Formula for unbiased bases

Kibler, Maurice R.

Kybernetika, Tome 46 (2010) no. 6, pp. 1122-1137

Abstract (VO)

The present paper deals with mutually unbiased bases for systems of qudits in $d$ dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of $1+p$ mutually unbiased bases is given for $d=p$ where $p$ is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group $SU(2)$. A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case when $d = p^e$ ($e \geq 2$), corresponding to the power of a prime integer, is briefly examined. Finally, complete sets of mutually unbiased bases are analysed through a Lie algebraic approach.

Détail

Classification : 81R05, 81R10, 81R15, 81R50
Keywords: mutually unbiased bases; Weyl pairs; phase states; Lie algebras

Bibliographie

[1] Albouy, O., Kibler, M. R.: SU(2) nonstandard bases: Case of mutually unbiased bases. SIGMA 3 (2007), 076 (22 pages). | MR | Zbl

[2] Aschbacher, M., Childs, A. M., Wocjan, P.: The limitations of nice mutually unbiased bases. J. Algebr. Comb. 25 (2007), 111–123. | DOI | MR | Zbl

[3] Atakishiyev, N. M., Kibler, M. R., Wolf, K. B.: SU(2) and SU(1,1) approaches to phase operators and temporally stable phase states: applications to mutually unbiased bases and discrete Fourier transforms. (in preparation)

[4] Balian, R., Itzykson, C.: Observations sur la mécanique quantique finie. C. R. Acad. Sci. (Paris) 303 (1986), 773–778. | MR | Zbl

[5] Bandyopadhyay, S., Boykin, P. O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34 (2002), 512–528. | DOI | MR | Zbl

[6] Bengtsson, I., Bruzda, W., Ericsson, Å., Larsson, J. Å., Tadej, W., Życkowski, K.: Mutually unbiased bases and Hadamard matrices of order six. J. Math. Phys. 48 (2007), 052106 (21 pages). | DOI | MR

[7] Berndt, B. C., Evans, R. J.: The determination of Gauss sums. Bull. Am. Math. Soc. 5 (1981), 107–130. | DOI | MR | Zbl

[8] Boykin, P. O., Sitharam, M., Tiep, P. H., Wocjan, P.: Mutually unbiased bases and orthogonal decompositions of Lie algebras. Quantum Inf. Comput. 7 (2007), 371–382. | MR | Zbl

[9] Brierley, S., Weigert, S.: Constructing mutually unbiased bases in dimension six. Phys. Rev. A 79 (2009), 052316 (13 pages). | DOI | MR

[10] Calderbank, A. R., Cameron, P. J., Kantor, W. M., Seidel, J. J.: Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. London Math. Soc. 75 (1997), 436–480. | MR

[11] Daoud, M., Kibler, M. R.: Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems. J. Phys. A: Math. Theor. 43 (2010), 115303 (18 pages). | DOI | MR | Zbl

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

[13] Diţă, P.: Some results on the parametrization of complex Hadamard matrices. J. Phys. A: Math. Gen. 37 (2004), 5355–5374. | DOI | MR | Zbl

[14] Gottesman, D., Kitaev, A., Preskill, J.: Encoding a qubit in an oscillator. Phys. Rev. A 64 (2001), 012310 (21 pages). | DOI

[15] Grassl, M.: Tomography of quantum states in small dimensions. Elec. Notes Discrete Math. 20 (2005), 151–164. | DOI | MR | Zbl

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

[17] Kibler, M. R.: Angular momentum and mutually unbiased bases. Int. J. Mod. Phys. B 20 (2006), 1792–1801. | DOI | MR | Zbl

[18] Kibler, M. R.: Variations on a theme of Heisenberg, Pauli and Weyl. J. Phys. A: Math. Theor. 41 (2008), 375302 (19 pages). | DOI | MR | Zbl

[19] Kibler, M. R.: An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group. J. Phys. A: Math. Theor. 42 (2009), 353001 (28 pages). | DOI | MR

[20] Kibler, M. R., Planat, M.: A SU(2) recipe for mutually unbiased bases. Int. J. Mod. Phys. B 20 (2006), 1802–1807. | DOI | MR | Zbl

[21] Lawrence, J., Brukner, Č., Zeilinger, A.: Mutually unbiased binary observable sets on N qubits. Phys. Rev. A 65 (2002), 032320 (5 pages). | DOI

[22] 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

[23] Pittenger, A. O., Rubin, M. H.: Wigner functions and separability for finite systems. J. Phys. A: Math. Gen. 38 (2005), 6005–6036. | DOI | MR | Zbl

[24] Šťovíček, P., Tolar, J.: Quantum mechanics in a discrete space-time. Rep. Math. Phys. 20 (1984), 157–170. | DOI | MR

[25] Šulc, P., Tolar, J.: Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions. J. Phys. A: Math. Gen. 40 (2007), 15099 (13 pages). | DOI | MR | Zbl

[26] Tadej, W., Życzkowski, K.: A concise guide to complex Hadamard matrices. Open Sys. Info. Dynamics 13 (2006), 133–177. | DOI | MR | Zbl

[27] Wocjan, P., Beth, T.: New construction of mutually unbiased bases in square dimensions. Quantum Inf. Comput. 5 (2005), 93–101. | MR | Zbl

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