Universality of Weyl Unitaries

TL;DR

This paper proves the universality of Weyl's unitary matrices in representing all unitary matrices with the same algebraic relations, and explores their extremal properties and limitations for higher orders.

Contribution

It establishes the universality of Weyl unitaries for matrices satisfying specific relations and analyzes their extremal and equivalence properties, extending understanding beyond the case p=2.

Findings

Weyl unitaries are universal for matrices with the same p-th power and commutation relations.

Any two pairs of Weyl matrices with the same relations are completely order equivalent.

The universality property for Pauli matrices (p=2) does not extend to p>2.

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