A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

On the Apparent Duality of the Kerdock and Preparata Codes

. The Kerdock and extended Preparata codes are something of an enigma in coding theory since they are both Hamming-distance invariant and have weight enumerators that are MacWilliams duals just as if they were dual linear codes. In this paper, we explain, by constructing in a natural way a Preparata-like code PL from the Kerdock code K, why the existence of a distance-invariant code with weight distribution that is the McWilliams transform of that of the Kerdock code is only to be expected. The construction involves quaternary codes over the ring ZZ4 of integers modulo 4. We exhibit a quaternary code Q and its quaternary dual Q ? which, under the Gray mapping, give rise to the Kerdock code K and Preparata-like code PL , respectively. The code PL is identical in weight and distance distribution to the extended Preparata code. The linearity of Q and Q ? ensures that the binary codes K and PL are distance invariant, while their duality as quaternary codes guarantees that K and PL ha..