Kybernetika 46 no. 6, 1122-1137, 2010

Formula for unbiased bases

Maurice R. Kibler

Abstract:

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.

Keywords:

mutually unbiased bases, Weyl pairs, phase states, Lie algebras

Classification:

81R05, 81R10, 81R15, 81R50

paper.pdf

References:

  1. O.~Albouy and M. R.~Kibler: SU(2) nonstandard bases: Case of mutually unbiased bases. SIGMA 3 (2007), 076 (22 pages).   CrossRef
  2. M.~Aschbacher, A. M.~Childs and P.~Wocjan: The limitations of nice mutually unbiased bases. J. Algebr. Comb. 25 (2007), 111-123.   CrossRef
  3. N. M.~Atakishiyev, M. R.~Kibler and K. B.~Wolf: 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)   CrossRef
  4. R.~Balian and C.~Itzykson: Observations sur la m\'ecanique quantique finie. C.~R.~Acad.~Sci. (Paris) 303 (1986), 773-778.   CrossRef
  5. S.~Bandyopadhyay, P. O.~Boykin, V.~Roychowdhury and F.~Vatan: A new proof for the existence of mutually unbiased bases. Algorithmica 34 (2002), 512-528.   CrossRef
  6. I.~Bengtsson, W.~Bruzda, \AA.~Ericsson, J. \AA.~Larsson, W.~Tadej and K. \.{Z}yckowski: Mutually unbiased bases and Hadamard matrices of order six. J. Math. Phys. 48 (2007), 052106 (21 pages).   CrossRef
  7. B. C.~Berndt and R. J.~Evans: The determination of Gauss sums. Bull. Am. Math. Soc. 5 (1981), 107-130.   CrossRef
  8. P. O.~Boykin, M.~Sitharam, P. H.~Tiep and P.~Wocjan: Mutually unbiased bases and orthogonal decompositions of Lie algebras. Quantum Inf. Comput. 7 (2007), 371-382.   CrossRef
  9. S.~Brierley and S.~Weigert: Constructing mutually unbiased bases in dimension six. Phys. Rev. A 79 (2009), 052316 (13 pages).   CrossRef
  10. A. R.~Calderbank, P. J.~Cameron, W. M.~Kantor and J. J.~Seidel: Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. London Math. Soc. 75 (1997), 436-480.   CrossRef
  11. M.~Daoud and M. R.~Kibler: Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems. J. Phys. A: Math. Theor. 43 (2010), 115303 (18 pages).   CrossRef
  12. P.~Delsarte, J. M.~Goethals and J. J.~Seidel: Bounds for systems of lines and Jacobi polynomials. Philips Res. Repts. 30 (1975), 91-105.   CrossRef
  13. P.~Di\c t\u a: Some results on the parametrization of complex Hadamard matrices. J. Phys. A: Math. Gen. 37 (2004), 5355-5374.   CrossRef
  14. D.~Gottesman, A.~Kitaev and J.~Preskill: Encoding a qubit in an oscillator. Phys. Rev. A 64 (2001), 012310 (21 pages).   CrossRef
  15. M.~Grassl: Tomography of quantum states in small dimensions. Elec. Notes Discrete Math. 20 (2005), 151-164.   CrossRef
  16. I. D.~Ivanovi\'c: Geometrical description of quantum state determination. J. Phys. A: Math. Gen. 14 (1981), 3241-3245.   CrossRef
  17. M. R.~Kibler: Angular momentum and mutually unbiased bases. Int. J. Mod. Phys. B 20 (2006), 1792-1801.   CrossRef
  18. M. R.~Kibler: Variations on a theme of Heisenberg, Pauli and Weyl. J. Phys. A: Math. Theor. 41 (2008), 375302 (19 pages).   CrossRef
  19. M. R.~Kibler: 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).   CrossRef
  20. M. R.~Kibler and M.~Planat: A SU(2) recipe for mutually unbiased bases. Int. J. Mod. Phys. B 20 (2006), 1802-1807.   CrossRef
  21. J.~Lawrence, \v{C}. Brukner and A.~Zeilinger: Mutually unbiased binary observable sets on N qubits. Phys. Rev. A 65 (2002), 032320 (5 pages).   CrossRef
  22. A. O.~Pittenger and M. H.~Rubin: Wigner functions and separability for finite systems. J. Phys. A: Math. Gen. 38 (2005), 6005-6036.   CrossRef
  23. P.~\v{S}\v{t}ov\'{\i}\v{c}ek and J.~Tolar: Quantum mechanics in a discrete space-time. Rep. Math. Phys. 20 (1984), 157-170.   CrossRef
  24. P.~\v{S}ulc and J.~Tolar: Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions. J. Phys. A: Math. Gen. 40 (2007), 15099 (13 pages).   CrossRef
  25. W.~Tadej and K.~\.Zyczkowski: A concise guide to complex Hadamard matrices. Open Sys. Info. Dynamics 13 (2006), 133-177.   CrossRef
  26. P.~Wocjan and T.~Beth: New construction of mutually unbiased bases in square dimensions. Quantum Inf. Comput. 5 (2005), 93-101.   CrossRef
  27. W. K.~Wootters and B. D.~Fields: Optimal state-determination by mutually unbiased measurements. Ann. Phys. (N.Y.) 191 (1989), 363-381.   CrossRef