{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:38:28Z","timestamp":1760243908000,"version":"build-2065373602"},"reference-count":58,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2010,7,12]],"date-time":"2010-07-12T00:00:00Z","timestamp":1278892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a group-theoretical approach to the generalized oscillator algebra A\u03ba recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case \u03ba \u2265 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the P\u00f6schl-Teller systems) while the case \u03ba &lt; 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for A\u03ba in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices.<\/jats:p>","DOI":"10.3390\/sym2031461","type":"journal-article","created":{"date-parts":[[2010,7,12]],"date-time":"2010-07-12T11:37:07Z","timestamp":1278934627000},"page":"1461-1484","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms"],"prefix":"10.3390","volume":"2","author":[{"given":"Natig M.","family":"Atakishiyev","sequence":"first","affiliation":[{"name":"Instituto de Matem\u00e1ticas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Av. Universidad s\/n, Cuernavaca, Morelos 62251, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maurice R.","family":"Kibler","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Lyon, 37 rue du repos, 69361 Lyon, France"},{"name":"Universit\u00e9 Claude Bernard and CNRS\/IN2P3, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France"},{"name":"Institut de Physique Nucl\u00e9aire, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kurt Bernardo","family":"Wolf","sequence":"additional","affiliation":[{"name":"Instituto de Ciencias F\u00edsicas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Av. Universidad s\/n, Cuernavaca, Morelos 62251, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2010,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2693","DOI":"10.1088\/0305-4470\/28\/9\/026","article-title":"Distorted Heisenberg algebra and coherent states for isospectral oscillator Hamiltonians","volume":"28","author":"Nieto","year":"1995","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3603","DOI":"10.1088\/0305-4470\/32\/19\/311","article-title":"Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states","volume":"32","author":"Hussin","year":"1999","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"10349","DOI":"10.1088\/0305-4470\/37\/43\/022","article-title":"Polynomial Heisenberg algebras","volume":"37","author":"Carballo","year":"2004","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1006\/aphy.1996.0012","article-title":"Deformed Heisenberg algebra, fractional spin fields, and supersymmetry without fermions","volume":"245","author":"Plyushchay","year":"1996","journal-title":"Ann. Phys. NY"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/S0550-3213(97)00065-5","article-title":"Deformed Heisenberg algebra with reflection","volume":"491","author":"Plyushchay","year":"1997","journal-title":"Nucl. Phys. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3679","DOI":"10.1142\/S0217751X00001981","article-title":"Hidden nonlinear supersymmetries in pure parabosonic systems","volume":"15","author":"Plyushchay","year":"2000","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_7","unstructured":"Horvathy, P.A., Plyushchay, M.S., and Valenzuela, M. Bosons, fermions and anyons in the plane, and supersymmetry. Ann. Phys. NY, in press."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0375-9601(98)00046-2","article-title":"C-lambda-extended harmonic oscillator and (para)supersymmetric quantum mechanics","volume":"240","author":"Quesne","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/S0375-9601(00)00457-6","article-title":"Spectrum generating algebra of the C-lambda-extended oscillator and multiphoton coherent states","volume":"272","author":"Quesne","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1023\/A:1003627217508","article-title":"C-lambda-extended oscillator algebras and some of their deformations and applications to quantum mechanics","volume":"39","author":"Quesne","year":"2000","journal-title":"Int. J. Theor. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1142\/S021773230300954X","article-title":"Fractional supersymmetric quantum mechanics, topological invariants and generalized deformed oscillator algebras","volume":"18","author":"Quesne","year":"2003","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_12","unstructured":"Engin, A. (1999, January 16\u201322). A fractional supersymmetric oscillator and its coherent states. Proceedings of the Sixth International Wigner Symposium, Istanbul, Turkey."},{"key":"ref_13","unstructured":"Lulek, T., Lulek, B., and Wal, A. (2001). Symmetry and Structural Properties of Condensed Matter, World Scientific."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.physleta.2003.12.027","article-title":"Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics","volume":"321","author":"Daoud","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"122108","DOI":"10.1063\/1.2401711","article-title":"Fractional supersymmetry and hierarchy of shape invariant potentials","volume":"47","author":"Daoud","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"115303","DOI":"10.1088\/1751-8113\/43\/11\/115303","article-title":"Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems","volume":"43","author":"Daoud","year":"2010","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1103\/PhysRevA.39.1665","article-title":"Phase properties of the quantized single-mode electromagnetic-field","volume":"39","author":"Pegg","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"L37","DOI":"10.1088\/1355-5111\/9\/6\/001","article-title":"Remarks on the construction of a Hermitian phase operator","volume":"9","author":"Roy","year":"1997","journal-title":"Quantum Semiclass. Opt."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1307","DOI":"10.1088\/0305-4470\/31\/4\/018","article-title":"Coherent states, even and odd coherent states in a finite-dimensional Hilbert space and their properties","volume":"31","author":"Roy","year":"1998","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Perelomov, A.M. (1986). Generalized Coherent States and Their Applications, Springer.","DOI":"10.1007\/978-3-642-61629-7"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gazeau, J.-P. (2009). Coherent States in Quantum Physics, Wiley-VCH.","DOI":"10.1002\/9783527628285"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Lulek, T., Lulek, B., and Wal, A. (1999). Symmetry and Structural Properties of Condensed Matter, World Scientific.","DOI":"10.1142\/9789814527354"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1135\/cccc20050771","article-title":"Representation theory and Wigner-Racah algebra of the group SU(2) in a noncanonical basis","volume":"70","author":"Kibler","year":"2005","journal-title":"Collect. Czech. Chem. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1792","DOI":"10.1142\/S0217979206034297","article-title":"Angular momentum and mutually unbiased bases","volume":"20","author":"Kibler","year":"2006","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1802","DOI":"10.1142\/S0217979206034303","article-title":"A SU(2) recipe for mutually unbiased bases","volume":"20","author":"Kibler","year":"2006","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_26","first-page":"076","article-title":"SU(2) nonstandard bases: Case of mutually unbiased bases","volume":"3","author":"Albouy","year":"2007","journal-title":"SIGMA"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s10946-007-0032-5","article-title":"A unified approach to SIC-POVMs and MUBs","volume":"28","author":"Albouy","year":"2007","journal-title":"J. Russ. Laser Res."},{"key":"ref_28","first-page":"092","article-title":"Miscellaneous applications of quons","volume":"3","author":"Kibler","year":"2007","journal-title":"SIGMA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1135\/cccc20081281","article-title":"Generalized spin bases for quantum chemistry and quantum information","volume":"73","author":"Kibler","year":"2008","journal-title":"Collect. Czech. Chem. Commun."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"375302","DOI":"10.1088\/1751-8113\/41\/37\/375302","article-title":"Variations on a theme of Heisenberg, Pauli and Weyl","volume":"41","author":"Kibler","year":"2008","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"353001","DOI":"10.1088\/1751-8113\/42\/35\/353001","article-title":"An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group","volume":"42","author":"Kibler","year":"2009","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_32","unstructured":"Kibler, M.R. Bases for qudits from a nonstandard approach to SU(2). Phys. Atom. Nucl., in press."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5887","DOI":"10.1088\/0305-4470\/29\/18\/018","article-title":"Factorization of analytic representations in the unit disc and number-phase statistics of a quantum harmonic oscillator","volume":"29","author":"Vourdas","year":"1996","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_34","unstructured":"Berndt, B.C., Evans, R.J., and Williams, K.S. (1998). Gauss and Jacobi Sums, Wiley."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3241","DOI":"10.1088\/0305-4470\/14\/12\/019","article-title":"Geometrical description of quantum state determination","volume":"14","year":"1981","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0003-4916(89)90322-9","article-title":"Optimal state-determination by mutually unbiased measurements","volume":"191","author":"Wootters","year":"1989","journal-title":"Ann. Phys. NY"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1088\/0034-4885\/67\/3\/R03","article-title":"Quantum systems with finite Hilbert space","volume":"67","author":"Vourdas","year":"2004","journal-title":"Rep. Prog. Phys."},{"key":"ref_38","unstructured":"Weyl, H. (1931). The Theory of Groups and Quantum Mechanics, Dover Publications."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1073\/pnas.46.4.570","article-title":"Unitary operator bases","volume":"46","author":"Schwinger","year":"1960","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"015302","DOI":"10.1088\/1751-8113\/41\/1\/015302","article-title":"Projective ring line on an arbitrary single qudit","volume":"41","author":"Saniga","year":"2008","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1007\/s10773-007-9541-9","article-title":"Multi-line geometry of qubit-qutrit and higher-order Pauli operators","volume":"47","author":"Planat","year":"2008","journal-title":"Int. J. Theor. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"072001","DOI":"10.1088\/1751-8113\/42\/7\/072001","article-title":"The isotropic line of \u2124d2","volume":"42","author":"Albouy","year":"2009","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1166\/jctn.2010.1541","article-title":"Unitary reflection groups for quantum fault tolerance","volume":"7","author":"Planat","year":"2010","journal-title":"J. Comput. Theor. Nanosci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1088","DOI":"10.1063\/1.528788","article-title":"Infinite-dimensional algebras and a trigonometric basis for the classical Lie-algebras","volume":"31","author":"Fairlie","year":"1990","journal-title":"J. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Gruber, B., and Ramek, M. (1998). Symmetries in Science X, Plenum Press.","DOI":"10.1007\/978-1-4899-1537-5"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1364\/JOSAA.24.000651","article-title":"Geometry and dynamics in the fractional discrete Fourier transform","volume":"24","author":"Wolf","year":"2007","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Ozaktas, H.M., Zalevsky, Z., and Kutay, M.A. (2001). Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley.","DOI":"10.23919\/ECC.2001.7076127"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1073\/pnas.23.3.158","article-title":"Immersion of the Fourier transform in a continuous group of functional transformations","volume":"23","author":"Condon","year":"1937","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1364\/JOSA.60.001168","article-title":"Lens-system diffraction integral written in terms of matrix optics","volume":"60","author":"Collins","year":"1970","journal-title":"J. Opt. Soc. Am."},{"key":"ref_50","unstructured":"Moshinsky, M., and Quesne, C. (1974). Proceedings of the 15th Solvay Conference in Physics, 1970, Gordon and Breach."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1364\/OL.22.001047","article-title":"Improved discrete fractional transform","volume":"22","author":"Pei","year":"1997","journal-title":"Opt. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1109\/78.757221","article-title":"Discrete fractional Fourier transform based on orthogonal projections","volume":"47","author":"Pei","year":"1999","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2209","DOI":"10.1088\/0305-4470\/33\/11\/304","article-title":"The discrete harmonic oscillator, Harper\u2019s equation, and the discrete fractional Fourier transform","volume":"33","author":"Barker","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1364\/JOSAA.27.000021","article-title":"Fast linear canonical transforms","volume":"27","author":"Healy","year":"2010","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1093\/imamat\/25.3.241","article-title":"The fractional order Fourier transform and its application to quantum mechanics","volume":"25","author":"Namias","year":"1980","journal-title":"J. Inst. Math. Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1063\/1.527619","article-title":"Eigenvalues and eigenvectors of the finite Fourier transform","volume":"28","author":"Mehta","year":"1987","journal-title":"J. Math. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"063507","DOI":"10.1063\/1.2209770","article-title":"Jacobi \u03d1-functions and discrete Fourier transforms","volume":"47","author":"Ruzzi","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"355212","DOI":"10.1088\/1751-8113\/42\/35\/355212","article-title":"Fractional discrete q-Fourier transforms","volume":"42","author":"Wolf","year":"2009","journal-title":"J. Phys. A Math. Theor."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/2\/3\/1461\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:02:54Z","timestamp":1760220174000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/2\/3\/1461"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,7,12]]},"references-count":58,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2010,9]]}},"alternative-id":["sym2031461"],"URL":"https:\/\/doi.org\/10.3390\/sym2031461","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2010,7,12]]}}}