Uncertainty principle and Zak transform on finite non Abelian group
Khole Timothy Poumai, Shankar Gulia, Nitin Sharma · Physica Scripta · 2025
Abstract We develop the discrete uncertainty principle of the Fourier transform on a finite non-Abelian group. Furthermore, we define the Zak basis and subsequently define the finite Zak transform on a finite non-Abelian group. Analogous to the conventional finite Zak transform, various properties of the Zak transform are studied, and the sparsity of the Zak transform is illustrated through its discrete uncertainty principle. It is proven that the Zak transform partially diagonalizes the left regular representation into a block diagonal matrix. A few applications of the Zak transform concerning quantum information theory and Cayley graphs are discussed. It is observed that the Zak transform characterizes a completely positive and trace-preserving operator. Finally, we prove that the Zak basis forms the eigenvectors of the adjacency operator of quasi-Abelian Cayley graphs.