On vector configurations that can be realized in the cone of positive matrices

Abstract

Let v1,…,vn be n vectors in an inner product space. Can we find a dimension d and positive (semidefinite) matrices A1,…,An∈Md(C) such that Tr(AkAl)=〈vk,vl〉 for all k,l=1,…,n? For such matrices to exist, one must have 〈vk,vl〉≥0 for all k,l=1,…,n. We prove that if n<5 then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from n=5 this is not so – even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal tracial state.The fact that the first such example occurs at n=5 is similar to what one has in the well-investigated problem of positive factorization of positive (semidefinite) matrices. If the matrix (〈vk,vl〉)(k,l) has a positive factorization, then matrices A1,…,An as above exist. However, as we show by a large class of examples constructed with the help of the Clifford algebra, the converse implication is false.