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October  2021, 15(5): 893-928. doi: 10.3934/ipi.2021021

Quantum tomography and the quantum Radon transform

Alberto Ibort 1, and  Alberto López-Yela 2,

1. 

Instituto de Ciencias Matemáticas (CSIC - UAM - UC3M - UCM) ICMAT and Depto. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain
2. 

Dpto. de Teoría de la señal y comunicaciones, Univ. Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain

Received  July 2019 Revised  February 2021 Published  October 2021 Early access  March 2021

A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of $ C^* $-algebras is presented. Given a $ C^* $-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on $ C^* $-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.

The abstract theory is realized by using dynamical systems, that is, groups represented on $ C^* $-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.

Keywords: Radon transform, quantum tomography, $ C^* $-algebras, group representations, positive-type functions.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021
References:
[1]

N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces, Dover Publ., New York, 1963.  Google Scholar

[2]

E. M. Alfsen and F. W. Shultz, State spaces of Jordan algebras, Acta Math., 140 (1978), 155-190.  doi: 10.1007/BF02392307.  Google Scholar

[3]

P. Aniello, V. I. Man'ko and G. Marmo, Frame transforms, star products and quantum mechanics on phase space, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 285304, 40 pp. doi: 10.1088/1751-8113/41/28/285304.  Google Scholar

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M. Asorey, A. Ibort, G. Marmo and F. Ventriglia, Quantum Tomography twenty years later, Phys. Scr., 90 (2015), 074031. doi: 10.1088/0031-8949/90/7/074031.  Google Scholar

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M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio and E. C. G. Sudarshan, Generalized tomographic maps and star-product formalism, Phys. Scr., 90 (2015), 065101. doi: 10.1088/0031-8949/90/6/065101.  Google Scholar

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J. Bertrand and P. Bertrand, A tomographic approach to Wigner's function, Foundations of Physics, 17 (1987), 397-405. doi: 10.1007/BF00733376.  Google Scholar

[8]

A. del Campo, V. I. Man'ko and G. Marmo, Symplectic tomography of ultracold gases in tight waveguides, Phys. Rev. A., 78 (2008), 025602. doi: 10.1103/PhysRevA.78.025602.  Google Scholar

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G. Cassinelli, G. M. D'Ariano, E. De Vito and A. Levrero, Group theoretical quantum tomography, Journal of Mathematical Physics, 41 (2000), 7940-7951.  doi: 10.1063/1.1323497.  Google Scholar

[10]

F. M. Ciaglia, F. Di Cosmo, A. Ibort and G. Marmo, Dynamical aspects in the quantizer-dequantizer formalism, Annals of Physics, 385 (2017), 769-781.  doi: 10.1016/j.aop.2017.08.025.  Google Scholar

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K. Banaszek, G. M. D'Ariano, P. Kumar and M. F. Sacchi, Maximum-likelihood estimation of the density matrix, Phys. Rev., A61 (1999), 010304(R). doi: 10.1103/PhysRevA.61.010304.  Google Scholar

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G. M. D'Ariano, M. G. A. Paris and M. F. Sacchi, Quantum tomography, Advances in Imaging and Electron Physics, 128 (2003), 205-308. arXiv: quant-ph/0302028. doi: 10.1016/S1076-5670(03)80065-4.  Google Scholar

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G. M. D'Ariano, L. Maccone and M. Paini, Spin tomography, J. Opt. B: Qiuantum Semiclass. Opt., 5 (2003), 77-84.  doi: 10.1088/1464-4266/5/1/311.  Google Scholar

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J. Diximier, $C^*$-Algebras, North Holand Publ. Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

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M. Duflo and C. C. Moore, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal., 21 (1976), 209-243.  doi: 10.1016/0022-1236(76)90079-3.  Google Scholar

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E. Ercolessi, G. Marmo, G. Morandi and N. Mukunda, Wigner distributions in Quantum Mechanics, J. Phys.: Conf. Ser., 87 (2007), 012010. doi: 10.1088/1742-6596/87/1/012010.  Google Scholar

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G. Esposito, G. Marmo and G. Sudarshan, From Classical to Quantum Mechanics, Cambridge Univ. Press, Cambridge 2004. doi: 10.1017/CBO9780511610929.  Google Scholar

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P. Facchi and M. Ligabò, Classical and quantum aspects of tomography, AIP Conference Proceedings., 1260 (2010), 3-34.   Google Scholar

[23]

F. Falceto, L. Ferro, A. Ibort and G. Marmo, Reduction of Lie-Jordan Banach algebras and quantum states, J. Phys. A: Math. Theor., 46 (2013), 015201, 14 pp. doi: 10.1088/1751-8113/46/1/015201.  Google Scholar

[24]

H.R. Fernández-Morales, A.G. García, M.A. Hernández-Medina and M.J. Muñoz-Bouzo, On Some Sampling-Related Frames in $U$-Invariant Spaces, Abstract and Applied Analysis (2013), 761620, 14 pp. doi: 10.1155/2013/761620.  Google Scholar

[25]

M. Frank and D. R. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebras, J. Operator Theory, 48 (2002), 273-314.   Google Scholar

[26]

A. Galindo and P. Pascual, Quantum Mechanics I, Springer-Verlag. Berlin 1990. doi: 10.1007/978-3-642-83854-5.  Google Scholar

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A. G. García, A brief walk through sampling theory, Advances in Imaging and Electron Physics, 124 (2002), 63-137.  doi: 10.1016/S1076-5670(02)80042-8.  Google Scholar

[28]

A. G. García, M. A. Hernández-Medina and G. Pérez-Villalón, Generalized sampling in shift-invariant spaces with multiple stable generators, Journal of Mathematical Analysis and Applications, 337 (2008), 69-84.  doi: 10.1016/j.jmaa.2007.03.083.  Google Scholar

[29]

A. G. García, M. A. Hernández-Medina and A. Ibort, Towards a quantum sampling theory: the case of finite groups, In: Marmo G., Martín de Diego D., Muñoz Lecanda M. (eds) Classical and Quantum Physics. Springer Proceedings in Physics, vol 229. Springer, Cham., (2019) 203-223. arXiv:1510.08134 [math-ph]. doi: 10.1007/978-3-030-24748-5_11.  Google Scholar

[30]

I. Gel'fand and M. Naimark, On the embedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S., 12 (1943), 197-217.   Google Scholar

[31]

A. Gleason, Measures on the closed subspaces of a Hilbert space, Indiana Univ. Math. J., 6 (1957), 885-893.  doi: 10.1512/iumj.1957.6.56050.  Google Scholar

[32]

R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer-Verlag. Berlin, 1996. doi: 10.1007/978-3-642-61458-3.  Google Scholar

[33]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventriglia, An introduction to the tomographic picture of quantum mechanics, Phys. Scr., 79 (2009), 065013. doi: 10.1088/0031-8949/79/06/065013.  Google Scholar

[34]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni, F. Ventrigilia, A generalized Wigner function on the space of irreducible representations of the Weyl-Heisenberg group and its transformation properties, J. Phys. A: Math. Theor., 42 (2009), 155302, 12 pp. doi: 10.1088/1751-8113/42/15/155302.  Google Scholar

[35]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventriglia, On the tomographic picture of quantum mechanics, Phys. Let. A., 374 (2010), 2614-2617.  doi: 10.1016/j.physleta.2010.04.056.  Google Scholar

[36]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventriglia, Pedagogical presentation of a $C^*$-algebraic approach to quantum tomography, Phys. Scr., 84 (2011), 065006. doi: 10.1088/0031-8949/84/06/065006.  Google Scholar

[37]

A. Ibort, A. López Yela and J. Moro, A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups, J. of Math. Phys., 58 (2017), 101702, 21 pp. doi: 10.1063/1.5004259.  Google Scholar

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[39]

A. W. Joshi, Elements of Group Theory for Physicists, Third edition. A Halsted Press Book. John Wiley & Sons, Inc., New York, 1982  Google Scholar

[40]

R. V. Kadison, A representation theory for commutative topological algebras, Mem. Amer. Math. Soc., 7 (1951), 39 pp.  Google Scholar

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G. Kaiser, A Friendly Guide to Wavelets, Birkhäuser Boston, Inc., Boston, MA, 1994.  Google Scholar

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A. López-Yela, On the Tomographic Description of Quantum Systems: Theory and Applications, Ph.D. Thesis. Universidad Carlos III de Madrid (2015). http://hdl.handle.net/10016/22629 Google Scholar

[43]

V. I. Man'ko and O. V. Man'ko, Spin state tomography, J. of Experimental and Theoretical Physics, 85 (1997), 430-434.  doi: 10.1134/1.558326.  Google Scholar

[44]

O. V. Man'ko, V. I. Man'ko and G. Marmo, Alternative commutation relations, star products and tomography, Journal of Physics A: Mathematical and General, 35 (2002), 699-719.  doi: 10.1088/0305-4470/35/3/315.  Google Scholar

[45]

V. I. Man'ko, G. Marmo, A. Simoni, A. Stern and E. C. G. Sudarshan, On the meaning and interpretation of tomography in abstract Hilbert spaces, Phys. Lett. A, 351 (2006) 1-12. doi: 10.1016/j.physleta.2005.10.063.  Google Scholar

[46]

M. A. Na${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$mark, Normed Rings, P. Noordhoff N. V., Groningen 1964.  Google Scholar

[47]

F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

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J. v. Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann., 102 (1930), 370-427.  doi: 10.1007/BF01782352.  Google Scholar

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D.-G. Welsch, T. Opatrny and W. Vogel, II Homodyne Detection and Quantum-State Reconstruction, Prog. Opt., vol. 39 (1999), 63-211. doi: 10.1016/S0079-6638(08)70389-5.  Google Scholar

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J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Proc. Sympos. Appl. Math., 27, Amer. Math. Soc., Providence, R.I., 1982. doi: 10.1090/psapm/027/692055.  Google Scholar

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show all references

References:
[1]

N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces, Dover Publ., New York, 1963.  Google Scholar

[2]

E. M. Alfsen and F. W. Shultz, State spaces of Jordan algebras, Acta Math., 140 (1978), 155-190.  doi: 10.1007/BF02392307.  Google Scholar

[3]

P. Aniello, V. I. Man'ko and G. Marmo, Frame transforms, star products and quantum mechanics on phase space, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 285304, 40 pp. doi: 10.1088/1751-8113/41/28/285304.  Google Scholar

[4]

J. Arthur, Characters, harmonic analysis and an $L^2$-Lefschetz formula, In The mathematical heritage of Hermann Weyl, Proc. Sym. Pure Math., 48 1988,167-179. doi: 10.1090/pspum/048/974334.  Google Scholar

[5]

M. Asorey, A. Ibort, G. Marmo and F. Ventriglia, Quantum Tomography twenty years later, Phys. Scr., 90 (2015), 074031. doi: 10.1088/0031-8949/90/7/074031.  Google Scholar

[6]

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio and E. C. G. Sudarshan, Generalized tomographic maps and star-product formalism, Phys. Scr., 90 (2015), 065101. doi: 10.1088/0031-8949/90/6/065101.  Google Scholar

[7]

J. Bertrand and P. Bertrand, A tomographic approach to Wigner's function, Foundations of Physics, 17 (1987), 397-405. doi: 10.1007/BF00733376.  Google Scholar

[8]

A. del Campo, V. I. Man'ko and G. Marmo, Symplectic tomography of ultracold gases in tight waveguides, Phys. Rev. A., 78 (2008), 025602. doi: 10.1103/PhysRevA.78.025602.  Google Scholar

[9]

G. Cassinelli, G. M. D'Ariano, E. De Vito and A. Levrero, Group theoretical quantum tomography, Journal of Mathematical Physics, 41 (2000), 7940-7951.  doi: 10.1063/1.1323497.  Google Scholar

[10]

F. M. Ciaglia, F. Di Cosmo, A. Ibort and G. Marmo, Dynamical aspects in the quantizer-dequantizer formalism, Annals of Physics, 385 (2017), 769-781.  doi: 10.1016/j.aop.2017.08.025.  Google Scholar

[11]

K. Banaszek, G. M. D'Ariano, P. Kumar and M. F. Sacchi, Maximum-likelihood estimation of the density matrix, Phys. Rev., A61 (1999), 010304(R). doi: 10.1103/PhysRevA.61.010304.  Google Scholar

[12]

G. M. D'Ariano, Universal quantum estimation, Phys. Lett., A268 (2000), 151-157.  doi: 10.1016/S0375-9601(00)00164-X.  Google Scholar

[13]

G. M. D'Ariano, M. G. A. Paris and M. F. Sacchi, Quantum tomography, Advances in Imaging and Electron Physics, 128 (2003), 205-308. arXiv: quant-ph/0302028. doi: 10.1016/S1076-5670(03)80065-4.  Google Scholar

[14]

G. M. D'Ariano, L. Maccone and M. Paini, Spin tomography, J. Opt. B: Qiuantum Semiclass. Opt., 5 (2003), 77-84.  doi: 10.1088/1464-4266/5/1/311.  Google Scholar

[15]

I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

[16]

E. B. Davies, Spectral Theory and Differential Operators, Cambridge Univ. Press, Cambridge, U.K. 1995. doi: 10.1017/CBO9780511623721.  Google Scholar

[17] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford university press, 1947.   Google Scholar
[18]

J. Diximier, $C^*$-Algebras, North Holand Publ. Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[19]

M. Duflo and C. C. Moore, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal., 21 (1976), 209-243.  doi: 10.1016/0022-1236(76)90079-3.  Google Scholar

[20]

E. Ercolessi, G. Marmo, G. Morandi and N. Mukunda, Wigner distributions in Quantum Mechanics, J. Phys.: Conf. Ser., 87 (2007), 012010. doi: 10.1088/1742-6596/87/1/012010.  Google Scholar

[21]

G. Esposito, G. Marmo and G. Sudarshan, From Classical to Quantum Mechanics, Cambridge Univ. Press, Cambridge 2004. doi: 10.1017/CBO9780511610929.  Google Scholar

[22]

P. Facchi and M. Ligabò, Classical and quantum aspects of tomography, AIP Conference Proceedings., 1260 (2010), 3-34.   Google Scholar

[23]

F. Falceto, L. Ferro, A. Ibort and G. Marmo, Reduction of Lie-Jordan Banach algebras and quantum states, J. Phys. A: Math. Theor., 46 (2013), 015201, 14 pp. doi: 10.1088/1751-8113/46/1/015201.  Google Scholar

[24]

H.R. Fernández-Morales, A.G. García, M.A. Hernández-Medina and M.J. Muñoz-Bouzo, On Some Sampling-Related Frames in $U$-Invariant Spaces, Abstract and Applied Analysis (2013), 761620, 14 pp. doi: 10.1155/2013/761620.  Google Scholar

[25]

M. Frank and D. R. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebras, J. Operator Theory, 48 (2002), 273-314.   Google Scholar

[26]

A. Galindo and P. Pascual, Quantum Mechanics I, Springer-Verlag. Berlin 1990. doi: 10.1007/978-3-642-83854-5.  Google Scholar

[27]

A. G. García, A brief walk through sampling theory, Advances in Imaging and Electron Physics, 124 (2002), 63-137.  doi: 10.1016/S1076-5670(02)80042-8.  Google Scholar

[28]

A. G. García, M. A. Hernández-Medina and G. Pérez-Villalón, Generalized sampling in shift-invariant spaces with multiple stable generators, Journal of Mathematical Analysis and Applications, 337 (2008), 69-84.  doi: 10.1016/j.jmaa.2007.03.083.  Google Scholar

[29]

A. G. García, M. A. Hernández-Medina and A. Ibort, Towards a quantum sampling theory: the case of finite groups, In: Marmo G., Martín de Diego D., Muñoz Lecanda M. (eds) Classical and Quantum Physics. Springer Proceedings in Physics, vol 229. Springer, Cham., (2019) 203-223. arXiv:1510.08134 [math-ph]. doi: 10.1007/978-3-030-24748-5_11.  Google Scholar

[30]

I. Gel'fand and M. Naimark, On the embedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S., 12 (1943), 197-217.   Google Scholar

[31]

A. Gleason, Measures on the closed subspaces of a Hilbert space, Indiana Univ. Math. J., 6 (1957), 885-893.  doi: 10.1512/iumj.1957.6.56050.  Google Scholar

[32]

R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer-Verlag. Berlin, 1996. doi: 10.1007/978-3-642-61458-3.  Google Scholar

[33]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventriglia, An introduction to the tomographic picture of quantum mechanics, Phys. Scr., 79 (2009), 065013. doi: 10.1088/0031-8949/79/06/065013.  Google Scholar

[34]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni, F. Ventrigilia, A generalized Wigner function on the space of irreducible representations of the Weyl-Heisenberg group and its transformation properties, J. Phys. A: Math. Theor., 42 (2009), 155302, 12 pp. doi: 10.1088/1751-8113/42/15/155302.  Google Scholar

[35]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventriglia, On the tomographic picture of quantum mechanics, Phys. Let. A., 374 (2010), 2614-2617.  doi: 10.1016/j.physleta.2010.04.056.  Google Scholar

[36]

A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventriglia, Pedagogical presentation of a $C^*$-algebraic approach to quantum tomography, Phys. Scr., 84 (2011), 065006. doi: 10.1088/0031-8949/84/06/065006.  Google Scholar

[37]

A. Ibort, A. López Yela and J. Moro, A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups, J. of Math. Phys., 58 (2017), 101702, 21 pp. doi: 10.1063/1.5004259.  Google Scholar

[38]

J. M. Jauch, Foundations of Quantum Mechanics, Reading, Mass.-London-Don Mills, Ont. 1968.  Google Scholar

[39]

A. W. Joshi, Elements of Group Theory for Physicists, Third edition. A Halsted Press Book. John Wiley & Sons, Inc., New York, 1982  Google Scholar

[40]

R. V. Kadison, A representation theory for commutative topological algebras, Mem. Amer. Math. Soc., 7 (1951), 39 pp.  Google Scholar

[41]

G. Kaiser, A Friendly Guide to Wavelets, Birkhäuser Boston, Inc., Boston, MA, 1994.  Google Scholar

[42]

A. López-Yela, On the Tomographic Description of Quantum Systems: Theory and Applications, Ph.D. Thesis. Universidad Carlos III de Madrid (2015). http://hdl.handle.net/10016/22629 Google Scholar

[43]

V. I. Man'ko and O. V. Man'ko, Spin state tomography, J. of Experimental and Theoretical Physics, 85 (1997), 430-434.  doi: 10.1134/1.558326.  Google Scholar

[44]

O. V. Man'ko, V. I. Man'ko and G. Marmo, Alternative commutation relations, star products and tomography, Journal of Physics A: Mathematical and General, 35 (2002), 699-719.  doi: 10.1088/0305-4470/35/3/315.  Google Scholar

[45]

V. I. Man'ko, G. Marmo, A. Simoni, A. Stern and E. C. G. Sudarshan, On the meaning and interpretation of tomography in abstract Hilbert spaces, Phys. Lett. A, 351 (2006) 1-12. doi: 10.1016/j.physleta.2005.10.063.  Google Scholar

[46]

M. A. Na${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$mark, Normed Rings, P. Noordhoff N. V., Groningen 1964.  Google Scholar

[47]

F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

[48]

J. v. Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann., 102 (1930), 370-427.  doi: 10.1007/BF01782352.  Google Scholar

[49]

D.-G. Welsch, T. Opatrny and W. Vogel, II Homodyne Detection and Quantum-State Reconstruction, Prog. Opt., vol. 39 (1999), 63-211. doi: 10.1016/S0079-6638(08)70389-5.  Google Scholar

[50] G. K. Pedersen, $C^*$-Algebras and Their Automorphism Groups, Academic Press, 1979.   Google Scholar
[51]

J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Proc. Sympos. Appl. Math., 27, Amer. Math. Soc., Providence, R.I., 1982. doi: 10.1090/psapm/027/692055.  Google Scholar

[52]

M. Reed and B. Simon, Methods of modern mathematical physics. Vol. 1: Functional analysis, revised and enlarged edition., Academic Press., USA 1980. Google Scholar

[53]

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Figure 1.  Left: Section of a body irradiated with gamma radiation and the measure of the tomogram $ \mathcal{W}(X, \theta) $. Right: Scheme of quantum state tomography. The quantum tomogram $ \mathcal{W}_\rho (X, \theta) $ obtained by homodyne measuring the quadrature operator $ \mathbf{X}_\theta = X -\textbf{Q}\cos\theta-\textbf{P}\sin\theta $
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Figure 2.  Tomographic problem
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Figure 3.  Tomographic map $ U $
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Figure 4.  Sampling diagram
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Figure 5.  Positive Transform diagram
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Figure 6.  Bloch's sphere representing states of a particle with spin $ 1/2 $
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