# SU(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quantizer–Dequantizer Operators and Structure Constants of Associative Products

## 3. Structure Constants of Lie Algebras and Their Relation to Associative Product Structure Constants

## 4. $\mathit{SU}\left(\mathbf{2}\right)$ Symmetry and the Probability Representation of Spin-1/2 States

## 5. The von Neumann Evolution Equation for the Density Matrix $\mathbf{\rho}$ in Vector Form

## 6. The Probability Representation of the von Neumann Equation

## 7. $\mathit{O}\left(\mathbf{3}\right)$ Transforms of Probabilities and Spin-Projection Mean Values

## 8. Transforms of Quantizer and Dequantizer Operators

## 9. Symplectic Tomographic Probability Distribution

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Schrödinger, E. Quantisierung als Eigenwertproblem (Erste Mitteilung). Ann. Phys.
**1926**, 384, 361–376. [Google Scholar] [CrossRef] - Schrödinger, E. Quantisierung als Eigenwertproblem (Zweite Mitteilung). Ann. Phys.
**1926**, 384, 489–527. [Google Scholar] [CrossRef] - Dirac, P.A.M. The Principles of Quantum Mechanics; Clarendon Press: Oxford, UK, 1981; ISBN 9780198520115. [Google Scholar]
- Landau, L. Das Dämpfungsproblem in der Wellenmechanik. Z. Phys.
**1927**, 45, 430–441. [Google Scholar] [CrossRef] - Von Neumann, J. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen Mathematisch-Physikalische Klasse
**1927**, 1927, 245–272. [Google Scholar] - Asorey, M.; Ibort, A.; Marmo, G.; Ventriglia, F. Quantum tomography twenty years later. Phys. Scr.
**2015**, 90, 074031. [Google Scholar] [CrossRef][Green Version] - Smithey, D.T.; Beck, M.; Raymer, M.G.; Faridani, A. 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.
**1993**, 70, 1244–1248. [Google Scholar] [CrossRef] [PubMed] - Lvovsky, A.I.; Raymer, M.G. Continuous-variable optical quantum-state tomography. Rev. Mod. Phys.
**2009**, 81, 299–332. [Google Scholar] [CrossRef] - Radon, J.; Parks, P.; Clark, C. On the determination of functions from their integral values along certain manifolds. IEEE Trans. Med. Imaging
**1986**, 5, 170–176. [Google Scholar] - Bertrand, J.; Bertrand, P. A tomographic approach to Wigner’s function. Found. Phys.
**1987**, 17, 397–405. [Google Scholar] [CrossRef] - Vogel, K.; Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A
**1989**, 40, 2847–2849. [Google Scholar] [CrossRef] [PubMed] - Mancini, S.; Man’ko, V.I.; Tombesi, P. Symplectic tomography as classical approach to quantum systems. Phys. Lett. A
**1996**, 213, 1–6. [Google Scholar] [CrossRef][Green Version] - Mancini, S.; Man’ko, V.I.; Tombest, P. Classical-like description of quantum dynamics by means of symplectic tomography. Found. Phys.
**1997**, 27, 801–824. [Google Scholar] [CrossRef][Green Version] - Korennoy, Y.A.; Man’ko, V.I. Probability representation of the quantum evolution and energy-level equations for optical tomograms. J. Russ. Laser Res.
**2011**, 32, 74–95. [Google Scholar] [CrossRef][Green Version] - Amosov, G.G.; Korennoy, Y.A.; Man’ko, V.I. Description and measurement of observables in the optical tomographic probability representation of quantum mechanics. Phys. Rev. A
**2012**, 85, 052119. [Google Scholar] [CrossRef] - Dodonov, V.V.; Man’ko, V.I. Positive distribution description for spin states. Phys. Lett. A
**1997**, 229, 335–339. [Google Scholar] [CrossRef] - Man’ko, V.I.; Man’ko, O.V. Spin state tomography. J. Exp. Theor. Phys.
**1997**, 85, 430–434. [Google Scholar] [CrossRef] - Amiet, J.-P.; Weigert, S. Reconstructing a pure state of a spin s through three Stern-Gerlach measurements. J. Phys. A Math. Gen.
**1999**, 32, 2777. [Google Scholar] [CrossRef][Green Version] - Amiet, J.-P.; Weigert, S. Coherent states and the reconstruction of pure spin states. J. Opt. B Quantum Semiclass. Opt.
**1999**, 1, L5–L8. [Google Scholar] [CrossRef] - D’Ariano, G.M.; Maccone, L.; Paini, M. Spin tomography. J. Opt. B Quantum Semiclass. Opt.
**2003**, 5, 77–84. [Google Scholar] [CrossRef] - Khrennikov, A. Probability and Randomness. Quantum versus Classical; World Scientific: Singapore, 2016. [Google Scholar] [CrossRef]
- Man’ko, O.V.; Man’ko, V.I.; Marmo, G. Alternative commutation relations, star products and tomography. J. Phys. A Math. Gen.
**2002**, 35, 699–719. [Google Scholar] [CrossRef] - Ciaglia, F.M.; Di Cosmo, F.; Ibort, A.; Marmo, G. Dynamical aspects in the quantizer-dequantizer formalism. Ann. Phys.
**2017**, 385, 769–781. [Google Scholar] [CrossRef][Green Version] - Terra-Cunha, M.O.; Man’ko, V.I.; Scully, M.O. Quasiprobability and probability distributions for spin-1/2 states. Found. Phys. Lett.
**2001**, 14, 103–117. [Google Scholar] [CrossRef] - Man’ko, V.I.; Marmo, G.; Ventriglia, F.; Vitale, P. Metric on the space of quantum states from relative entropy. Tomographic reconstruction. J.Phys. A Math. Gen.
**2017**, 50, 335302. [Google Scholar] [CrossRef][Green Version] - Wigner, E. On the quantum correction for thermodynamic equilibrium. Phys. Rev.
**1932**, 40, 749–759. [Google Scholar] [CrossRef] - Husimi, K. Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn.
**1940**, 22, 264–314. [Google Scholar] [CrossRef] - Glauber, R.J. Coherent and incoherent states of the radiation field. Phys. Rev.
**1963**, 131, 2766–2788. [Google Scholar] [CrossRef] - Sudarshan, E.C.G. Equivalence of semiclassical and quantum-mechanical descriptions of statistical light beams. Phys. Rev. Lett.
**1963**, 10, 277–279. [Google Scholar] [CrossRef] - Scully, M.O.; Wodkiewicz, K. Spin quasidistribution functions. Found. Phys.
**1994**, 24, 85–107. [Google Scholar] [CrossRef] - Stratonovich, R.L. On distributions in representation space. J. Exp. Theor. Phys.
**1957**, 4, 891–898. [Google Scholar] - Heisenberg, W. Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys.
**1927**, 43, 172–198. [Google Scholar] [CrossRef] - Schrödinger, E. Zum Heisenbergschen Unscharfeprinzip; Berliner Königlich Akademie und die Wissenschaft: Berlin, Germany, 1930; pp. 296–303. [Google Scholar]
- Robertson, H.R. The uncertainty principle. Phys. Rev.
**1929**, 34, 163–164. [Google Scholar] [CrossRef] - Klimov, A.B.; Romero, J.L.; Björk, G.; Sánchez-Soto, L.L. Geometrical approach to mutually unbiased bases. J. Phys. A Math. Gen.
**2007**, 40, 3987–3998. [Google Scholar] [CrossRef] - Lizzi, F.; Vitale, P. Matrix bases for star-products: A review. Symmetry Integr. Geom. Methods Appl.
**2014**, 10, 86. [Google Scholar] [CrossRef][Green Version] - Mancini, S.; Man’ko, O.V.; Man’ko, V.I.; Tombesi, P. The Pauli equation for probability distributions. J. Phys. A Math. Gen.
**2001**, 34, 3461–3476. [Google Scholar] [CrossRef][Green Version] - Man’ko, M.A.; Man’ko, V.I. Observables, interference phenomenon and Born’s rule in the probability representation of quantum mechanics. Int. J. Quantum Inform.
**2020**, 18, 1941021. [Google Scholar] [CrossRef] - Bender, C.M.; Boettcher, S. Real spectra in non-hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett.
**1998**, 80, 5243–5246. [Google Scholar] [CrossRef][Green Version] - Mostafazadeh, A. Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys.
**2010**, 7, 1191–1306. [Google Scholar] [CrossRef][Green Version] - Sergi, A.; Zloshchastiev, K.G. Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments. Int. J. Mod. Phys. B
**2013**, 27, 1350163. [Google Scholar] [CrossRef] - Sergi, A.; Giaquinta, P.V. Linear quantum entropy and non-Hermitian Hamiltonians. Entropy
**2016**, 18, 451. [Google Scholar] [CrossRef][Green Version] - Kolmogorov, A.N. Foundation of the Theory of Probability; Chelsea: New York, NY, USA, 1956; ISBN1 10:0821826484. ISBN2 13/9780821826485. [Google Scholar]
- Malkin, I.A.; Man’ko, V.I. Symmetry of the hydrogen atom. JETP Lett.
**1965**, 2, 146–148. [Google Scholar] - Barut, A.O.; Kleinert, H. Transition probabilities of the hydrogen atom from noncompact dynamical groups. Phys. Rev.
**1967**, 156, 1541–1545. [Google Scholar] [CrossRef] - Barut, A.; Bohm, A.; Neeman, Y. Dynamical Groups and Spectrum Generating Algebras; World Scientific: Singapore, 1986; ISBN1 978-9971-5-0147-1. ISBN2 9971-5-0147-3. [Google Scholar]
- Andreev, V.A.; Malkin, I.A.; Man’ko, V.I. Dynamical Symmetries of Magnetic Monopole; Preprint No. 1; Lebedev Physical Institute: Moscow, Russia, 1971. [Google Scholar]
- Chernega, V.N.; Man’ko, O.V.; Man’ko, V.I. Triangle geometry of the qubit state in the probability representation expressed in terms of the Triada of Malevich’s Squares. J. Russ. Laser Res.
**2017**, 38, 141–149. [Google Scholar] [CrossRef][Green Version] - Chernega, V.N.; Man’ko, O.V.; Man’ko, V.I. Probability representation of quantum observables and quantum states. J. Russ. Laser Res.
**2017**, 38, 324–333. [Google Scholar] [CrossRef][Green Version] - Chernega, V.N.; Man’ko, O.V.; Man’ko, V.I. Triangle geometry for qutrit states in the probability representation. J. Russ. Laser Res.
**2017**, 38, 416–425. [Google Scholar] [CrossRef][Green Version] - Sudarshan, E.C.G. Search for purity and entanglement. J. Russ. Laser Res.
**2003**, 24, 195–203. [Google Scholar] [CrossRef] - Adam, P.; Andreev, V.A.; Isar, A.; Man’ko, M.A.; Man’ko, V.I. Continuous sets of dequantizers and quantizers for one-qubit states. J. Russ. Laser Res.
**2016**, 37, 544–555. [Google Scholar] [CrossRef] - Adam, P.; Andreev, V.A.; Isar, A.; Man’ko, M.A.; Man’ko, V.I. Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems. Phys. Lett. A
**2017**, 381, 2778–2782. [Google Scholar] [CrossRef][Green Version] - Adam, P.; Andreev, V.A.; Man’ko, M.A.; Man’ko, V.I. Nonnnegative discrete symbols and their probabilistic interpretation. J. Russ. Laser Res.
**2017**, 38, 491–506. [Google Scholar] [CrossRef] - Adam, P.; Andreev, V.A.; Man’ko, M.A.; Man’ko, V.I. Symbols of multiqubit states admitting a physical interpretation. J. Russ. Laser Res.
**2018**, 39, 360–375. [Google Scholar] [CrossRef] - Man’ko, M.A.; Man’ko, V.I. New entropic inequalities and hidden correlations in quantum suprematism picture of qudit states. Entropy
**2018**, 20, 692. [Google Scholar] [CrossRef][Green Version]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Adam, P.; Andreev, V.A.; Man’ko, M.A.; Man’ko, V.I.; Mechler, M.
*SU*(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics. *Symmetry* **2020**, *12*, 1099.
https://doi.org/10.3390/sym12071099

**AMA Style**

Adam P, Andreev VA, Man’ko MA, Man’ko VI, Mechler M.
*SU*(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics. *Symmetry*. 2020; 12(7):1099.
https://doi.org/10.3390/sym12071099

**Chicago/Turabian Style**

Adam, Peter, Vladimir A. Andreev, Margarita A. Man’ko, Vladimir I. Man’ko, and Matyas Mechler.
2020. "*SU*(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics" *Symmetry* 12, no. 7: 1099.
https://doi.org/10.3390/sym12071099