Almost Hadamard matrices: general theory and examples

Teodor Banica (1) , Ion Nechita (2, 3) , Karol Zyczkowski (4, 5)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 ICQ - Cohérence Quantique (LPT)
3 LPT - Laboratoire de Physique Théorique
4 Centrum Fizyki Teoretycznej
5 Institute of Physics

We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1-norm on O(N). Our study includes a detailed discussion of the circulant case ($H_{ij}=\gamma_{j-i}$) and of the two-entry case ($H_{ij}\in\{x,y\}$), with the construction of several families of examples, and some 1-norm computations.

Domaines: Combinatoire [math.CO]; Physique Quantique [quant-ph]

Teodor Banica, Ion Nechita, Karol Zyczkowski. Almost Hadamard matrices: general theory and examples. Open Systems and Information Dynamics, 2012, 19 (4), 1250024 [26 p.]. ⟨10.1142/S1230161212500242⟩. ⟨hal-00686061⟩