Gerhard Zauner

  • Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie, Ph.D. Thesis, University of Vienna, 1999 - Download (German, pdf, 77 pages, 442 KB).

  • Quantum Designs: Foundations of a non-commutative Design Theory, English translation of the Ph.D. Thesis (including a new preface 2010) - Download (pdf, 79 pages, 468 KB).

    Published in the International Journal of Quantum Information (IJQI), Volume 9, Issue 1, February 2011, page 445-507. [DOI No: 10.1142/S0219749911006776], © Copyright World Scientific Publishing Company, see IJQI.

    The thesis contains an overall presentation of quantum designs, based on the terminology given here.

    A list of Citations (external link).

  • Orthogonale Lateinische Quadrate und Anordnungen, Verallgemeinerte Hadamard-Matrizen und Unabhängigkeit in der Quanten-Wahrscheinlichkeitstheorie, Master (Diploma) Thesis, University of Vienna, 1991 - Download (German, pdf, 87 pages, 7273 KB)

    This paper contains a presentation and analysis of the concept of independent observables - a generalization of affine designs. The spectral decomposition

    of k self-adjoint matrices A1, ..., Ak defines k complete orthogonal classes on the quantum design built up from all the projections. The orthogonal classes, resp. obervables, Ai are said to be mutually independent (w.r.t. the density matrix (1/d)I), if

    For regular quantum design this reduced to the definition of affine designs, and with the case r=1 to MUBs. This definition also covers the (basic) mutually independence of Ai = Bi I and Aj = I Bj for any selfadjoint matrices Bi, Bj.

    This definition also allows a generalization to infinite dimensional Hilbert Spaces: Let A and B be two self-adjoint operators over a seperable Hilbert-Space with spectrum respectively. Let further be the characteristic functions of Borel subsets E, F of the spectrum of A and B, hence be spectral projections of A and B.
    We say that A and B are independent if there are Borel measures on the spectrum of A and B such, that for any two compact subset E and F  the following relation holds:

    Please notice, that for arbitrary projections P1 and P2 in infinite-dimensional Hilbert spaces tr(P1P2P1) = tr(P2P1P2) holds (the expression tr(P1P2) is generally not defined).
    It can be shown, that the position operator X and the impuls operator P are mutually independent, as well as linear-combinations of these two.
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Copyright © 2010-2012 Gerhard Zauner