Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p xp 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 uand v, satisfy u(p)= v(p)= 1p( the p xpidentity matrix) and the commutation relation u(v) =zeta vu, where.is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if uand vare any d xdunitary matrices such that u(p) = v(p)= 1(d) and uv= sigma vu, then there exists a unital completely positive linear map phi : Mp(C) -> M-d(C) such that f(u) = uand f(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, a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory. (C) 2021 Elsevier Inc. All rights reserved.
Weyl unitaryPauli matrixOperator systemComplete order equivalenceMatrix rangeExtreme points
Farenick, Douglas、Ojo, Oluwatobi Ruth、Plosker, Sarah
NSTL
Elsevier
