Universality of Weyl Unitaries

Abstract

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper on group theory and quantum mechanics, are $p\times p$ unitary matrices given by the diagonal matrix whose entries are the $p$-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by $\mathfrak{u}$ and $\mathfrak{v}$, satisfy ${\mathfrak{u}}^p={\mathfrak{v}}^p=1_p$ (the $p\times p$ identity matrix) and the commutation relation $\mathfrak{uv}=\zeta \mathfrak{vu}$, where $\zeta$ is a primitive $p$-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if $u$ and $v$ are any $d\times d$ unitary matrices such that $u^p=v^p=1_d$ and $uv=\zeta vu$, then there exists a unital completely positive linear map $\varphi :{\mathcal{M}}_p(\mathbb{C})\to {\mathcal{M}}_d(\mathbb{C})$ such that $\varphi (\mathfrak{u})=u$ and $\varphi (\mathfrak{v})=v$. We also show, moreover, that any two pairs of $p$-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent. When $p=2$, the Weyl matrices are two of the three Pauli matrices from quantum mechanics. It was recently shown that $g$-tuples of Pauli-Weyl-Brauer unitaries are universal for all $g$-tuples of anticommuting selfadjoint unitary matrices; however, we show here that the analogous result fails for positive integers $p>2$. Finally, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

14 pages