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

Quantum tomography and the quantum Radon transform

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.

Citation: Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and 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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[9]

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

[10]

F. M. CiagliaF. Di CosmoA. 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.

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

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

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

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[28]

A. G. GarcíaM. 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.

[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.

[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. 

[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.

[32]

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

[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.

[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.

[35]

A. IbortV. I. Man'koG. MarmoA. 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.

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[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

[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.

[44]

O. V. Man'koV. 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.

[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.

[46]

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

[47]

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

[48]

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

[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.

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

[52]

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

[53]

I. E. Segal, Irreducible representations of operator algebras, Bull. Am. Math. Soc., 53 (1947), 73-88.  doi: 10.1090/S0002-9904-1947-08742-5.

[54]

D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum, Physical Review Letters, 70 (1993), 1244. doi: 10.1103/PhysRevLett.70.1244.

[55]

M. H. Stone, On one-parameter unitary groups in Hilbert Space, Ann. Math., 33 (1932), 643-648.  doi: 10.2307/1968538.

[56]

K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase, Physical Review A, 40 (1989), 2847. doi: 10.1103/PhysRevA.40.2847.

[57]

E. Wigner, On the Quantum Correction for Thermodinamic Equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[9]

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

[10]

F. M. CiagliaF. Di CosmoA. 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.

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

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

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

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[28]

A. G. GarcíaM. 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.

[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.

[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. 

[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.

[32]

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

[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.

[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.

[35]

A. IbortV. I. Man'koG. MarmoA. 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.

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[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

[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.

[44]

O. V. Man'koV. 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.

[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.

[46]

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

[47]

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

[48]

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

[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.

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

[52]

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

[53]

I. E. Segal, Irreducible representations of operator algebras, Bull. Am. Math. Soc., 53 (1947), 73-88.  doi: 10.1090/S0002-9904-1947-08742-5.

[54]

D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum, Physical Review Letters, 70 (1993), 1244. doi: 10.1103/PhysRevLett.70.1244.

[55]

M. H. Stone, On one-parameter unitary groups in Hilbert Space, Ann. Math., 33 (1932), 643-648.  doi: 10.2307/1968538.

[56]

K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase, Physical Review A, 40 (1989), 2847. doi: 10.1103/PhysRevA.40.2847.

[57]

E. Wigner, On the Quantum Correction for Thermodinamic Equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.

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 $
Figure 2.  Tomographic problem
Figure 3.  Tomographic map $ U $
Figure 4.  Sampling diagram
Figure 5.  Positive Transform diagram
Figure 6.  Bloch's sphere representing states of a particle with spin $ 1/2 $
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