The limitations of nice mutually unbiased bases
Michael Aschbacher1
, Andrew M. Childs2
and Paweł Wocjan2
1California Institute of Technology Department of Mathematics Pasadena CA 91125 USA Pasadena CA 91125 USA
2California Institute of Technology Institute for Quantum Information Pasadena CA 91125 USA Pasadena CA 91125 USA
2California Institute of Technology Institute for Quantum Information Pasadena CA 91125 USA Pasadena CA 91125 USA
DOI: 10.1007/s10801-006-0002-y
Abstract
Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets.
Pages: 111–123
Keywords: keywords quantum information theory; mutually unbiased bases; quantum designs
Full Text: PDF
References
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2. C. Archer, There is no generalization of known formulas for mutually unbiased bases, J. Math. Phys. 46 (2005), 022106, arxiv.org/quant-ph/0312204.
3. M. Aschbacher, Finite Group Theory, 2nd edition, Cambridge University Press, Cambridge,
2000. Springer
4. S. Bandyopadhyay, P.O. Boykin, V. Roychowdhury, and F. Vatan, A new proof of the existence of mutually unbiased bases, Algorithmica 34 (2002), 512-528, arxiv.org/quant-ph/0103162.
5. C.H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in: Proc. IEEE Intl. Conf. Computers, Systems, and Signal Processing, 1984, pp. 175-179.
6. 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.
7. P. Delsarte, J.M. Goethals, and J.J. Seidel, Bounds for systems of lines and Jacobi polynomials, Philips Res. Rep. 30 (1975), 91-105.
8. K.S. Gibbons, M.J. Hoffman, and W.K. Wootters, Discrete phase space based on finite fields, Phys. Rev. A 70 (2004), 062101, arxiv.org/quant-ph/0401155.
9. M. Grassl, On SIC-POVMs and MUBs in dimension 6, in: Proc. ERATO Conference on Quantum Information Science, 2004, pp. 60-61, arxiv.org/quant-ph/0406175.
10. S.G. Hoggar, t-designs in projective spaces, Europ. J. Combin. 3 (1982), 233-254.
11. I.D. Ivanovic, Geometrical description of quantal state determination, J. Phys. A 14 (1981), 3241-3245.
12. G.A. Kabatiansky and V.I. Levenshtein, Bounds for packings on a sphere and in space, Problems Inform. Transmission 14(1) (1978), 1-17.
13. A. Klappenecker and M. R\ddot otteler, Beyond stabilizer codes I: Nice error bases, IEEE Trans. Inf. Theory 48 (2002), 2392-2395, arxiv.org/quant-ph/0010082.
14. A. Klappenecker and M. R\ddot otteler, Constructions of mutually unbiased bases, in: Proc. International Conference on Finite Fields and Applications, 2003, pp. 137-144, arxiv.org/quant-ph/0309120.
15. A. Klappenecker and M. R\ddot otteler, On the monomiality of nice error bases, Tech. Report CORR 2003-04, Department of Combinatorics and Optimization, University of Waterloo, April 2003, arxiv.org/quantph/0301078.
16. E. Knill, Non-binary unitary error bases and quantum codes, Tech. Report LAUR-96-2717, Los Alamos National Laboratory, 1996, arxiv.org/quant-ph/9608048.
17. R. Lidl and H. Niederreiter, Introduction to Finite Fields and Applications, Cambridge University Press, Cambridge, 1986.
18. J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.
19. H.F. MacNeish, Euler squares, Ann. of Math. 23(3) (1922), 221-227.
20. W.R. Scott, Group Theory, Prentice Hall, Englewood Cliffs, 1964.
21. Y. Watatani, Latin squares, commuting squares, and intermediate subfactors, Subfactors, 1994, pp. 85- 104.
22. R.F. Werner, All teleportation and dense coding schemes, J. Phys. A 34 (2001), 7081-7094, arxiv.org/quant-ph/0003070.
23. P. Wocjan and Th. Beth, New construction of mutually unbiased bases in square dimensions, Quantum Inform. Comput. 5(2) (2005), 93-101, arxiv.org/quant-ph/0407081.
24. W.K. Wootters and B.D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Physics 191 (1989), 363-381.
25. G. Zauner, Quantendesigns: Grundz\ddot uge einer nichtkommutativen Designtheorie, Ph.D. thesis, Universit\ddot at Wien, 1999.