$\mathrm{SU}_2$ Nonstandard Bases: Case of Mutually Unbiased Bases

Olivier Albouy ; Maurice R. Kibler

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of $\mathrm{SU}_2$ corresponding to an irreducible representation of $\mathrm{SU}_2$. The representation theory of $\mathrm{SU}_2$ is reconsidered via the use of two truncated deformed oscillators. This leads to replacement of the familiar scheme $\{j^2, j_z\}$ by a scheme $\{j^2,v_{ra} \}$, where the two-parameter operator $v_{ra}$ is defined in the universal enveloping algebra of the Lie algebra $\mathrm{su}_2$. The eigenvectors of the commuting set of operators $\{j^2,v_{ra}\}$ are adapted to a tower of chains $\mathrm{SO}_3\supset C_{2j+1}$ ($2j\in\mathbb N^{\ast}$), where $C_{2j+1}$ is the cyclic group of order $2j+1$. In the case where $2j+1$ is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.

Détail

Keywords: symmetry adapted bases; truncated deformed oscillators; angular momentum; polar decomposition of su2; finite quantum mechanics; cyclic systems; mutually unbiased bases; Gauss sums.

Bibliographie

[1] Melvin M. A., “Simplification in finding symmetry-adapted eigenfunctions”, Rev. Modern Phys., 28 (1956), 18–44 | DOI | MR | Zbl

[2] Altmann S. L., Cracknell A. P., “Lattice harmonics. I. Cubic groups”, Rev. Modern Phys., 37 (1965), 19–32 | DOI | MR | Zbl

[3] Altmann S. L., Bradley C. J., “Lattice harmonics. II. Hexagonal close-packed lattice”, Rev. Modern Phys., 37 (1965), 33–45 | DOI | MR

[4] Kibler M., “Ionic and paramagnetic energy levels: algebra. I”, J. Mol. Spectrosc., 26 (1968), 111–130 | DOI

[5] Kibler M., “Energy levels of paramagnetic ions: algebra. II”, Int. J. Quantum Chem., 3 (1969), 795–822 | DOI

[6] Michelot F., Moret-Bailly J., “Expressions algébriques approchées de symboles de couplage et de fonctions de base adaptés à la symétrie cubique”, J. Phys. (Paris), 36 (1975), 451–460 | MR

[7] Champion J. P., Pierre G., Michelot F., Moret-Bailly J., “Composantes cubiques normales des tenseurs sphériques”, Can. J. Phys., 55 (1977), 512–520 | MR | Zbl

[8] Patera J., Winternitz P., “A new basis for the representations of the rotation group. Lamé and Heun polynomials”, J. Math. Phys., 14 (1973), 1130–1139 | DOI | Zbl

[9] Patera J., Winternitz P., “On bases for irreducible representations of $O(3)$ suitable for systems with an arbitrary finite symmetry group”, J. Chem. Phys., 65 (1976), 2725–2731 | DOI | MR

[10] Michel L., “Invariants polynomiaux des groupes de symétrie moléculaire et cristallographique”, Group Theoretical Methods in Physics, eds. R. T. Sharp and B. Kolman, Academic Press, New York, 1977, 75–91 | MR

[11] Kibler M. R., Planat M., “A $\mathrm{SU}(2)$ recipe for mutually unbiased bases”, Internat. J. Modern Phys. B, 20 (2006), 1802–1807 ; quant-ph/0601092 | DOI | MR | Zbl

[12] Arik M., Coon D. D., “Hilbert spaces of analytic functions and generalized coherent states”, J. Math. Phys., 17 (1976), 524–527 | DOI | MR

[13] Biedenharn L. C., “The quantum group $\mathrm{SU}_q(2)$ and a $q$-analogue of the boson operators”, J. Phys. A: Math. Gen., 22 (1989), L873–L878 | DOI | MR | Zbl

[14] Sun C.-P., Fu H.-C., “The $q$-deformed boson realisation of the quantum group $\mathrm{SU}(n)_q$ and its representations”, J. Phys. A: Math. Gen., 22 (1989), L983–L986 | DOI | MR | Zbl

[15] Macfarlane A. J., “On $q$-analogues of the quantum harmonic oscillator and the quantum group $\mathrm{SU}(2)_q$”, J. Phys. A: Math. Gen., 22 (1989), 4581–4588 | DOI | MR | Zbl

[16] Daoud M., Kibler M., “Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics”, Phys. Lett. A, 321 (2004), 147–151 ; math-ph/0312019 | DOI | MR | Zbl

[17] Daoud M., Hassouni Y., Kibler M., “The $k$-fermions as objects interpolating between fermions and bosons”, Symmetries in Science X, eds. B. Gruber and M. Ramek, Plenum Press, New York, 1998, 63–67 ; quant-ph/9710016 | MR

[18] Daoud M., Hassouni Y., Kibler M., “Generalized supercoherent states”, Phys. Atom. Nuclei, 61 (1998), 1821–1824 ; quant-ph/9804046 | MR

[19] Daoud M., Kibler M. R., “Fractional supersymmetry and hierarchy of shape invariant potentials”, J. Math. Phys., 47 (2006), 122108, 11 pp., ages ; quant-ph/0609017 | DOI | MR | Zbl

[20] Kibler M., Daoud M., “Variations on a theme of quons: I. A non standard basis for the Wigner–Racah algebra of the group $\mathrm{SU}(2)$”, Recent Res. Devel. Quantum Chem., 2 (2001), 91–99

[21] Daoud M., Kibler M., “Fractional supersymmetric quantum mechanics”, Phys. Part. Nuclei, 33 Suppl. 1 (2002), S43–S51

[22] Schwinger J., “On angular momentum”, Quantum Theory of Angular Momentum, eds. L. C. Biedenharn and H. van Dam, Academic Press, New York, 1965, 229–279 | MR

[23] Lévy-Leblond J.-M., “Azimuthal quantization of angular momentum”, Rev. Mex. Fís., 22 (1973), 15–23

[24] Chaichian M., Ellinas D., “On the polar decomposition of the quantum $\mathrm{SU}(2)$ algebra”, J. Phys. A: Math. Gen., 23 (1990), L291–L296 | DOI | MR | Zbl

[25] Ellinas D., “Phase operators via group contraction”, J. Math. Phys., 32 (1991), 135–141 | DOI | MR | Zbl

[26] Ellinas D., “Quantum phase angles and $\mathrm{su}(\infty)$”, J. Mod. Opt., 38 (1991), 2393–2399 | DOI

[27] Ellinas D., Floratos E. G., “Prime decomposition and correlation measure of finite quantum systems”, J. Phys. A: Math. Gen., 32 (1999), L63–L69 ; quant-ph/9806007 | DOI | MR | Zbl

[28] Fairlie D. B., Fletcher P., Zachos C. K., “Infinite-dimensional algebras and a trigonometric basis for the classical Lie algebras”, J. Math. Phys., 31 (1990), 1088–1094 | DOI | MR | Zbl

[29] Patera J., Zassenhaus H., “The Pauli matrices in $n$ dimensions and finest gradings of simple Lie algebras of type $A_{n-1}$”, J. Math. Phys., 29 (1988), 665–673 | DOI | MR | Zbl

[30] Weyl H., The theory of groups and quantum mechanics, Dover Publications, New York, 1950

[31] Delsarte P., Goethals J. M., Seidel J. J., “Bounds for systems of lines and Jacobi polynomials”, Philips Res. Repts., 30 (1975), 91–105 | MR | Zbl

[32] Ivanović I. D., “Geometrical description of quantum state determination”, J. Phys. A: Math. Gen., 14 (1981), 3241–3245 | DOI | MR

[33] Wootters W. K., “Quantum mechanics without probability amplitudes”, Found. Phys., 16 (1986), 391–405 | DOI | MR

[34] Wootters W. K., “A Wigner-function formulation of finite-state quantum mechanics”, Ann. Phys. (N.Y.), 176 (1987), 1–21 | DOI | MR

[35] Wootters W. K., Fields B. D., “Optimal state-determination by mutually unbiased measurements”, Ann. Phys. (N.Y.), 191 (1989), 363–381 | DOI | MR

[36] Gibbons K. S., Hoffman M. J., Wootters W. K., “Discrete phase space based on finite fields”, Phys. Rev. A, 70 (2004), 062101, 23 pp., ages ; quant-ph/0401155 | DOI | MR

[37] Wootters W. K., “Picturing qubits in phase space”, IBM J. Res. Dev., 48 (2004), 99, 12 pp., ages ; quant-ph/0306135 | DOI

[38] Wootters W. K., “Quantum measurements and finite geometry”, Found. Phys., 36 (2006), 112–126 ; quant-ph/0406032 | DOI | MR | Zbl

[39] Calderbank A. R., Cameron P. J., Kantor W. M., Seidel J. J., “Zopf 4-Kerdock codes orthogonal spreads and extremal euclidean line-sets”, Proc. London Math. Soc., 75 (1997), 436–480 | DOI | MR | Zbl

[40] Edmonds A. R., Angular momentum in quantum mechanics, Princeton University Press, Princeton, 1960 | MR

[41] Racah G., “Theory of complex spectra. II”, Phys. Rev., 62 (1942), 438–462 | DOI

[42] Grenet G., Kibler M., “On the operator equivalents”, Phys. Lett. A, 68 (1978), 147–150 | DOI

[43] Kibler M., Grenet G., “On the $\mathrm{SU}_2$ unit tensor”, J. Math. Phys., 21 (1980), 422–439 | DOI | MR | Zbl

[44] Berndt B. C., Evans R. J., Williams K. S., Gauss and Jacobi sums, Wiley, New York, 1998 | MR | Zbl

[45] Bandyopadhyay S., Boykin P. O., Roychowdhury V., Vatan F., “A new proof for the existence of mutually unbiased bases”, Algorithmica, 34 (2002), 512–528 ; quant-ph/0103162 | DOI | MR | Zbl

[46] Lawrence J., Brukner Č., Zeilinger A., “Mutually unbiased binary observable sets on $N$ qubits”, Phys. Rev. A, 65 (2002), 032320, 5 pp., ages ; quant-ph/0104012 | DOI

[47] Klimov A. B., Sánchez-Soto L. L., de Guise H., “Multicomplementary operators via finite Fourier transform”, J. Phys. A: Math. Gen., 38 (2005), 2747–2760 ; quant-ph/0410155 | DOI | MR | Zbl

[48] Klimov A. B., Sánchez-Soto L. L., de Guise H., “A complementary-based approach to phase in finite-dimensional quantum systems”, J. Opt. B: Quantum Semiclass. Opt., 7 (2005), 283–287 ; quant-ph/0410135 | DOI | MR

[49] Romero J. L., Björk G., Klimov A. B., Sánchez-Soto L. L., “Structure of the sets of mutually unbiased bases for $N$ qubits”, Phys. Rev. A, 72 (2005), 062310, 8 pp., ages ; quant-ph/0508129 | DOI | MR

[50] Le Bellac M., Physique quantique, EDP Sciences/CNRS Éditions, Paris, 2003

[51] Chaturvedi S., “Aspects of mutually unbiased bases in odd-prime-power dimensions”, Phys. Rev. A, 65 (2002), 044301, 3 pp., ages ; quant-ph/0109003 | DOI

[52] Pittenger A. O., Rubin M. H., “Mutually unbiased bases, generalized spin matrices and separability”, Linear Algebr. Appl., 390 (2004), 255–278 ; quant-ph/0308142 | DOI | MR | Zbl

[53] Pittenger A. O., Rubin M. H., “Wigner functions and separability for finite systems”, J. Phys. A: Math. Gen., 38 (2005), 6005–6036 ; quant-ph/0501104 | DOI | MR | Zbl

[54] Saniga M., Planat M., Rosu H., “Mutually unbiased bases and finite projective planes”, J. Opt. B: Quantum Semiclass. Opt., 6 (2004), L19–L20 ; math-ph/0403057 | DOI | MR

[55] Saniga M., Planat M., “Sets of mutually unbiased bases as arcs in finite projectives planes”, Chaos Solitons Fractals, 26 (2005), 1267–1270 ; quant-ph/0409184 | DOI | MR | Zbl

[56] Saniga M., Planat M., “Hjelmslev geometry of mutually unbiased bases”, J. Phys. A: Math. Gen., 39 (2006), 435–440 ; math-ph/0506057 | DOI | MR | Zbl

[57] Planat M., Rosu H., “Mutually unbiased phase states, phase uncertainties, and Gauss sums”, Eur. Phys. J. D, 36 (2005), 133–139 | DOI

[58] Planat M., Saniga M., “Galois algebras of squeezed quantum phase states”, J. Opt. B: Quantum Semiclass. Opt., 7 (2005), S484–S489 | DOI | MR

[59] Saniga M., Planat M., “Finite geometries in quantum theory: from Galois (fields) to Hjelmslev (rings)”, Internat. J. Modern Phys. B, 20 (2006), 1885–1892 | DOI | MR | Zbl

[60] Vourdas A., “$\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ phase states”, Phys. Rev. A, 41 (1990), 1653–1661 | DOI

[61] Vourdas A., “The angle-angular momentum quantum phase space”, J. Phys. A: Math. Gen., 29 (1996), 4275–4288 | DOI | MR | Zbl

[62] Vourdas A., “Quantum systems with finite Hilbert space”, Rep. Prog. Phys., 67 (2004), 267–320 | DOI | MR

[63] Vourdas A., “Galois quantum systems”, J. Phys. A: Math. Gen., 38 (2005), 8453–8471 | DOI | MR | Zbl

[64] Vourdas A., “Galois quantum systems, irreducible polynomials and Riemann surfaces”, J. Math. Phys., 47 (2006), 092104, 15 pp., ages | DOI | MR | Zbl

[65] Klappenecker A., Rötteler M., “Constructions of mutually unbiased bases”, Lect. Notes Comput. Sci., 2948 (2004), 137–144 ; quant-ph/0309120 | MR | Zbl

[66] Klappenecker A., Rötteler M., Shparlinski I. E., Winterhof A., “On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states”, J. Math. Phys., 46 (2005), 082104, 17 pp., ages ; quant-ph/0503239 | DOI | MR | Zbl

[67] Grassl M., “On SIC-POVMs and MUBs in dimension 6”, Proceedings ERATO Conference on Quantum Information Science (EQIS 2004), ed. J. Gruska, Tokyo, 2005, 60–61; quant-ph/0406175

[68] Durt T., “If 1=2+3, then 1=2.3: Bell states, finite groups, and mutually unbiased bases, a unifying approach”, J. Phys. A: Math. Gen., 38 (2005), 5267–5283 ; quant-ph/0401046 | DOI | MR | Zbl

[69] Colin S., Corbett J., Durt T., Gross D., “About SIC POVMs and discrete Wigner distributions”, J. Opt. B: Quantum Semiclass. Opt., 7 (2005), S778–S785 | DOI | MR

[70] Durt T., “About the mean king's problem and discrete Wigner distributions”, Internat. J. Modern Phys. B, 20 (2006), 1742–1760 | DOI | MR | Zbl

[71] Wocjan P., Beth T., “New construction of mutually unbiased bases in square dimensions”, Quantum Inf. Comput., 5 (2005), 93–101 ; quant-ph/0407081 | MR | Zbl

[72] Archer C., “There is no generalization of known formulas for mutually unbiased bases”, J. Math. Phys., 46 (2005), 022106, 11 pp., ages ; quant-ph/0312204 | DOI | MR | Zbl

[73] Bengtsson I., MUBs, polytopes, and finite geometries, quant-ph/0406174 | MR

[74] Bengtsson I., Ericsson Å., “Mutually unbiased bases and the complementary polytope”, Open Syst. Inf. Dyn., 12 (2005), 107–120 ; quant-ph/0410120 | DOI | MR | Zbl

[75] Bengtsson I., Bruzda W., Ericsson Å., Larsson J.-Å., Tadej W., .{Z}yckowski K., Mubs and Hadamards of order six, quant-ph/0610161

[76] Bengtsson I., Three ways to look at mutually unbiased bases, quant-ph/0610216 | MR

[77] Boykin P. O., Sitharam M., Tiep P. H., Wocjan P., Mutually unbiased bases and orthogonal decompositions of Lie algebras, quant-ph/0506089 | MR

[78] Hayashi A., Horibe M., Hashimoto T., “Mean king's problem with mutually unbiased bases and orthogonal Latin squares”, Phys. Rev. A, 71 (2005), 052331, 4 pp., ages ; quant-ph/0502092 | DOI | MR