CanaDAM 2011
Université de Victoria, 31 mai - 3 juin 2011

Generalized Hadamard Matrices and Applications
Org: Aidan Roy and Ada Chan (University of Waterloo and York University)

ROBERT CRAIGEN, University of Manitoba
Some Circulant Generalized Weighing matrices  [PDF]

An elementary construction produces new classes of circulant generalized weighing matrices of so-called "Butson type" (i.e., whose entries are complex roots of unity) with parameters GW$(nN,n^2)$ for all positive integers $n, N$ such that $n \leq N$. We also discuss some generalizations and context. This was work done with Warwick de Launey before his recent death.

HADI KHARAGHANI, University of Lethbridge
Mutually unbiased complex weighing matrices  [PDF]

A $k$-complex weighing matrix of order $n$ and weight $p$ is a matrix $CW(n,p)$ of order $n$ with entries consisting of the $k$-th root of unity and $WW^*=pI_n$. Two $k$-complex $CW(n,p)$, $H,K$ are called {\it unbiased} if the absolute value of the entries of $HK^*$ equal $\sqrt{p}$. The class of mutually unbiased $k$-complex $CW(n,p)$s for small values of $n$ and $p$ will be discussed. This is a joint work with D. Best and H. Ramp.

AIDAN ROY, University of Waterloo
Generalized Hadamard matrices and quantum measurements  [PDF]

The columns of a unitary matrix $M$ may be thought of as a von Neumann measurement in quantum mechanics. When the entries of $M$ are highly structured, such as in a generalized Hadamard matrix, the measurement may prove particularly useful in quantum computing. I will give three instances of this phenomenon and describe the combinatorics involved in each: mutually unbiased bases, weighted complex $2$-designs, and the entanglement-assisted capacity of a graph.

ALYSSA SANKEY, University of New Brunswick
Type-II matrices associated with 2-graphs and weighted strongly regular graphs  [PDF]

The class of type-II matrices includes Hadamard matrices and spin models. The Nomura algebra of a type-II matrix is the Bose-Mesner algebra of an association scheme. Since spin models are contained in their Nomura algebras, we consider type-II matrices associated with known schemes. Indeed, they exist in connection with strongly regular graphs, certain distance-regular graphs, and other combinatorial objects.

In this talk we investigate type-II matrices, 2-graphs and weighted strongly regular graphs.

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Centre de recherches mathématiques Fields Institute MITACS Pacific Institute for the Mathematical Sciences Société mathématique du Canada University of Victoria