# What is the motivation for Weyl matrices in quantum information theory?

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:

1. 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?

2. 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?

• As an example of their usage, see my answer on teleportation of qudits. Jun 14, 2019 at 8:06
• Didn't read the paper fully, but they have been used a lot in this paper: arxiv.org/abs/0901.4729. The last line of the paper: When decomposing density matrices into operator bases the Weyl operator basis is the optimal one for all our calculations. The reason is that entangled states – the maximally entangled Bell states – are in fact easily constructed by unitary operators à la Weyl Jun 14, 2019 at 8:09