Constructions of Mutually Unbiased Bases

Two orthonormal bases B and B′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b′ 〉|2 = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of ℂ d cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

[1]  William O. Alltop,et al.  Complex sequences with low periodic correlations , 1980 .

[2]  Howard Barnum,et al.  Information-disturbance tradeoff in quantum measurement on the uniform ensemble , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[3]  P. Oscar Boykin,et al.  A New Proof for the Existence of Mutually Unbiased Bases , 2002, Algorithmica.

[4]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[5]  C. Carlet.  One-weight Z4-linear Codes , 2000 .

[6]  Arthur O. Pittenger,et al.  Mutually Unbiased Bases, Generalized Spin Matrices and Separability , 2003 .

[7]  Christian Kurtsiefer,et al.  Ascertaining the Values of σx, σy, and σz of a Polarization Qubit , 2003 .

[8]  P. K. Aravind.  Solution to the King’s Problem in Prime Power Dimensions , 2002 .

[9]  T. Helleseth,et al.  On the weight hierarchy of Kerdock codes over Z4 , 1996, IEEE Trans. Inf. Theory.

[10]  Zhe-Xian X. Wan,et al.  Quaternary Codes , 1997 .

[11]  I. D. Ivonovic.  Geometrical description of quantal state determination , 1981 .

[12]  Vaidman,et al.  How to ascertain the values of sigmax, sigma y, and sigma z of a spin-1/2 particle. , 1987, Physical review letters.

[13]  P. K. Aravind.  Best Conventional Solutions to the King’s Problem , 2003 .

[14]  J. Schwinger.  UNITARY OPERATOR BASES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[15]  S. Chaturvedi,et al.  Aspects of mutually unbiased bases in odd-prime-power dimensions , 2001, quant-ph/0109003.

[16]  S. G. Hoggar,et al.  t-Designs in Projective Spaces , 1982, Eur. J. Comb..

[17]  Richard M. Wilson,et al.  Concerning the number of mutually orthogonal latin squares , 1974, Discret. Math..

[18]  Berthold-Georg Englert,et al.  The Mean King's Problem: Spin , 2001, quant-ph/0101065.

[19]  H. Bechmann-Pasquinucci,et al.  Quantum Cryptography using larger alphabets , 1999, quant-ph/9910095.

[20]  Y. Aharonov,et al.  The mean king's problem: Prime degrees of freedom , 2001, quant-ph/0101134.

[21]  J. Seidel,et al.  BOUNDS FOR SYSTEMS OF LINES, AND JACOBI POLYNOMIALS , 1975 .

[22]  G. Mullen,et al.  Discrete Mathematics Using Latin Squares , 1998, The Mathematical Gazette.

[23]  William O. Alltop,et al.  Complex sequences with low periodic correlations (Corresp.) , 1980, IEEE Trans. Inf. Theory.

[24]  C. Archer.  There is no generalization of known formulas for mutually unbiased bases , 2003, quant-ph/0312204.

[25]  Paul Erdös,et al.  On the Maximal Number of Pairwise Orthogonal Latin Squares of a Given Order , 1960, Canadian Journal of Mathematics.

[26]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .