A Generalised Hadamard Transform

A Generalised Hadamard Transform for multi-phase or multilevel signals is introduced, which includes the Fourier, Generalised, Discrete Fourier, Walsh-Hadamard and Reverse Jacket Transforms. The jacket construction is formalised and shown to admit a tensor product decomposition. Primary matrices under this decomposition are identified. New examples of primary jacket matrices of orders 8 and 12 are presented.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

How much complementarity?

Bohr placed complementary bases at the mathematical centre point of his view of quantum mechanics. On the technical side then my question translates into that of classifying complex Hadamard matrices. Recent work (with Barros e Sa) shows that the answer depends heavily on the prime number decomposition of the Hilbert space. By implication so does the geometry of quantum state space.Comment: 6 pages; talk at the Vaxjo conference on Foundations of Probability and Physics, 201

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.Comment: Topical review (40 pages). Dedicated to the memory of Yurii Fedorovich Smirno

Complex Hadamard matrices of order 6: a four-parameter family

In this paper we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the new, generic family G together with Karlsson's degenerate family K and Tao's spectral matrix S form an exhaustive list of complex Hadamard matrices of order 6. Such a complete characterization might finally lead to the solution of the famous MUB-6 problem.Comment: 17 pages; Contribution to the workshop "Quantum Physics in higher dimensional Hilbert Spaces", Traunkirchen, Austria, July 201

Minkowski sums and Hadamard products of algebraic varieties

We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.Comment: 25 pages, 7 figure