On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six

Andrew Maxwell, STEPHEN D. BRIERLEY

Paul Butterley, William Hall

Lin Chen, Li Zhi Yu

André Ranchin

We show that $K_6^{(2)} \cap X_6^{(2)} = D_6^{(1)} \cup C_6^{(1)}$ up to standard Hadamard equivalence, where $C_6^{(1)}$ is a new non-affine one-parameter family of complex Hadamard matrices. This demonstrates that the intersection of the two-parameter Hadamard families of Karlsson and Szöllősi is strictly larger than the known affine Diţă curve $D_6^{(1)}$. Our new parametrisation of the Szöllősi family $X_6^{(2)}$ in Karlsson coordinates and our constructive classification of the Karlsson-Szöllősi intersection shed light on the intersection of block-reducible forms and hypocycloid constraints. This may support the search for mutually unbiased bases in six dimensions.

Daniel McNulty, Stefan Weigert

We show that a complete set of seven mutually unbiased bases in dimension six, if it exists, cannot contain more than one product basis.

JamingPhilippe, MatolcsiMáté, MóraPéter