Universality of Weyl unitaries

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 u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v 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 × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p ( C ) → M d ( 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.