Universality of Weyl unitaries

Abstract

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p×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 u and v, satisfy up=vp=1p (the p×p identity matrix) and the commutation relation uv=ζvu, 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×d unitary matrices such that up=vp=1d and uv=ζvu, then there exists a unital completely positive linear map ϕ:Mp(C)→Md(C) such that ϕ(u)=u and ϕ(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, but that the assertion for three such unitaries fails.There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63]. This standard construction is generalised herein to the case p≥3, producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.