Quantum Entanglement and Geometry — Andreas Gabriel (2010) — Sec: 2.3.4 ~p. 11
Another basis for $d\times d$-dimensional matrices that has proven to be quite useful in quantum information theory is the Weyl operator basis, which consists of $d^2$ unitary and mutually orthogonal matrices $W_{m,n}$ defined as $$W_{m,n}=\sum_{k=0}^{d-1}e^{\frac{2\pi i}{d}kn}|k\rangle\langle k+m|$$ where $0\leq m.n \leq d-1$ and $(k+m)$ is understood to modulo $d$. Note that $$U_{00}=\Bbb 1, U_{0,1}=\sigma_1, U_{1,0}=\sigma_3, U_{1,1}=i\sigma_2.$$
Questions:
It seems Weyl matrices are generalizations to the Pauli matrices. So how and where exactly are they useful? Do they have any special properties like Pauli matrices? And what motivated the definition of Weyl matrices?
Weyl matrices are only finite-dimensional; I suppose there must be an infinite dimensional operator analogue to them such that the summation would be replaced by an integral?
