Universality of Weyl Unitaries

Abstract

Weyl's unitaries are p×p unitary matrices given by a diagonal matrix having primitive p-th roots of unity as its entries and a cyclic shift matrix. The Weyl unitaries, which we denote by u and v, satisfy u^p= v^p=1_p(the p×p identity matrix) and the commutation relation uv=ζvu, where ζ is a primitive p-th root of unity. In this work, we prove that the Weyl unitaries are universal in the sense that if u and v are any d×d unitary matrices such that u^p=v^p=1_d and uv=ζvu, for some ζ, 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. Also, we show that any two pairs of p-th order unitary matrices (not just the Weyl unitaries) satisfying the commutation relation are completely order equivalent. However, we show in this work that the analogous result does not hold for triples of p-th order unitary matrices satisfying the Weyl commutation relation. In conclusion, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.