The Gale-Berlekamp game for complex Hadamard matrices

Abstract

Associated to a complex Hadamard matrix $H\in M_N(\mathbb{C})$ is the complex probability measure $\mu \in \mathcal{P}(\mathbb{C})$ describing the distribution of $\phi (a,b)=<a,Hb>$, where $a,b\in {\mathbb{T}}^N$ are random. This measure is called "glow" of the matrix, due to the analogy with the Gale-Berlekamp switching game, where $H,a,b$ are real. We prove here that: (1) $\mu$ becomes complex Gaussian in the $N\to \infty$ limit, (2) the universality holds as well at order 2, (3) the order 3 term seems to be quite interesting, particularly for the master Hadamard matrices, (4) in the Fourier matrix case, some of the higher order terms control counting problems for circulant Hadamard matrices.

Withdrawn by the author - the main findings in this paper are now part of arXiv:1403.2108

Cited by in corpus (2)

• Quantum invariants of deformed Fourier matrices
• Algebraic invariants of truncated Fourier matrices