be given for different types of distance in the space of probability distributions. We consider the problem with partially known tomographic data as a stability problem for appropriately chosen distances. This approach allows us to give a solution of the so-called computer tomography paradox. (c) The relation of quantum mechanics to computer tomography. An intriguing method for ‘measuring' wavefunctions by tomographic methods (CAT scans) opens a new approach to various problems in quantum mechanics. Using the method outlined for the solution of the computer tomography paradox, we derive inequalities that estimate the amount of information on the wavefunctions resulting from real CAT scans, i.e. CAT scans based on the finite number of measured marginals (projections) of the Wigner distributions. In conclusion, we propose a new version of the mathematical justification of CAT scans.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported by the ISF Grant No NXL 300 (Dr Klebanov) and by the Alexander von Humboldt Research Award for Senior U.S. Scientists (Dr Rachev), during his stay at Institute für Mathematische Stochastik, University of Freiburg.

References

[1] Guttmann, S., Kemperman, J. H. B., Reeds, J. A. and Shepp, L. A. (1991) Existence of probability measures with given marginals. Ann. Prob. 19, 17811791.CrossRefGoogle Scholar
[2] Hudson, R. L. (1974) When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249252.CrossRefGoogle Scholar
[3] Kakosyan, A. V., Klebanov, L. B. and Rachev, S. T. (1988) Quantitative Criteria for Convergence of Probability Measures. Ayastan, Yerevan. (In Russian.) Google Scholar
[4] Karlin, S. and Studden, W. J. (1966) Tchebysheff Systems. Interscience, New York.Google Scholar
[5] Khalfin, L. A. and Klebanov, L. B. (1994) A solution of the computer tomography paradox and estimation of the distances between the densities of measures with the same marginals. Ann. Prob. 22, 22352241.CrossRefGoogle Scholar
[6] Logan, B. F. (1975) The uncertainty principle in reconstructing functions from projections. Duke Math. J. 42, 661706.Google Scholar
[7] Natterer, F. (1986) The Mathematics of Computerized Tomography. Wiley, New York.CrossRefGoogle Scholar
[8] Radon, J. (1917) über die bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte Süchsische Akad. Wissenschaften, Leipzig 69, 262267.Google Scholar
[9] Shepp, L. A. and Kruskal, J. B. (1978) Computerized tomography: the new medical X-ray technology. Amer. Math. Monthly 85, 420439.CrossRefGoogle Scholar
[10] Smithey, D. T., Beck, M., Raymer, M. G. and Faridani, A. (1993) Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum. Phys. Rev. Lett. 70, 12441247.CrossRefGoogle ScholarPubMed
[11] Vogel, K. and Risken, H. (1989) Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 28472849.CrossRefGoogle ScholarPubMed
[12] Zolotarev, V. M. (1986). Modern Theory for Summation of Independent Random Variables. Nauka, Moscow.Google Scholar