{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,26]],"date-time":"2025-10-26T15:04:09Z","timestamp":1761491049155,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T00:00:00Z","timestamp":1593648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In view of the probabilistic quantizer\u2013dequantizer operators introduced, the qubit states (spin-1\/2 particle states, two-level atom states) realizing the irreducible representation of the     S U ( 2 )     symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1\/2 particle projections     m = \u00b1 1 \/ 2     onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schr\u00f6dinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer\u2013dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg\u2013Weyl symmetry properties.<\/jats:p>","DOI":"10.3390\/sym12071099","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T11:07:42Z","timestamp":1594033662000},"page":"1099","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["SU(2) Symmetry of Qubit States and Heisenberg\u2013Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics"],"prefix":"10.3390","volume":"12","author":[{"given":"Peter","family":"Adam","sequence":"first","affiliation":[{"name":"Institute for Solid State Physics and Optics, Wigner Research Center for Physics, P.O. Box 49, H-1525 Budapest, Hungary"},{"name":"Institute of Physics, University of P\u00e9cs, Ifj\u00fas\u00e1g \u00fatja 6, H-7624 P\u00e9cs, Hungary"}]},{"given":"Vladimir","family":"Andreev","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"}]},{"given":"Margarita","family":"Man\u2019ko","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"}]},{"given":"Vladimir","family":"Man\u2019ko","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"},{"name":"Moscow Institute of Physics and Technology, State University, Institutskii per. 9, Dolgoprudnyi, Moscow Region 141700, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5491-8915","authenticated-orcid":false,"given":"Matyas","family":"Mechler","sequence":"additional","affiliation":[{"name":"Institute of Physics, University of P\u00e9cs, Ifj\u00fas\u00e1g \u00fatja 6, H-7624 P\u00e9cs, Hungary"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1002\/andp.19263840404","article-title":"Quantisierung als Eigenwertproblem (Erste Mitteilung)","volume":"384","year":"1926","journal-title":"Ann. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1002\/andp.19263840602","article-title":"Quantisierung als Eigenwertproblem (Zweite Mitteilung)","volume":"384","year":"1926","journal-title":"Ann. Phys."},{"key":"ref_3","unstructured":"Dirac, P.A.M. (1981). The Principles of Quantum Mechanics, Clarendon Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1007\/BF01343064","article-title":"Das D\u00e4mpfungsproblem in der Wellenmechanik","volume":"45","author":"Landau","year":"1927","journal-title":"Z. Phys."},{"key":"ref_5","first-page":"245","article-title":"Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik","volume":"1927","year":"1927","journal-title":"Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen Mathematisch-Physikalische Klasse"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"074031","DOI":"10.1088\/0031-8949\/90\/7\/074031","article-title":"Quantum tomography twenty years later","volume":"90","author":"Asorey","year":"2015","journal-title":"Phys. Scr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1244","DOI":"10.1103\/PhysRevLett.70.1244","article-title":"Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum","volume":"70","author":"Smithey","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1103\/RevModPhys.81.299","article-title":"Continuous-variable optical quantum-state tomography","volume":"81","author":"Lvovsky","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1109\/TMI.1986.4307775","article-title":"On the determination of functions from their integral values along certain manifolds","volume":"5","author":"Radon","year":"1986","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/BF00733376","article-title":"A tomographic approach to Wigner\u2019s function","volume":"17","author":"Bertrand","year":"1987","journal-title":"Found. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2847","DOI":"10.1103\/PhysRevA.40.2847","article-title":"Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase","volume":"40","author":"Vogel","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0375-9601(96)00107-7","article-title":"Symplectic tomography as classical approach to quantum systems","volume":"213","author":"Mancini","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1007\/BF02550342","article-title":"Classical-like description of quantum dynamics by means of symplectic tomography","volume":"27","author":"Mancini","year":"1997","journal-title":"Found. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1007\/s10946-011-9191-5","article-title":"Probability representation of the quantum evolution and energy-level equations for optical tomograms","volume":"32","author":"Korennoy","year":"2011","journal-title":"J. Russ. Laser Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"052119","DOI":"10.1103\/PhysRevA.85.052119","article-title":"Description and measurement of observables in the optical tomographic probability representation of quantum mechanics","volume":"85","author":"Amosov","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/S0375-9601(97)00199-0","article-title":"Positive distribution description for spin states","volume":"229","author":"Dodonov","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1134\/1.558326","article-title":"Spin state tomography","volume":"85","year":"1997","journal-title":"J. Exp. Theor. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2777","DOI":"10.1088\/0305-4470\/32\/15\/006","article-title":"Reconstructing a pure state of a spin s through three Stern-Gerlach measurements","volume":"32","author":"Amiet","year":"1999","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"L5","DOI":"10.1088\/1464-4266\/1\/5\/101","article-title":"Coherent states and the reconstruction of pure spin states","volume":"1","author":"Amiet","year":"1999","journal-title":"J. Opt. B Quantum Semiclass. Opt."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1088\/1464-4266\/5\/1\/311","article-title":"Spin tomography","volume":"5","author":"Maccone","year":"2003","journal-title":"J. Opt. B Quantum Semiclass. Opt."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2016). Probability and Randomness. Quantum versus Classical, World Scientific.","DOI":"10.1142\/p1036"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1088\/0305-4470\/35\/3\/315","article-title":"Alternative commutation relations, star products and tomography","volume":"35","author":"Marmo","year":"2002","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1016\/j.aop.2017.08.025","article-title":"Dynamical aspects in the quantizer-dequantizer formalism","volume":"385","author":"Ciaglia","year":"2017","journal-title":"Ann. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1023\/A:1012373419313","article-title":"Quasiprobability and probability distributions for spin-1\/2 states","volume":"14","author":"Scully","year":"2001","journal-title":"Found. Phys. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"335302","DOI":"10.1088\/1751-8121\/aa7d7d","article-title":"Metric on the space of quantum states from relative entropy. Tomographic reconstruction","volume":"50","author":"Marmo","year":"2017","journal-title":"J.Phys. A Math. Gen."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the quantum correction for thermodynamic equilibrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_27","first-page":"264","article-title":"Some formal properties of the density matrix","volume":"22","author":"Husimi","year":"1940","journal-title":"Proc. Phys. Math. Soc. Jpn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2766","DOI":"10.1103\/PhysRev.131.2766","article-title":"Coherent and incoherent states of the radiation field","volume":"131","author":"Glauber","year":"1963","journal-title":"Phys. Rev."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1103\/PhysRevLett.10.277","article-title":"Equivalence of semiclassical and quantum-mechanical descriptions of statistical light beams","volume":"10","author":"Sudarshan","year":"1963","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF02053909","article-title":"Spin quasidistribution functions","volume":"24","author":"Scully","year":"1994","journal-title":"Found. Phys."},{"key":"ref_31","first-page":"891","article-title":"On distributions in representation space","volume":"4","author":"Stratonovich","year":"1957","journal-title":"J. Exp. Theor. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1007\/BF01397280","article-title":"Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik","volume":"43","author":"Heisenberg","year":"1927","journal-title":"Z. Phys."},{"key":"ref_33","unstructured":"Schr\u00f6dinger, E. (1930). Zum Heisenbergschen Unscharfeprinzip, Berliner K\u00f6niglich Akademie und die Wissenschaft."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1103\/PhysRev.34.163","article-title":"The uncertainty principle","volume":"34","author":"Robertson","year":"1929","journal-title":"Phys. Rev."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3987","DOI":"10.1088\/1751-8113\/40\/14\/014","article-title":"Geometrical approach to mutually unbiased bases","volume":"40","author":"Klimov","year":"2007","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_36","first-page":"86","article-title":"Matrix bases for star-products: A review","volume":"10","author":"Lizzi","year":"2014","journal-title":"Symmetry Integr. Geom. Methods Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3461","DOI":"10.1088\/0305-4470\/34\/16\/314","article-title":"The Pauli equation for probability distributions","volume":"34","author":"Mancini","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1941021","DOI":"10.1142\/S0219749919410211","article-title":"Observables, interference phenomenon and Born\u2019s rule in the probability representation of quantum mechanics","volume":"18","year":"2020","journal-title":"Int. J. Quantum Inform."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"5243","DOI":"10.1103\/PhysRevLett.80.5243","article-title":"Real spectra in non-hermitian Hamiltonians having PT symmetry","volume":"80","author":"Bender","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1142\/S0219887810004816","article-title":"Pseudo-Hermitian representation of quantum mechanics","volume":"7","author":"Mostafazadeh","year":"2010","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1350163","DOI":"10.1142\/S0217979213501634","article-title":"Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments","volume":"27","author":"Sergi","year":"2013","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Sergi, A., and Giaquinta, P.V. (2016). Linear quantum entropy and non-Hermitian Hamiltonians. Entropy, 18.","DOI":"10.3390\/e18120451"},{"key":"ref_43","unstructured":"Kolmogorov, A.N. (1956). Foundation of the Theory of Probability, Chelsea."},{"key":"ref_44","first-page":"146","article-title":"Symmetry of the hydrogen atom","volume":"2","author":"Malkin","year":"1965","journal-title":"JETP Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.1103\/PhysRev.156.1541","article-title":"Transition probabilities of the hydrogen atom from noncompact dynamical groups","volume":"156","author":"Barut","year":"1967","journal-title":"Phys. Rev."},{"key":"ref_46","unstructured":"Barut, A., Bohm, A., and Neeman, Y. (1986). Dynamical Groups and Spectrum Generating Algebras, World Scientific."},{"key":"ref_47","unstructured":"Andreev, V.A., Malkin, I.A., and Man\u2019ko, V.I. (1971). Dynamical Symmetries of Magnetic Monopole, Lebedev Physical Institute. Preprint No. 1."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s10946-017-9628-6","article-title":"Triangle geometry of the qubit state in the probability representation expressed in terms of the Triada of Malevich\u2019s Squares","volume":"38","author":"Chernega","year":"2017","journal-title":"J. Russ. Laser Res."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1007\/s10946-017-9648-2","article-title":"Probability representation of quantum observables and quantum states","volume":"38","author":"Chernega","year":"2017","journal-title":"J. Russ. Laser Res."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1007\/s10946-017-9662-4","article-title":"Triangle geometry for qutrit states in the probability representation","volume":"38","author":"Chernega","year":"2017","journal-title":"J. Russ. Laser Res."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1023\/A:1024090024283","article-title":"Search for purity and entanglement","volume":"24","author":"Sudarshan","year":"2003","journal-title":"J. Russ. Laser Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1007\/s10946-016-9606-4","article-title":"Continuous sets of dequantizers and quantizers for one-qubit states","volume":"37","author":"Adam","year":"2016","journal-title":"J. Russ. Laser Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2778","DOI":"10.1016\/j.physleta.2017.06.042","article-title":"Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems","volume":"381","author":"Adam","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/s10946-017-9673-1","article-title":"Nonnnegative discrete symbols and their probabilistic interpretation","volume":"38","author":"Adam","year":"2017","journal-title":"J. Russ. Laser Res."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1007\/s10946-018-9730-4","article-title":"Symbols of multiqubit states admitting a physical interpretation","volume":"39","author":"Adam","year":"2018","journal-title":"J. Russ. Laser Res."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Man\u2019ko, M.A., and Man\u2019ko, V.I. (2018). New entropic inequalities and hidden correlations in quantum suprematism picture of qudit states. Entropy, 20.","DOI":"10.3390\/e20090692"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1099\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:46:18Z","timestamp":1760175978000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1099"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,2]]},"references-count":56,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071099"],"URL":"https:\/\/doi.org\/10.3390\/sym12071099","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,7,2]]}}}