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 $\mathfrak u\mathfrak v=ζ\mathfrak v\mathfrak u$, where $ζ$ 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 $ u v=ζvu$, then there exists a unital completely positive linear map $ϕ:\mathcal M_p(\mathbb C)\rightarrow\mathcal M_d(\mathbb C)$ such that $ϕ(\mathfrak u)= u$ and $ϕ(\mathfrak v)=v$. 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.