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

Abstract

Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d \times d$ such that ${\rm Tr}(A_kA_l)= <v_k, v_l>$ for all $k,l=1,..., n$? For such matrices to exist, one must have $<v_k, v_l> \geq 0$ 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 $(<v_k, v_l>)$ has a positive factorization, then matrices $A_1$,..., $A_n$ 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.