Ferenc Szöllősi

A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6

We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F(a, b). However, in the main result of this paper we also prove that for any values of the parameters (a, b), the standard basis and F(a, b) cannot be extended to a MUB-quartet. The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.

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 (4) 6 together with Karlsson’s degenerate family K (3) 6 and Tao’s spectral matrix S (0) 6 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. 2010 Mathematics Subject Classification. Primary 05B20, secondary 46L10.

Towards a classification of 6 × 6 complex hadamard matrices

Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţăs matrix D6. This answers a recent question of Bengtsson et al. [3].

Exotic complex Hadamard matrices and their equivalence

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe and Jones (C R Acad Sci Paris 311(Série I): 147–150, 1990), and Munemasa and Watatani (C R Acad Sci Paris 314(Série I): 329–331, 1992) and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (nu2009−u20098)×(nu2009−u20098) minors of real Hadamard matrices (Koukouvinos et al., Linear Algebra Appl 371:111–124, 2003).Abstract. In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of [7], [10] and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n− 8)× (n− 8) minors of real Hadamard matrices [13].

State tomography for two qubits using reduced densities

The optimal state determination (or tomography) is studied for a composite system of two qubits when measurements can be performed on one of the qubits and interactions of the two qubits can be implemented. The goal is to minimize the number of interactions to be implemented. The algebraic method used in the paper leads to an extension of the concept of mutually unbiased measurements.

Enumeration of Seidel matrices

Abstract In this paper Seidel matrices are studied, and their spectrum and several related algebraic properties are determined for order n ≤ 13 . Based on this Seidel matrices with exactly three distinct eigenvalues of order n ≤ 23 are classified. One consequence of the computational results is that the maximum number of equiangular lines in R 12 with common angle 1 ∕ 5 is exactly 20 .

The Hunt for Weighing Matrices of Small Orders

In this note we use a variety of techniques to construct new weighing matrices of small orders. In particular, we construct new examples of W(n, 9) for n ∈ { 14, 18, 19, 21} and W(n, n − 1) for n ∈ { 42, 46}. We also discuss two possible approaches for constructing a W(66, 65), and show nonexistence of these under certain assumptions.